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Home , Mechanics, Quantum, Quantum beats, Quantum mechanics

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Joachim Burgdorfer¨ and Stefan Rotter

1.1 Introduction 3 1.2 - and the 4 1.3 Schrodinger¨ 6 1.4 Boundary Conditions and 8 1.5 Angular in 9 1.6 Formalism of Quantum Mechanics 12 1.7 Solution of the Schrodinger¨ Equation 16 1.7.1 Methods for Solving the -Dependent Schrodinger¨ Equation 16 1.7.1.1 Time-Independent Hamiltonian 16 1.7.1.2 Time-Dependent Hamiltonian 17 1.7.2 Methods for Solving the Time-Independent Schrodinger¨ Equation 19 1.7.2.1 Separation of Variables 19 1.7.2.2 Variational Methods 23 1.7.3 24 1.7.3.1 Stationary Perturbation Theory 25 1.7.3.2 Time-Dependent Perturbation Theory 25 1.8 Quantum 27 1.8.1 Born Approximation 28 1.8.2 Partial-Wave Method 29 1.8.3 30 1.9 Semiclassical Mechanics 31 1.9.1 The WKB Approximation 31 1.9.2 The EBK Quantization 33 1.9.3 Gutzwiller Trace Formula 34 1.10 Conceptual Aspects of Quantum Mechanics 35 1.10.1 Quantum Mechanics and Physical 36 1.10.2 Quantum 38 1.10.3 Decoherence and Process 39 1.11 Relativistic Wave 41 1.11.1 The Klein–Gordon Equation 41 1.11.2 The 42

Encyclopedia of Applied High and Particle .EditedbyReinhardStock Copyright  2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40691-3 2 1 Quantum Mechanics

Glossary 43 References 44 Further Reading 45 3

1.1 characteristic v of the constituents Introduction of the under study. The domain of nonrelativistic QM is characterized by Quantum mechanics (QM), also called that are small compared with the wave mechanics, is the modern theory of speed of : , of , , , and of the interaction of electromagnetic fields v c (1) with matter. It supersedes the embodied in Newton’s laws where c = 3 × 108 ms−1 is the speed of and contains them as a limiting case (the light. For v approaching c, relativistic exten- so-called of QM). Its develop- sions are necessary. They are referred to ment was stimulated in the early part of the as relativistic wave equations.Becauseofthe twentieth century by the failure of classical possibility of production and destruction of mechanics and classical electrodynamics to by conversion of relativistic energy explain important discoveries, such as the E into M (or vice versa) according to , the spectral density of the relation (see chapter 2) blackbody , and the electromag- netic absorption and emission spectra of E = Mc2 (2) atoms. QM has proven to be an extremely successful theory, tested and confirmed a satisfactory formulation of relativistic to an outstanding degree of accuracy for quantum mechanics can only be devel- physical ranging in size from sub- oped within the framework of quantum atomic particles to macroscopic samples field theory. This chapter focuses primar- of matter, such as superconductors and ily on nonrelativistic quantum mechanics, superfluids. Despite its undisputed success its formalism and techniques as well as in the description of physical phenomena, applications to atomic, molecular, optical, QM has continued to raise many concep- and condensed-matter physics. A brief dis- tual and philosophical questions, in part cussion of extensions to relativistic wave due to its counterintuitive character, defy- equations is given at the end of the chapter. ing common-sense interpretation. The new ‘‘quantum ’’ will be One distinguishes between nonrelativis- contrasted with the old world of classical tic quantum mechanics and . The bridge between the latter quantum mechanics depending on the and the former is provided by the

Encyclopedia of Applied High Energy and .EditedbyReinhardStock Copyright  2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40691-3 4 1 Quantum Mechanics

semiclassical mechanics, whose rigorous unifies the description of distinct entities formulation and application have only of : mechanics and elec- recently been developed. tromagnetism. Particles, the fundamental constituents of matter, possess, in to mechanical properties such as momen- 1.2 tum and , wave charac- Particle-Wave Duality and the Uncertainty ter. Conversely, electromagnetic radiation, Principle successfully described by Maxwell’s the- ory of electromagnetic , takes on, The starting point of modern quantum under appropriate conditions, the prop- mechanics is the hypothesis of the particle- erty of , the , carrying wave duality put forward by de Broglie and momentum. The (1924): the of each particle of mass particle and wave character are comple- M is associated with a wave (‘‘’’) mentary to each other. An object behaves of either like a particle or like a wave. The hypothesis of particle-wave duality h λ = (3a) has been experimentally verified in remark- p able detail. Interference of matter waves due to the indistinguishability of paths where h = 6.63 × 10−34 J s is Planck’s con- was demonstrated first for (Davis- stant, discovered by M. Planck in 1900 in son and Germer, 1927), later for , his investigation of blackbody radiation, and, most recently, for neutral atoms. The and p = Mv is the classical momentum of particle-wave duality of electromagnetic the particle. Similarly, the ν of radiation and its complementarity have the wave is associated with the mechan- been explored through modern versions ical energy of the particle through the of Young’s double-slit experiment using expression state-of-the-art technology and fast E switches. Hellmuth and colleagues (1987) ν = (3b) h succeeded, for example, in switching from particle to wave character and vice versa These relations, ascribing wave properties even after the has passed through to particles, are the logical complements the slit (in the experiment, a beam split- to postulates put forward 20 years earlier ter) by activating a switch (a Pockels cell) by Einstein (1905), assigning the charac- in one of the two optical paths of the teristics of particles (‘‘photons,’’ particles interferometer. By use of the switch, the of zero rest mass) to electromagnetic radi- interference patterns are made to appear ation. In fact, Einstein employed, in his or disappear, as expected for a wavelike or explanation of the ‘‘photoelectric effect,’’ particlelike behavior of the photon, respec- the expression E = hν (Eq. 3b) to find the tively. This ‘‘delayed choice’’ experiment energy E of each photon associated with not only demonstrates the particle-wave electromagnetic radiation of frequency ν. duality for photons but also elucidates Photons can eject electrons from the atoms important conceptual aspects of the nonlo- of a material provided that the energy cality of quantum mechanics. The decision exceeds the binding energy of the elec- as to whether the photon will behave tron. This particle-wave duality of QM like a particle or a wave need not be 1.2 Particle-Wave Duality and the 5 made at the double slit itself but can be propagation. The physical interpretation of spatially separated and temporarily post- the amplitude A was provided by Born poned. (1926) in terms of a density The consequences of the wave-particle duality for the description of physical phe- P(x) =|ψ(x)|2 =|A|2 (5) nomena are far reaching and multifaceted. One immediate consequence that repre- sents the most radical departure from for finding the particle at the coordinate x. classical physics is the uncertainty princi- The matter wave (Eq. 4) is further charac- = =  ple (Heisenberg, 1927). One deeply rooted terized by the wave number k 2π/λ p/ = =  notion of classical physics is that all dynam- and the ω 2πv E/ .  = ical can be measured, at least The ‘‘rationalized’’ Planck’s constant = × −34 in principle, with arbitrary accuracy. Uncer- h/2π 1.05 10 J s is frequently used tainties in the measurement are a matter in QM. of experimental imperfection that, in prin- The uncertainty principle is inherent to ciple, can be overcome. The wave all wave phenomena. Piano tuners have of particles in QM imposes, however, fun- exploited it for centuries. They a damental limitations on the simultaneous vibrating tuning fork of standard frequency accuracy of of dynamical in unison with a piano note of the same variables that cannot be overcome, no mat- nominal frequency and listen to a beat tone ter how much the measurement process between the struck tune and the tuning = can be improved. This uncertainty princi- fork. For a fork frequency of ν 440 Hz = ple is an immediate consequence of the and a frequency of ν 441 Hz, wave behavior of matter. Consider a wave one beat tone will be heard per second of the form (δν = ν − ν = 1Hz).Thegoalofthetuner is to reduce the number of beats as much − ψ(x) = Aei(kx ωt) (4) as possible. To achieve an accuracy of δν = 0.010.01 Hz, the tuner has to wait for at traveling along the x coordinate least about δt = 100 s (in fact, only a fraction (Figure 1.1). Equation (4) is called a of this time since the frequency mismatch because the wave fronts form can be detected prior to completion of planes perpendicular to the direction of one beat ) to be sure no beat had

Traveling matter wave Reψ(x, t)

0 0

1

2 t x Fig. 1.1 Matter wave traveling 3 along the x direction; the figure 10 shows part of ψ(x), 0 4 Eq. (4) (after Brandt and Dah- − 10 men, 1989). 6 1 Quantum Mechanics

occurred. Piano tuning therefore relies on implications become obvious only in the the frequency–time uncertainty microscopic world of atoms, molecules, electrons, nuclei, and elementary parti- ν t  1(6)cles. More generally, whenever the size of an object or mean between con- Measurement of the frequency with infi- stituents becomes comparable to the de nite accuracy (ν → 0) requires an infinite Broglie wavelength λ, the wave nature of time period of measurement (t →∞), or matter comes into play. The size of λ also equivalently, it is impossible within any provides the key for quantum effects in finite period of time to determine the fre- macroscopic systems such as superfluids, quency of the string exactly since time superconductors, and solids. and frequency are complementary variables. Similarly, the variables characterizing the matter wave (Eq. 4) are complementary: 1.3 Schrodinger¨ Equation k x ≥ 1 ω t ≥ 1(7)The Schrodinger¨ equation for matter waves ψ describes the dynamics of quantum These relations, well known from classical particles. In the realm of quantum physics, theory of waves and , have it plays a role of similar importance to that unexpected and profound consequences which Newton’s equation of motion does when combined with the de Broglie for classical particles. As with Newton’s hypothesis (Eq. 3) for quantum objects: laws, the Schrodinger¨ equation cannot be rigorously derived from some underlying, more fundamental . Its form can p x ≥  be made plausible, however, by combining E t ≥  (8) the Hamiltonian function of classical mechanics, The momentum and the of a particle, its energy and the time, H = T + V = E (9) and more generally, any other pair of complementary variables, such as angular which equals the mechanical energy, with momentum and angle, are no longer the de Broglie hypothesis of matter waves simultaneously measurable with infinite (Eq. 4). The is denoted by precision. The challenge to the classical V and the by T = p2/2M. laws of physics becomes obvious when Formally multiplying the one contemplates the fact that Newton’s ψ(x,t)byEq.(9)yields equation of motion uses the simultaneous knowledge of the position and of the p2 Hψ(x, t) = ψ(x, t) + Vψ(x, t) (10) change of momentum. 2M Since  is extremely small, the uncer- tainty principle went unnoticed during In order to connect the de Broglie relations the era of classical physics for hun- for energy and momentum appearing in dreds of years and is of little conse- the arguments of the plane wave (Eq. 4) quence for everyday life. Its far-reaching with the energy and momentum in the 1.3 Schrodinger¨ Equation 7

Hamiltonian function, H and p in Eq. (10) classical Hamiltonian function (Eq. 9), must be taken as differential operators, through the replacement of the classi- cal by the equivalents ∂ H ←→ i (Eq. 11). ∂t The replacement of dynamical variables  ∂ p ←→ (11) by differential operators leads to the i ∂x noncommutativity of pairs of conjugate This substitution into Eq. (10) leads to the variables in QM, for example, time-dependent Schrodinger¨ equation  ∂  ∂ [x, p ] ≡ x − x = i (14) ∂ x i ∂x i ∂x i ψ(x, t) = Hψ(x, t) ∂t   2 ∂2 with similar relations for other comple- = − + V(x) 2M ∂x2 mentary pairs, such as Cartesian coordi- × ψ(x, t) (12) nates of position and momentum (y, py), (z, pz), or energy and time. It is the noncom- which describes the of the mutativity that provides the basis for a for- matter wave. For physical systems that mal proof of the uncertainty principle, Eq. are not explicitly time dependent, i.e., that (8). The Schrodinger¨ equation (Eqs. 12 and have time-independent potentials V(x), the 13) is a linear, homogeneous, and (in gen- energy is conserved and i ∂/∂t can be eral) partial . One con- replaced by E, giving the time-independent sequence of this fact is that matter waves Schrodinger¨ equation ψ satisfy the .Ifψ1   and ψ2 are two acceptable solutions of the 2 ∂2 Schrodinger¨ equation, called states,alinear Eψ(x) = − + V(x) ψ(x) (13) 2M ∂x2 combination

If no interaction potential is present ψ = c ψ + c ψ (15) (V(x) = 0), Eq. (12) is referred to as the 1 1 2 2 free-particle Schr¨odinger equation.Theplane waves of Eq. (4) are solutions of the is an acceptable solution, as well. The phys- free-particle Schrodinger¨ equation with ical implication of Eq. (15) is that the quan- eigenenergies E = (k)2/2M. tum particle can be in a ‘‘coherent superpo- The ,firstfor- sition of states’’ or ‘‘,’’ a fact mulated by Bohr, provides a guide to that leads to many strange features of QM, quantizing a mechanical system. It states such as destructive and constructive inter- that in the limit of large quantum num- ference, the collapse of the wave function, bers or small de Broglie wavelength, the and the revival of wave packets. quantum-mechanical result should be the A solution of a linear, homogeneous same, or nearly the same, as the classi- differential equation is determined only cal result. One consequence of this pos- up to an overall amplitude factor, which tulate is that the quantum-mechanical is arbitrary. The probability interpreta- Hamiltonian operator in the Schrodinger¨ tion of the matter wave (Eq. 5), however, equation (Eq. 12) should be derived from removes this arbitrariness. The probabil- its corresponding classical analog, the ity for finding the particle somewhere 8 1 Quantum Mechanics

is unity, time-independent Schrodinger¨ equation:   2 2  ∂ dxP(x) = dx|ψ(x)| = 1 (16) − ψ(φ) = Eψ(φ) (18) 2I ∂φ2

Equation (16) and its generalization to where φ is the angle of about the z three are referred to as the axis. Among all solutions of the form normalization condition on the wave func- + tion and eliminate the arbitrariness of the ψ(φ) = Aei(mφ φ0) (19) modulus of the wave function. A wave function that can be normalized according only those satisfying the periodic boundary to Eq. (16) is called square integrable.The conditions only arbitrariness remaining is the overall ‘‘ factor’’ of the wave function, eiφ, ψ(φ + 2π) = ψ(φ) (20) i.e., a of modulus 1.

are admissible in order to yield a uniquely 1.4 defined single-valued function ψ(φ). (The Boundary Conditions and Quantization implicit assumption is that the rotator is spinless, as discussed below.) The In order to constitute a physically accept- magnetic m therefore =−∞ − able solution of the Schrodinger¨ equation, takes only integer values m , ..., 1, ∞ i.e., to represent a state, ψ must satisfy 0, 1, ... . Accordingly, the z component appropriate boundary conditions. While of the angular momentum their detailed forms depend on the coor- =  dinates and of the problem Lzψ(φ) m ψ(φ) (21) at hand, their choice is always dictated by the requirement that the wave func- takesononlydiscretevaluesofmultiples tion be normalizable and unique and that of the rationalized Planck’s constant . physical observables take on only real The on the right-hand side values (more precisely, real expectation that results from the operation of a values) despite the fact that the wave func- on the wave function tion itself can be complex. Take, as an is called the eigenvalue of the operator. example, the Schrodinger¨ equation for a The wave functions satisfying equations rotator with a of I,con- that are of the form of Eqs (18) or (21) strained to rotate about the z axis. The are called . The eigenvalues = 2 classical Hamiltonian function H Lz/2I, and eigenfunctions contain all relevant with Lz the z component of the angular information about the physical system; in momentum, particular, the eigenvalues represent the measurable quantities of the systems.  ∂ The amplitude of the Lz = xpy − ypx = (17) i ∂φ follows from the normalization condition  2π gives rise, according to the corre- dφ|ψ(φ)|2 = 1 (22) spondence principle, to the following 0 1.5 Angular Momentum in Quantum Mechanics 9 as A = (2π)−1/2. The eigenvalues of E in along the z axis becomes 2 2 Eq. (18) are Em =  m /2I.Justlikefor a vibrating string, it is the imposition −γ BLzψ(φ) = Emψ(φ) (24) of boundary conditions (in this case, periodic boundary conditions) that leads to ‘‘quantization,’’ that is, the selection of an The eigenenergies infinite but discrete set of modes of matter waves, the eigenstates of matter, each of Em =−γ Bm (25) which is characterized by certain quantum numbers. Different eigenstates may have the same are quantized and depend linearly on the energy eigenvalues. One speaks then of . Depending . In the present case, eigenstates on the sign of γ , states with positive m with positive and negative quantum num- for γ>0, or negative m for γ<0, corre- spond to states of lower energy and are bers ±m of Lz are degenerate since the Hamiltonian depends only on the square preferentially occupied when the atomic is in thermal equilib- of Lz. Degeneracies reflect the underlying symmetries of the Hamiltonian physical rium. This leads to an alignment of the system. Reflection of coordinates at the magnetic moments of the atoms along one particular direction and to x–z plane, (y →−y, py →−py and there- of matter when its constituents carry a net fore Lz →−Lz) is a symmetry operation, i.e., it leaves the Hamiltonian invariant. In magnetic moment. more general cases, the underlying symme- The an experiences as try may be dynamical rather than geometric it passes through an inhomogeneous in origin. magnetic field depends on the gradient of the magnetic field, ∇B.Becauseof the directional quantization of the angular 1.5 momentum, a beam of particles passing Angular Momentum in Quantum Mechanics through an inhomogeneous magnetic field will be deflected into a set of The quantization of the projection of the discrete directions determined by the angular momentum, Lz, is called directional quantum number m. A deflection pattern quantization. The Hamiltonian function consisting of a few spots rather than for the interaction of an atom having a a continuous distribution of a beam magnetic moment µ = γ L with an external of silver atoms was observed by Stern magnetic field B reads and Gerlach (1921, 1922) and provided thefirstdirectevidenceofdirectional H =−µ · B =−γ L · B (23) quantization. In addition to the projection of the angu- The proportionality constant γ between lar momentum along one particular axis, the magnetic moment and the angular for example, the z axis, the total angu- momentum is called the gyromagnetic ratio. lar momentum L plays an important role According to the correspondence principle, in QM. Here, one encounters a concep- the Schrodinger¨ equation for the magnetic tual difficulty, the noncommutativity of moment in a field with field lines oriented different angular momentum components. 10 1 Quantum Mechanics

Application of Eq. (14) leads to − l, ..., −1, 0, 1, ... l. The total number of +     quantized projections of Lz is 2l 1. The L , L = iL , L , L = iL eigenvalue of the L2 operator is, however,  x y z y z x 2 + 2 2 Lz, Lx = iLy (26) l(l 1) and not l . The latter can be understood in terms of the uncertainty As noncommutativity is the formal expres- principle and can be illustrated with the sion of the uncertainty principle, the help of a vector model (Figure 1.2). Because implication of Eq. (26) is that different of the uncertainty principle, the projections components of the angular momentum Ly and Lx do not take a well-defined value cannot be simultaneously determined with and cannot be made to be exactly zero, arbitrary accuracy. The maximum num- when Lz possesses a well-defined quantum ber of independent quantities formed by number. Therefore, the z projection m components of the angular momentum will always be smaller than the magnitude that have simultaneously ‘‘sharply’’ defined of L =  l(l + 1) in order to accommodate eigenvalues is two. Specifically, the nonzero fluctuations in the remaining components. 2 [L , Lz] = 0 (27) The are given in terms of associated Legendre functions 2 = 2 + 2 + 2 m with L Lx Ly Lz being the square of Pl (cos θ)by the total angular momentum. Therefore,   1/2 one projection of the angular momentum (2l + 1) (l − m)! Ym(θ, φ) = (−1)m vector (conventionally, one chooses the z l 4π (l + m)! projection) and the L2 operator can have × m imφ Pl (cos θ)e (29) simultaneously well-defined eigenvalues,

unaffected by the uncertainty principle. The eigenfunctions of Lz (Eq. 19) can be The differential operator of L2 in spherical recognized as one factor in the expression coordinates gives rise to the following m | m|2 for Yl . The probability densities Yl of eigenvalue equation: the first few eigenfunctions are represented    by the polar plots in Figure 1.3. 1 ∂ ∂ L2Y(θ, φ) =−2 sin θ In addition to the orbital angular sin θ ∂θ ∂θ  L,whichisthe 1 ∂2 + Y(θ, φ) quantum-mechanical counterpart to the sin2 θ ∂φ2 classical angular momentum, another type = 2l(l + 1)Y(θ, φ) (28) of angular momentum, the angu- lar momentum, plays an important role m The solutions of Eq. (28), Yl (θ, ψ), are for which, however, no classical analog called spherical harmonics.Thesuperscript exists. The spin is referred to as an inter- denotes the magnetic quantum number nal degree of freedom.TheStern–Gerlach of the z component, as above, while experiment discussed above provided the the quantum number l = 0, 1, ...∞ is major clue for the existence of the called the angular momentum quantum spin (Goudsmit and Uhlenbeck, 1925). The number.Since|Lz |≤|L|,wehavethe orbital angular momentum quantization restriction |m |≤l,thatis,forgivenangular allows only for an odd number of dis- momentum quantum number l,thez crete values of Lz (2l + 1 = 1, 3, 5, ...) component varies between −l and l : m = and therefore an odd number of spots in 1.5 Angular Momentum in Quantum Mechanics 11

Lz Fig. 1.2 Vector model of angular momentum√ L for quantum number l = 2and|L|= 6 .

2h

6 h h L

6 h

0 6 h

6 h −h

6 h

−2h

the deflection pattern. Stern and Gerlach by relativistic quantum mechanics (see observed, however, two discrete directions Section 1.11). The identification of the spin of deflection (see schematic illustration in as an angular momentum rests on the Figure 1.14). The spin hypothesis solved validity of the commutation rules (Eq. 26), this mystery. The outermost electron of     S , S = iS , S , S = iS , a silver atom has a total orbital angular  x y z y z x momentum of zero but has a spin angular Sz, Sx = iSy (30) = 1 momentum of s 2 . The number of dif- ferent projections of the spin is therefore identical to those of the orbital angular 2s + 1 = 2 in accordance with the num- momentum. Any set of the components ber of components into which the beam of a vector operator satisfying Eq. (26) was magnetically split. Within nonrela- has a of eigenvalues (quantum tivistic quantum mechanics, the spin must numbers) of an angular momentum with = 1 = be introduced as an additional degree of quantum numbers j 0, 2 ,1,...and m freedom on empirical grounds. A concep- −j, −j + 1 ..., + j.Thequantumnumber tually satisfactory description is provided j is an integer in systems with an even 12 1 Quantum Mechanics

θ Fig. 1.3 Polar plots of the probability | m |2 distributions Yl (θ, φ) of angular momentum eigenfunctions. = 0

m = 0

= 1

m = 0 m = ±1

= 2

m = 0 m = ±1 m = ±2

1 an infinite-dimensional vector , the number of spin 2 particles (including zero) and half-integer in systems with , an odd number of spin 1 particles. The 2 −→ | Hamiltonian for the interaction of an ψ ψ (32) electron spin with an external magnetic field is given in direct analogy to Eq. (23) with the vectors represented by a ‘‘ket.’’ by The ket notation was first introduced by Dirac (1930). The coherent superposition illustrated by Eq. (15) can be recognized as H =−γ gS · B (31) vector addition in Hilbert space. Physical observables such as the Hamiltonian or the The only difference is the ‘‘anomalous’’ angular momentum can be represented by gyromagnetic ratio represented by the linear and Hermitian operators. Linearity factor g. An approximate value of g can be of an operator B means deduced from the relativistic (Dirac’s equation) for the electron (see B(c1|ψ1 +c2|ψ2 ) = c1B|ψ1 +c2B|ψ2 Section 1.11). (33) for all complex numbers c1, c2 and all vectors |ψ1 , |ψ2 . Hermitian operators are 1.6 defined as those for which Formalism of Quantum Mechanics B† = B (34) The quantum-mechanical description out- lined in the previous section with the help that is, the operator is equal to its Hermi- of specific examples can be expanded to tian adjoint. The Hermiticity assures that a general formalism of quantum mechan- theeigenvalueequation ics. The starting point is the identifica- | = | tion of wave functions with vectors in B ψ λ ψ (35) 1.6 Formalism of Quantum Mechanics 13 possesses real eigenvalues λ, which can be containing only its eigenstates {ψi} and its identified with measurable quantities. The eigenvalues λi. Two operators possess the time-independent Schrodinger¨ equation, same eigenbasis when they commute with Eq. (13), is of the form in Eq. (35) with each other. Physical observables whose B being the Hamiltonian operator H and operators commute with each other are λ being the eigenenergy E.Theresultof called compatible observables.Astateofa a measurement of a physical observable physical system is therefore specified to yields an eigenvalue λ of the corresponding the maximum extent possible, consistent operator, B. with the uncertainty principle, by one The noncommutativity of operators (Eq. common eigenstate of a maximum set 14) corresponds to the well-known non- of compatible observables. For example, { 2 } commutativity of matrices in linear alge- the set L ,Lz is the maximum set of bra. The overlap of wave functions can be compatible observables for the angular identified with the scalar product in Hilbert momentum with the common eigenstate space: |lm .  The for finding the ∗ system in a particular state |ψ is given by φ|ψ ≡ d3r φ (r)ψ(r) (36) 1 the projection of the state vector onto this state, i.e., by the scalar product ψ1 | ψ . The adjoint of a vector in Hilbert space The corresponding probability is ψ| is called a bra and the scalar product a (Dirac, 1930). The P(ψ ) =| ψ |ψ |2 (39) normalization requirement imposed on 1 1 thewavefunctioncanbeexpressedin terms of the scalar product as ψ | ψ =1. Formally, Eq. (39) can be viewed as the The requirement of normalizability (Eq. expectation value of the projection operator 22) implies that the Hilbert space is P =|ψ1 ψ1| in the state |ψ .Theprocess spanned by square-integrable functions. of a measurement is therefore associated Since eigenstates of Hermitian operators with a projection in Hilbert space. This belonging to different eigenvalues are projection is sometimes referred to as orthogonal to each other, each physical the collapse of the wave function.The dynamical possesses a complete process of measurement thus influences orthonormal set of eigenstates: the quantum system to be measured. The state of a physical system is often 1 i = j not completely specified by a ‘‘pure’’ ψi|ψj =δij = (37) 0 i = j state described by a single Dirac ket |ψ or a coherent superposition of kets ∼ | + | The operator B is a diagonal in c1 ψ1 c2 ψ2 . The limited informa- a basis consisting of its eigenstates, the tion available may only allow us to specify a statistical mixture of states. Such a mixture ‘‘eigenbasis’’ |ψi with i = 1, ..., ∞.Inits eigenbasis, the operator B possesses the is described by the statistical operator or following ‘‘spectral representation’’: density operator

B = λi|ψi ψi| (38) ρ = |ψi ψi| (40) i i 14 1 Quantum Mechanics

with Pi denoting the probability for the with the strength of the singularity such system to be found in a particular state that the area underneath the peak is |ψi . The result of the measurement of an  observable B is then given by the statistical dxδ(x) = 1 (45) expectation value While the δ function is not an ordinary B =Tr(ρB) = ψi|B|ψi Pi (41) (Riemann integrable) function, it can be i approximated as a limit of Riemann integrable functions, for example, i.e., by the diagonal elements of the opera- 1 ε tor B, each weighted with the probability to lim = δ(x) (46) ε−→ 0 π 2 + ε2 find the system in the corresponding state. x | The preferred set of states, ψi ,forwhich Equation (46) satisfies Eq. (45) provided the is ‘‘diagonal,’’ that is, of one evaluates the integral first and takes the form in Eq. (40), is sometimes referred the limit ε → 0afterward.Astheorderof to as the set of pointer states. the operations of integration and taking the In light of the expression for the limit is not interchangeable, Eq. (46) is said probability as a projection, Eq. (39), we to converge ‘‘weakly’’ toward a δ function can rewrite Eq. (5) as in the limit ε → 0. δ functions play an important role in the of P(r) =|ψ(r)|2 =| r|ψ |2 (42) scattering. The time evolution of the Hilbert-space vectors is given by Schrodinger’s¨ equation where |r is formally treated as a vector written in Dirac’s ket notation: (‘‘ket’’) in Hilbert space. This definition leads to the difficulty that |r cannot be ∂ i |ψ =H|ψ (47) defined as a square-integrable function. ∂t Nevertheless, it can be consistently incor- For systems with a classical analog, the porated into the formalism by employing Hamiltonian operator is given, according the concept of generalized functions (‘‘dis- to the correspondence principle, by the tributions’’; see Lighthill, 1958; Schwartz, classical Hamiltonian function H(r, p) 1965). One defines a scalar product upon treating the canonical variables (r, p) through as noncommuting operators (see Eq. 14). There are, however, additional dynamical r|r =δ(r − r ) variables in quantum mechanics, such as = δ(x − x )δ(y − y )δ(z − z ) (43) spin, for which no classical counterparts exist. A vector in Hilbert space can be expanded where the right-hand side is called Dirac’s in terms of any basis (a ‘‘representation’’), δ function.Theδ function is zero almost

everywhere: |ψ = ci|φi = φi|ψ |φi (48) i i 0, x = 0 δ(x) = (44) where c = φ | ψ are the expansion coef- ∞ = i i , x 0 ficients of the ket |ψ in the basis 1.6 Formalism of Quantum Mechanics 15

{|φi }. Switching from one representation motion depends on the initial conditions to another |ψ →|ψ =U | ψ amounts x(t = t0), p(t = t0), which are not subject to a unitary transformation, U,since to any symmetry constraint. They may thenormofastatevectormustbe ‘‘break’’ the symmetry and usually do so. In conserved, quantum mechanics, however, the of a state vector becomes a dynamical ψ |ψ = ψ|U†U|ψ observable. = ψ|ψ (49) We can define a parity operator pˆ through

pHˆ (x, p)pˆ† = H(−x, −p) (52) or U†U = 1. Transformations among rep- resentations play an important role in QM. Since pˆ2 is the unit operator (two successive While the maximum number of compat- reflections restore the original function), ible observables is determined by com- we have pˆ = pˆ† = pˆ−1.Eigenfunctionsof mutation rules, the choice of observables pˆ, ψ(x) = x|ψ , can now be characterized that are taken to be diagonal or to have by their parity: ‘‘good’’ quantum numbers depends on the specific problem under consideration. x|pˆ|ψ = −x|ψ =± x|ψ (53) Unitary transformations provide the tool to switch from a basis within which one The ± sign in Eq. (53) is referred to set of observables is diagonal to another. as indicating positive or negative parity.A The transformation law for state vectors wave function cannot always be assigned a |ψ =U | ψ implies the transformation well-defined parity quantum number ±1. law for operators However, when pˆ commutes with H, A = UAU† (50) pHˆ (x, p)pˆ = H(x, p) (54)

The latter follows from the invariance there exists a common eigenbasis in | | = of the expectation value ψ A ψ which both the Hamiltonian operator | † | ψ U A U ψ under unitary transfor- and the parity operator are diagonal and, mations. consequently, ψ is an eigenstate of pˆ with a Classical Hamiltonian functions as well-defined parity. A similar argument can well as quantum-mechanical Hamiltonian be developed for symmetry with respect operators possess discrete symmetries. For to the time-reversal operator T: t →−t, example, the Hamiltonian function of the p →−p, r → r. , An additional discrete symmetry, which is of fundamental importance for the p2 H(x, p) = + 1 Mω2x2 (51) quantum mechanics of , 2M 2 is the symmetry. Consider is invariant under reflection of all spatial a wave function describing the state coordinates (x →−x, p →−p)sinceH of two identical particles ψ(r1, r2). The depends only on the square of each permutation operator pˆ12 exchanges the canonical variable. In classical mechanics, two particles in the wave function: this symmetry is of little consequence = since the solution of Newton’s equation of pˆ12ψ(r1, r2) ψ(r2, r1) (55) 16 1 Quantum Mechanics

The spin-statistics postulate of quantum that mechanics makes a general statement   − − concerning the permutation symmetry. A iH(t t0) U(t, t0) = exp (57) many-body system consisting of identical  particles of integer spin (‘‘’’) pos- sesses wave functions that are even under where the exponential function of the oper- all interchanges of two particles, whereas ator H is defined through its -series thewavefunctionisoddforhalf-integer expansion spin particles (‘‘fermions’’). The eigenvalue

of pˆij, the operator of permutation for any t − t0 U(t, t0) = 1 − i H two particles of an N-body system, is +1for    − 2 bosons and −1forfermions.Wavefunc- 1 t t0 − H · H +··· (58) tions are totally symmetric under particle 2  exchange for bosons and totally antisym- metric for fermions. An immediate con- Ifoneisabletosolvethetime-independent sequence is that two fermions cannot be Schrodinger¨ equation in identically the same state for which all quantum numbers agree (the so-called H|ψn =En|ψn (59) Pauli exclusion principle) since the anti- symmetric wave function would vanish identically. where n labels the eigenstate, one can cal- culate the time evolution of the eigenstate:

− −  iH(t t0)/ 1.7 U(t, t0)|ψn =e |ψn − −  Solution of the Schrodinger¨ Equation iEn(t t0)/ = e |ψn (60)

1.7.1 Methods for Solving the Time-Dependent using the definition in Eq. (58). The phase Schrodinger¨ Equation factor in Eq. (60) is called the dynamical phase. The explicit solution for an arbitrary 1.7.1.1 Time-Independent Hamiltonian state can then be found by expanding | If the Hamilton operator is not explic- ψ(t0) in terms of eigenstates of the itly time dependent, i.e., the total Hamilton operator: energy is conserved, the time-dependent Schrodinger¨ equation can be reduced to the |ψ(t0) = |ψn ψn|ψ(t0) time-independent Schrodinger¨ equation. n |ψ = |ψ Equation (47) can be formally solved by (t) U (t, t0) (t0) − −  iEn(t t0)/ = e |ψn n × ψ |ψ |ψ(t) =U(t, t0)|ψ(t0) (56) n (t0) (61)

where U(t, t0) is called the time-evolution Hence, for time-independent systems, the operator, a unitary transformation describ- solution of the energy eigenvalue problem ing the translation of the system in time. (Eq. 59) is sufficient to determine the time Substitution of Eq. (56) into Eq. (47) shows evolution for any initial state. 1.7 Solution of the Schrodinger¨ Equation 17

1.7.1.2 Time-Dependent Hamiltonian The additional phase γ n(t) is called For a time-dependent Hamiltonian H(t), Berry’s phase (Berry, 1984) or the geometric the solution of the time-evolution operator phase, since it records the information on U in terms of the exponential function the excursion of the Hamiltonian H(t)in Eq. (57) must be redefined in order to parameter space in the case of adiabatic take into account that the Hamiltonian evolution. The role of γ n(t)istorelate operator at different instances of time may the phases of the adiabatic wave function not commute, [H(t), H(t )] = 0. We have |ψn(t) at different points in time. Insertion of Eq. (63) into Schrodinger’s¨ equation  t i yields the equation that defines the real U(t, t0) = 1 − dt H(t )  phase function γ n(t),  t0 t 1 − dt H(t ) d 2 γ˙n(t) − i φn,t φn,t = 0 (64) t0 dt  t × dt H(t ) ··· The analogy to the parallel transport of vec- t0    tors on a sphere (Figure 1.4) can give a t = − i glimpse of the concept underlying Berry’s T exp  H(t )dt (62) t0 phase. Consider a quantum-mechanical state to be represented by a vector at point Equation (62) is called the time-ordered A. By infinitely slowly (adiabatically) vary- exponential function and T is called ing an external parameter, for example, the the time-ordering operator.Intwolim- direction of the magnetic field, one can iting cases, the evolution operator for ‘‘transport’’ state vectors successively from time-dependent Hamiltonian systems can point A to B,thentoC, and finally back be easily calculated by reducing the prob- to A. After completion of the closed loop, lem to the time-independent Hamiltonian. the vector does not coincide with its orig- One is the adiabatic limit. If the temporal inal orientation but is rotated by an angle changes in H(t) are very slow compared γ that is proportional to the angle with the changes in the dynamical phases subtended by the loop. This angle gives of Eq. (60) for all t,theU matrix is diagonal rise to a , exp(iγ ), which in the slowly changing eigenbasis of H(t) depends on the of a loop, but at each instant t, unlike the dynamical phase, Eq. (60), it is    independent of the time it takes to com- i t |ψ = iγn(t) − |φ plete the transport of the state along the n(t) e exp  dt En(t ) n,t t0 closed loop. If one is now able to form in a (63) physical system a coherent superposition of The dynamical is determined two states, one accumulating the geomet- by En(t ),theenergyeigenvalueofthe ric phase along the adiabatic change of the stationary Schrodinger¨ equation for the external field while the other is subject to a Hamiltonian operator H(t )attimet .InEq. constant field, an interference pattern will (63), the state |ψn(t) denotes the evolved arise. The Berry phase has been observed state and the state |φn,t represents the in interference experiments involving neu- stationary eigenstate for the Hamiltonian tron spin rotation, in molecules, in light H = H(t)atthetimet,wherethetimeplays transmission through twisted optical-fiber theroleofaparameter. cables, and in the . 18 1 Quantum Mechanics

Fig. 1.4 Parallel transport of a vector on a sphere along a closed loop leads to a rotation of the vector. The vector’s orientation relative to its local environment C remains unchanged during parallel transport. The angle of rotation is related to Berry’s phase.

γ B A

In the opposite limit, where a sudden The projection onto a final state |f change of the Hamiltonian from H(t) = representing, for example, the atom in H0 for all t < 0toanotherHamiltonian the ground state and the emission of a H(t) = H0 for all t > 0 occurs, no dynamical photon, or geometric phase can be accumulated during the instantaneous change of the P (t) =| f |ψ(t) |2 Hamiltonian at t = 0. The initial eigenstate f 2 2 2 2 =|c1| | f |ψ1 | +|c2| | f |ψ2 | of H0, ψn, in which the system is prepared   at t = 0, evolves for t > 0as + ∗ it − c1c2 exp  (E2 E1)t   ∗ i εn t × f |ψ1 f |ψ2 |ψ = − |ψ ψ |ψ n(t) exp  n n n + n complex conjugate (67) (65) In the sudden limit, the matrix U is non- will display oscillations (‘‘’’) diagonal. The eigenstate belonging to the due to the time dependence of the relative ‘‘old’’ Hamiltonian forms a coherent super- phases of the two states. These quantum position of eigenstates |ψ of the new n beats due to dynamical phases have been Hamiltonian. The ‘‘sudden’’ formation of observed, for example, in the Lyman-α (n = coherent states causes ‘‘quantum beats,’’ 2 → n = 1) radiation of excited in observable oscillations in amplitudes and the n = 2 level subsequent to fast collisions . with a foil (Figure 1.5). Consider, for example, an atom that is The standard computational method of excited at the time t = 0toastatethatisa solving the time-dependent Schrodinger¨ coherent superposition of a state |ψ ,with 1 equation for time-dependent interactions energy E , and a state |ψ ,withenergyE , 1 2 2 consists of expanding a trial wave function that is, |ψ =c | ψ +c | ψ .Atalater 1 1 2 2 |ψ(t) in terms of a conveniently chosen point in time, t, the state acquires a phase truncated time-independent basis of finite factor as a function of time according to size: Eq. (65):

−  −  N iE1t/ iE2t/ |ψ(t) =c1e |ψ1 +c2e |ψ2 |ψ(t) = an(t)|ψn (68) (66) n=1 1.7 Solution of the Schrodinger¨ Equation 19

Fig. 1.5 Quantum beats in the Lyman-α photon intensity of hydrogen after collision with a carbon foil: •, experimental data from Andra¨ (1974); – , theory from Burgdorfer¨ (1981). The theory curve is shifted relative to the data for clarity. Intensity

0 10 20 30 40 50 mm with unknown, time-dependent coeffi- 1.7.2.1 Separation of Variables cients an(t). Insertion of Eq. (68) into Eq. The energy eigenvalue problem, Eq. (59), (47) leads to a system of coupled first-order requires the solution of a second-order par- differential equations, which can be written tial differential equation. It can be reduced in matrix notation as to an ordinary second-order differential equation if the method of separation of ia˙ = H(t) a(t) (69) variables can be applied, i.e., if the wave function can be written in a suitable coor- using the vector notation a = (a1, ...an) dinate system (q1, q2, q3)infactorizedform with H the matrix of the Hamiltonian in the basis {|ψn }, Hn, n = ψn | H | ψn . The solution of Eq. (69) is thereby reduced ψ(r) = f (q1)g(q2)h(q3) (70) to a standard problem of numerical mathe- matics, the integration of a set of N coupled with obvious generalizations for an arbi- first-order differential equations with vari- trary number of degrees of freedom. able (time-dependent) coefficients. Equation (70) applies only to a very spe- cial but important class of problems for 1.7.2 which the Hamiltonian possesses as many Methods for Solving the Time-Independent constants of motion or compatible observ- Schrodinger¨ Equation ables as there are degrees of freedom (in our example, three). In this case, each fac- The solution of the stationary Schrodinger¨ tor function in Eq. (70) is associated with equation, Eq. (59), proceeds mostly in one ‘‘good’’ quantum number belonging the {|r } representation, i.e., using wave to one compatible observable. This class functions ψ(r). Standard methods include of Hamiltonians permits an exact solution the method of separation of variables and of the eigenvalue problem and is called variational methods. separable (or integrable). 20 1 Quantum Mechanics

Consider the example of the isotropic all contributions in the inte- three-dimensional harmonic oscillator. gral (Eq. 36) be made to cancel. The wave The Hamiltonian is functions for excited states of a harmonic oscillatoraregivenintermsofHermite 2 2 2 p py p polynomials H by H = x + + z n 2M 2M 2M   + 1 2 2 + 2 + 2 1/4 Mω (x y z ) (71) − Mω 2 f (x) = (2nn!) 1/2 n π   In this case, one suitable set of coordinates 2 ω × −(Mω/2π)x M for the separation are the Cartesian e Hn  x (75) coordinates (q1 = x, q2 = y, q3 = z). Each of the three factor functions of Eq. (70) Examples of the first few eigenstates are satisfies a one-dimensional Schrodinger¨ showninFigure1.6Withincreasingn, equation of the form 2 the probability density |f n(x) | begins to   2 2 resemble the classical probability density ∂ 1 2 2 − + Mω x f (x) = Exf (x) for finding the oscillating particle near the 2M ∂x2 2 coordinate x, P (x) (Figure 1.7). The latter (72) cl is proportional to the time the particle and E = Ex + Ey + Ez. The simplest solu- tion of Eq. (72) satisfying the boundary spends near x or inversely proportional conditions f (x →±∞) → 0 for a normal- to its speed: izable wave function is a Gaussian wave     ω −1/2 P (x) = 2E /M − ω2x2 (76) function of the form c1 π x   1/4 Mω −  2 f (x) = e (Mω/2 )x (73) π The increasing similarity between the clas- sicalandquantalprobabilitydensitiesas It corresponds to the lowest possible energy n →∞ isatthecoreofthecorrespon- = 1  eigenstate, with Ex 2 ω.Ground-state dence principle. This example highlights wave functions are, in general, the nonuniformity of the convergence to free, i.e., without a zero at any finite x. the classical limit: the quantum probabil- The Schrodinger¨ equation possesses an ity density oscillates increasingly rapidly infinite number of admissible solutions around the classical value with decreasing = with an increasing number n 0, 1, ...of de Broglie λ (compared with nodes of the wave function and increasing the size of the classically allowed region). eigenenergies Furthermore, the wave function penetrates   the classically forbidden region (‘‘tunnel- =  + 1 Ex ω n 2 (74) ing’’). While the amplitude decreases expo- nentially, ψ(x) possesses significant non- The fact that wave functions belonging to vanishing values outside the domain of higher states of excitation have an increas- classically allowed of the same ing number of zeros can be understood as a energy over a distance of the order of the consequence of the orthogonality require- de Broglie wavelength λ.Onlyuponaver- ment, Eq. (37). Only through the additional aging over regions of the size of λ can change of sign of the wave function can the classical–quantum correspondence be 1.7 Solution of the Schrodinger¨ Equation 21

E Fig. 1.6 Wave functions f n(x)forthefirstfive ψ n(X) eigenstates of the harmonic oscillator. 5

4

3

2

1

0 x −4 −2024

0.4 Fig. 1.7 Probability density | |2 n = 39 f n(x) of a harmonic oscillator in n = 39 compared with the 0.3 classical probability density Classical probability density 2 2 −1/2 Pcl(x) = (ω/π)(2E/M − ω x ) indicated by the dashed line. 0.2 Classical turning points are denoted by a and b. Quantum probability density

0.1

0.0 Turning point a Turning point b

−0.1 −10 0 10 x recovered. As λ tends to zero in the classi- The energy depends only on the sum of the cal limit, classical dynamics emerges. It three quantum numbers, is the smallness of λ that renders the macroscopic world as being classical for n = nx + ny + nz (79) all practical purposes. Thecompletewavefunctionforthe three-dimensional isotropic oscillator (Eq. but not on the individual quantum num- 71) consists of a product of three wave bers nx, ny,andnz, separately. Different functions each of which is of the form of combinations of quantum numbers yield Eq. (75), the same energy; the spectrum is therefore degenerate. The degeneracy factor gn,the = ψnxnynz (x, y, z) fnx (x) fny (y) fnz (z) (77) number of different states having the same energy En,is with eigenenergies   + + = (n 1)(n 2) = ω + + + 3 gn (80) Enxnynz nx ny nz 2 (78) 2 22 1 Quantum Mechanics

Only the (n = 0) is nonde- equation is separable in spherical coordi- generate, a feature generic to quantum nates and the wave function can be writ- = m systems and the origin of the third law ten as a product ψ(r, θ, φ) Rnl(r)Ye (θ, φ) (Nernst’s theorem) of statistical mechan- with one factor being the spherical har- m ics. The degeneracy signals that alternative monics Yl and the Rnl(r)beingtheradial complete sets of commuting observables wave function satisfying the second-order (instead of the in each of the three differential equation Cartesian coordinates Ex, Ey,andEz)and    −2 2 2 + corresponding alternate sets of coordinates ∂ + 2 ∂ + l(l 1) 2m ∂r2 r ∂r 2m r2 should exist in which the problem is sep- e  e arable. One such set of coordinates is the e2 − Rnl(r) = ERnl(r) (83) spherical coordinates. 4πε0r Consider the example of the . One of the most profound early The negative-energy eigenvalues of bound successes of quantum mechanics was the states can be found as interpretation of the line spectrum emitted 4 =− mee =−EH from the simplest atom, the hydrogen En 2 2 2 2 (84) 2 n (4πε0) 2n atom, consisting of an electron and a 4 2 2 . The Schrodinger¨ equation (in the where EH = mee / (4πε0) is called a center-of-mass frame) for the electron Hartree or an atomic unit of energy.The interacting with the proton through a energy depends only on the ‘‘principal’’ =− 2 Coulomb potential V(r) e /4πε0r reads quantum number n, being independent of as the angular momentum   −2∇2 e2 = − − ψ(r) = Eψ(r) (81) l 0, 1, ..., n 1 (85) 2me 4πε0r and of the magnetic quantum number m. = × −31 where me 9.1 10 kg is the reduced The m independence can be found for any mass of the electron, r is the vector that isotropic potential V(|r|) that depends only points from the nucleus to the electron, on the distance of the interacting particles. −19 e = 1.6 × 10 C is the electric of The degeneracy in l is a characteristic the nucleus, and −e isthechargeofthe feature of the Coulomb potential (and of the electron. If we express the isotropic oscillator), and it reflects the fact 2 2 2 2 2 2 ∇ (defined as ∇ = ∂ /∂x + ∂ /∂y + that the Schrodinger¨ equation is separable 2 2 ∂ /∂z ) in polar spherical coordinates (r, not only in polar spherical coordinates but θ, φ), the Schrodinger¨ equation becomes also in parabolic coordinates. The Coulomb    problem possesses a dynamical symmetry −2 2 2 ∂ + 2 ∂ + L that allows the choice of alternative 2m ∂r2 r ∂r 2m r2 e  e complete sets of compatible observables. 2 − e = The separability in spherical coordinates ψ(r, θ, φ) Eψ(r, θ, φ) (82) { 2 } 4πε0r is related to the set H, L , Lz while the separability in parabolic coordinates Since the angular coordinates enter only is related to the set {H, Lz, Az} where through the square of the angular momen- A is the Runge–Lenz (or perihelion) tum operator (see Eq. 28), the Schrodinger¨ vector, which, as is well known from 1.7 Solution of the Schrodinger¨ Equation 23

Fig. 1.8 The Runge–Lenz vector A and the angular L momentum L for the classical in an attractive −1/r potential.

A Nucleus v L×A

Electron v , is a for a nonlinearly on parameters (c1, ...,ck). The 1/r potential (Figure 1.8). The emission trial function is optimized by minimizing spectrum follows from Eq. (84) as the energy in the k-dimensional parameter space according to (E − E ) ν = n n0 h d   E =0(1≤i≤k) (87) = EH 1 − 1 dci 2 2 (86) 2h n0 n Provided the trial function is sufficiently with n0 = 1andn = 2, 3, ... giving rise to flexibletoapproximateacompletesetof the Lyman series of spectral lines, n0 = 2 states in Hilbert space, ψ becomes exact in and n = 3, 4, ... giving rise to the Balmer the limit as k →∞. The ground state of the series, and so on. The radial probability atom consisting of an α particle as 2 2 = densities r Rnl(r) are shown in Figure 1.9 a nucleus (nuclear charge number Z 2) and two electrons provides one of the 1.7.2.2 Variational Methods most striking illustrations of the success of For all but the few separable Hamilto- the variational method. The Hamiltonian nians, the time-independent Schrodinger¨ (neglecting the motion of the nucleus) equation is only approximately and numer- 2 2 ically solvable. The accuracy of approxi- =− ∇2 +∇2 − Ze H ( 1 2 ) mation can, however, be high. It may 2me 4πε0r1 only be limited by the preci- Ze2 e2 − + (88) sion of modern computers. The most 4πε0r2 4πε0|r1 − r2| successful and widely used method in generating accurate numerical solutions is nonseparable, an analytic solution is is the variational method. The solution unknown, and no perturbation solution of the Schrodinger¨ equation can be rec- is possible since all interactions are ognized as a solution to the variational of comparable strength. The variational problem of finding an extremum (in prac- method pioneered by Hylleraas (1929) tice, a minimum) of the energy and further developed by Pekeris (1958, E = ψ | H | ψ / ψ | ψ by varying |ψ 1959) achieves, however, a virtually exact until E reaches a stationary point at an result for the ground-state energy, E = approximateeigenvalue.Thewavefunction 2.903 724 377 03 ...EH, for which the accu- at the minimum is an approximate solution racy is no longer limited by the method to the Schrodinger¨ equation. In practice, of solution but rather by the approxi- one chooses functional forms called trial mate nature of the Hamiltonian itself functions ψ(r) that depend linearly and/or (for example, the neglect of the finite 24 1 Quantum Mechanics

E Fig. 1.9 Radial densities r2R (r)forn = 1, 2, r 2R2 (r), l = 0 nl n,l 3andl = 0, 1, and 2 of the hydrogen atom. 0

−0.1

−0.2

−0.3

−0.4

−0.5

E 2 2 = r Rn,l(r), l 1 0

−0.05

−0.1

E 2 2 = r Rn,l(r), l 2 0

−0.05

−0.1

r 0 5 10 15 20 25

size of the nucleus and of relativistic perturbation gV, effects). H = H0 + gV (89)

1.7.3 which is nonseparable, but ‘‘small’’ such Perturbation Theory that it causes only a small modification of the spectrum and of the eigenstates of A less accurate method, which is, however, H0. The control parameter g (0 ≤ g ≤ 1) easier to implement, is perturbation theory. is used as a convenient formal device for The starting point is the assumption that controlling the strength of the perturbation the Hamiltonian H can be decomposed and keeping track of the order of the into a ‘‘simple,’’ separable (or, at least, perturbation. At the end of the calculation, numerically solvable) part H0 plus a g can be set equal to unity if V itself is small. 1.7 Solution of the Schrodinger¨ Equation 25

Two cases can be distinguished: stationary Alternative perturbation series can be perturbation theory and time-dependent devised. For example, in the Brillouin– perturbation theory. Wigner expansion, the perturbed E0(g), rather than the unperturbed energy E0, 1.7.3.1 Stationary Perturbation Theory appears in the energy denominator of Eq. The Rayleigh–Schrodinger¨ perturbation (94). The energy is therefore only implicitly theory involves the expansion of both the given by this equation. perturbed energy eigenvalue E0(g)andthe perturbed eigenstate |ψ0(g) in powers of 1.7.3.2 Time-Dependent Perturbation the order parameter g (or, equivalently, in Theory powers of V), Time-dependent perturbation theory deals with the iterative solution of the system of (1) 2 (2) E0(g) = E0 + gE + g E +··· coupled equations, Eq. (69), representing (1) 2 (2) the time-dependent Schrodinger¨ equation. |ψ0(g) =|ψ0 +g|ψ +g |ψ +··· One assumes again that the Hamiltonian (90) consists of an unperturbed part H0 Inserting Eq. (90) into the Schrodinger¨ and a time-dependent perturbation V(t). equation Expansion in the basis of eigenstates of the unperturbed Hamiltonian (Eq. 68) yields a H(g)|ψ (g) =E (g)|ψ (g) (91) 0 0 0 matrix H consisting of a diagonal matrix i | H0 | j =Ejδij and a time-dependent yields recursion relations ordered in perturbation matrix Vij = i | V(t) | j . ascending powers of g for the successive The dynamical phases associated with |ψ corrections to E0(g)and 0(g) .Inprac- the unperturbed evolution can be easily tice, the first few terms are most important: included in the perturbative solution of Eq. Thus the first-order corrections for a non- (69) by making the phase transformation degenerate eigenvalue E0 are −  = iEjt/ (1) aj(t) cj(t)e (95) E = ψ0|V|ψ0 (92) This transformation, often referred to and as the transformation to the interaction | | representation, removes the diagonal matrix (1) ψn V ψ0 |ψ = |ψn (93) − i | H0 | j from Eq. (69), resulting in the = E0 En n 0 new system of equations {| } where the sum over ψn n=0 extends d i C(t) = V (t) C(t) (96) over the complete set of eigenstates of dt I the unperturbed Hamiltonian excluding the state |ψ0 for which the perturbation with matrix elements expansion is performed. The second-order   correction to the eigenenergy is | | = i − | | i VI(t) j exp  t(Ei Ej) i V(t) j

| ψ |V|ψ |2 (97) (2) = ψ | |ψ(1) = 0 n E 0 V and C = (c1,c2, ...,cN). The perturbative E0 − En n=0 solution of Eq. (96) proceeds now by (94) assuming that initially, say at t = 0, the 26 1 Quantum Mechanics

f (ω, t) Fig. 1.10 The function f (ω, t)asa function of ω.

t 2

−6π −4π −2π 2π 4π 6π 0 ω t t t t t t

system is prepared in the unperturbed state with ωij = (εi − εj)/. Plotted as a function 2 2 |j with unit amplitude, cj(0) = 1. Since of ω, the function f (ω,t) = 4sin(ωt/2)/ω the perturbation is assumed to be weak, (Figure 1.10) is strongly peaked around the system at a later time t will still be ω = 0. The peak height is t2 and the approximately in this state. However, small width of the central peak is given by admixtures of other states will develop with ω  2π/t.Inotherwords,f describes amplitudes the energy–time uncertainty governing  the transition: The longer the available t 1 time, the more narrowly peaked is the c (t) = dt i|V (t )|j c (t ) i  I j ω − ω = i 0 transition probability around i j 0, t ≈ 1 | | that is, for transition into states that are  dt i VI(t ) j (98) i 0 almost degenerate with the initial state. In the limit t →∞, the time derivative of Therefore, to first order, the time- Eq. (100), that is, the transition probability dependent perturbation induces during the per unit time, or transition rate w,hasthe time interval [0, t] transitions |j →|i with well-defined limit probabilities d 2π 2  w = Pi(t)t−→ ∞ = | i|V|j | δ(εi − εj) t 2  1 dt P (t) =|c (t)|2 = dt i|V (t )|j i i 2 I (101) 0 where the representation of the δ function, (99) πδ(ω) = lim →∞ (sinωt/ω), has been used. In the special case that V is time t Equation (101) is often called Fermi’s golden independent, Eq. (99) gives rule and provides a simple, versatile tool to calculate transition rates to first order. 1 |eiωijt − 1|2 P (t) = | i|V|j |2 The physical meaning of the δ function is i 2 ω2 ij such that in the limit t →∞ only those 2 1 4sin (ωijt/2) transitions take place for which energy is = | i|V|j |2 (100) 2 2 ωij conserved, as we would have expected since 1.8 Quantum Scattering Theory 27 the interaction in Eq. (100) was assumed to function. Nevertheless, most of the time be time-independent (after being switched one can manipulate scattering states using on at time t = 0). For example, in scattering thesamerulesvalidforHilbert-spacevec- processes, many final states with energies tors (see Section 1.6) except for replacing very close to that of the initial states are the ordinary orthonormalization condition, accessible. The δ function is then evaluated Eq.(37),bytheδ normalization condition, within the sum (or integral) over these final Eq. (102). states. The strategy for solving the Schrodinger¨ equation, which governs both bounded and unbounded motion, is different for 1.8 scattering states. Rather than determining Quantum Scattering Theory the energy eigenvalues or the entire wave function (which is, in many cases, Scattering theory plays, within the frame- of no interest), one aims primarily at of quantum mechanics, a special determining the asymptotic behavior of and important role. Its special role is the wave function emerging from the derived from the fact that scattering deals interaction region, which determines the with unbound physical systems: two (or flux of scattered particles. The central more) particles approach each other from quantity of both classical and quantum infinity, scatter each other, and finally scattering is the σ ,definedas separate to an infinite distance. Obvi- ously, one cannot impose upon the wave function the boundary condition ψ(r = number of scattered particles per unit time |r2 − r2 |→∞) → 0 since the probability σ = of finding the scattered particles at large number of incident particles per unit area and per unit time interparticle r is, in fact, nonva- (103) nishing. This has profound consequences: A cross section σ has the of energies are not quantized since the corre- an area and can be visualized as an effec- sponding boundary condition is missing. tive area that the target presents to the The energy spectrum consists of a con- incoming . Using the probabilis- tinuum instead of a set of discrete states. tic description of quantum mechanics, one The states describing the unbound motion, can replace in the definition Eq. (103) all the ‘‘scattering states,’’ are not square particle currents by corresponding prob- integrable and do not, strictly speaking, ability currents. Accordingly, σ can be belong to the Hilbert space. They can expressed in terms of the transition rate δ only be orthonormalized in terms of a w as function, w σ = (104) | = − ψE ψE δ(E E ) (102) j

Mathematically, they belong to the space where j =|ψ | 2v represents the incoming of distributions and can be associated with density. For an incom- states in Hilbert space only upon form- ing plane particle wave (Eq. 4), j = k/M ing ‘‘wave packets,’’ that is, by convoluting (with M the reduced mass and the ampli- the scattering state with a well localized tude A set equal to unity). Calculations of 28 1 Quantum Mechanics

total cross sections σ proceed typically by and vanishes sufficiently rapidly at large first calculating differential cross sections distances for the matter wave to become a for scattering into a particular of final plane wave. In such cases, the matrix ele- states and then summing over all possible ment in Eq. (107) can be expressed as the final states. For elastic scattering of parti- Fourier component of the potential, cles with incident wave vectors kin pointing  3 ir · (kout−k ) in the z direction, a group of final states kout|V|kin = d re in V(r) is represented by a kout of the (108) outgoing particles pointing in a solid angle

centered around the spherical angles (θ, φ), 2 For a Coulomb potential V(r) = Z1Z2e / 4πε0r representing two charged particles dσ M with charges Z1e or Z2e scattering each = −→ wkin kout (105) d kin other, Eq. (107) yields – coincidentally – the Rutherford cross section Equation (105) gives the differential cross   dσ Z2Z2e4 section per unit solid angle. = 1 2 (109) 2 2 4 d R 16E (4πε0) sin (θ/2) 1.8.1 Born Approximation This expression provided essential clues as to an internal structure and charge The evaluation of dσ /d is most straight- distribution of atoms. Two remarkable forward in the Born approximation. The features should be noted: Prior to the first Born approximation is nothing but advent of quantum theory, Rutherford the application of Fermi’s ‘‘golden rule,’’ (1911) used classical mechanics instead of Eq. (101), to scattering: quantum mechanics to derive Eq. (109). He could do so since, for Coulomb scattering,  dσ 2πM ∞ k2 classical and happen to = out dk (106) 2 3 out d kin 0 (2π) agree. Keeping in that the classical 2 limit entails the limit  → 0, the QM | kout|V|kin | × δ(Ein − Eout) and classical cross sections should agree,  where the integral is taken over the since (dσ /d)R does not depend on . The equivalence of the classical and the magnitude of kout but not over the 2 3 QM Rutherford cross sections is due to solid angle and where koutdkout d/(2π) represents the number of final states the long range (in fact, infinite range) available for the scattered particles in of the Coulomb potential. In this case, 2 = the de Broglie wavelength of the particle the interval dkoutd.Sincekoutdkout 2 wave is negligibly small compared to the (M/ )koutdEout,wehave ‘‘size’’ of the target at all energies. A dσ M2 second coincidence lies in the fact that = | k |V|k |2 (107) d 4(2π)2 out in the first Born approximation yields the exact quantum result even though the with |kout |=|kin|. The first Born approx- Born approximation is, strictly speaking, imation to elastic scattering, Eq. (107), is not even applicable in the case of a only valid if the potential V is sufficiently Coulomb potential. The infinite range of weak to allow for a perturbative treatment the Coulomb potential distorts the matter 1.8 Quantum Scattering Theory 29 wave at arbitrarily large distances so that potential V(r), the wave function will be it never becomes the plane wave that is modified within the region of the nonvan- assumed in Eq. (108). It turns out that ishing potential. At large distances r →∞, all high-order corrections to the Born however, when V(r) → 0, the wave func- approximation can be summed up to give tion will again become u(r) = sin(kr+δ0) a phase factor that drops out from the since in this region it must satisfy Eq. expression for the cross section. (111). The only effect that the potential can have on the scattered wave function 1.8.2 at large distances is to introduce a phase Partial-Wave Method shift δ0 relative to the unperturbed wave. For elastic scattering by a spherically sym- Simple arguments give a clue as to the metric potential, that is, a potential that origin and the sign of the phase shift. For depends only on the distance between the an attractive potential (negative V), the particles, V(r) = V(|r|), phase-shift analy- wave number k(r) = 2M(E − V(r))/2 sis provides a versatile nonperturbative increases locally and the de Broglie method for calculating differential and wavelength λ(r) = 2π/k(r) becomes shorter integral cross sections. For each angular compared with that of a of momentum l (which is a good quantum the same energy. The phase of the matter number for a spherically symmetric poten- wave is therefore advanced compared with tial), the phase shift for a radial wave thewaveofafreeparticleandδ0 is function u(r) = rR(r)canbedetermined positive (Figure 1.11). Similarly, if V from the radial Schrodinger¨ equation is a repulsive potential, λ increases in the interaction region causing a phase   delay and, hence, a negative phase shift 2 d2 2 l(l + 1) − + + δ0. Including all angular momenta, or 2 2 V(r) u(r) 2M dr 2M r all ‘‘partial waves,’’ the differential cross = Eu(r) (110) section can be expressed in terms of the partial-wave phase shifts δl as The idea underlying the phase-shift anal- ∞ 2 ysis can be best illustrated for the phase dσ 1 = + iδl shift δ for l = 0. In absence of the scat- (2l 1)e sin δlPl(cos θ) l d k2 tering potential V(r), Eq. (110) represents l=0 a one-dimensional Schrodinger¨ equation (112) A particularly simple expression can be of a free particle in the radial coordinate, found for the total cross section: ∞ 2 2 4π  d σ = + 2 δ − = 2 (l 1) sin l (113) 2 u(r) Eu(r) (111) k 2M dr l=0

2 whose solution is u(r) = sin(kr)withk = For short-ranged potentials, δl decreases 2ME/2. Since the matter wave cannot with increasing l such that only a small reach the unphysical region of negative number of terms must be included in r,wehavetoimposetheboundarycondi- this sum. The scattering phase shifts δl(E) tion u(0) = 0, which excludes the alternative depend sensitively on the energy E of the solution ∼cos kr. In the presence of the incident particle. 30 1 Quantum Mechanics

Fig. 1.11 Comparison between δ 0 the scattered wave in an Free–particle wave 1 attractive potential and a Scattered wave free-particle wave (the scattering phase δ0 is shown modulo π).

0

−1 Potential

01020 r

1.8.3 the rapid change of the scattering phase Resonances near the can be modeled by

Often, one phase shift δl(E)increasesby −  π within a narrow energy range. This δ (E)  tan 1 (115) l 2(E − E) corresponds to a phase shift of one-half res of a wave length, or to the addition of and that the energy dependence of the non- one node to the wave function. A sudden resonant (smooth part) of the phase shift increase of the scattering phase by π is can be neglected. The term  denotes the a signature of a ‘‘resonance’’ in the cross width over which the phase jump occurs. section. A resonance is characterized by a Inserting Eq. (115) into the lth term of Eq. sharp maximum in the partial-wave cross (113) yields, for the shape of the resonant = 2 2 section at the energy Eres kres/2M: cross section, the Breit–Wigner formula   4π (σ ) = (2l + 1) (114) l max k2 2/4 res σ = σ l(E) ( l)res 2 2 (116) (Eres − E) +  /4 where the phase shift passes through π/2 (or an odd multiple thereof). Resonances The shapes of resonant cross sections appear when the scattering potential allows become more complicated when inter- for the existence of ‘‘virtual levels’’ or ferences between the resonant and the quasi-bound states inside the potential nonresonant parts of the scattering ampli- well, whose probability density can leak tude come into play. The cross section out by tunneling through the potential can then exhibit a variety of shapes well. The additional node that the scattering (Beutler–Fano profile; Fano, 1961), includ- wave function acquires as the energy passes ing asymmetric peaks or dips (‘‘win- through Eres reflects the presence of a dow resonance’’) depending on the rel- quasi-. An isolated resonance ative phases of the amplitudes. In the can be parametrized by a formula due limit that a large number of reso- to Breit and Wigner. One assumes that nances with energies Eres,i (i = 1, ...,N) 1.9 Semiclassical Mechanics 31 are situated within the width  (‘‘over- canbewrittenas lapping resonances’’), the cross section   displays irregular fluctuations (‘‘Ericson iS(r) ψ(r) = A exp (118) fluctuations’’) (see Chapter 10). These  play an important role in scattering in complex systems, for example, in where the unknown function S(r)replaces nuclear compound resonances (Ericson the exponent k · r of a plane wave (Eq. and Meyer-Kuckuk, 1966) as well as in 4) for a free particle in the presence of scattering in chaotic systems Burgdorfer¨ et the potential. The idea underlying the al., 1995). semiclassical approximation is that in the limit  → 0, or likewise, in the limit of small de Broglie wavelength λ → 0 (compared 1.9 with the distance over which the potential Semiclassical Mechanics changes), ψ(r) should behave like a plane wave with, however, a position-dependent While the correspondence principle pos- de Broglie wavelength. Inserting Eq. (118) tulates that in the limit of large quan- into Eq. (117) leads to tum numbers (or, equivalently,  → 0), quantum mechanics converges to classi- 1 i cal mechanics, the approach to this limit E = [∇S(r)]2 + V(r) − ∇2S(r) is highly nonuniform and complex. The 2M 2M (119) description of dynamical systems in the with p =∇S. Equation (119) is well limit of small but nonzero  is called suited for taking the classical limit  → 0 semiclassical mechanics. Its importance is and yields the classical Hamilton–Jacobi derived not only from conceptual but also equation. In this limit, S is the or from practical aspects since a calculation Hamilton’s principal function. Quantum of highly excited states with large quan- corrections enter through the term linear tum numbers involved becomes difficult in . Approximate solutions of Eq. (119) becauseoftherapidoscillationsofthe can be found by expanding S in ascending wave function (see Figure 1.7). Semi- powers of . The apparent simplicity of the classical mechanics has been developed classical limit is deceiving: the principal only recently. The old Bohr–Sommerfeld function S appears inside the exponent Eq. quantization rules that preceded the devel- (118) with the prefactor −1 and this causes opment of quantum mechanics are now the wave function to oscillate infinitely understood as an approximate form of rapidly as  → 0. semiclassical mechanics applicable to sep- arable (or integrable) systems. 1.9.1 The wave function of the time-indepen- The WKB Approximation dent Schrodinger¨ equation for a single particle, For problems that depend only on one coordinate, retention of the first two terms   2 of the expansion of Eq. (119), Eψ(r) =− ∇2ψ(r) + V(r)ψ(r) 2M (117) S(x) = S0(x) + S1(x) 32 1 Quantum Mechanics

with traveling wave:

 WKB A x ψ (x) =  sin =± |p(x)| S0(x) p(x )dx    x 1 π × p(x )dx + (122) = 1 | |+  S1(x) 2 i ln p(x) c (120) a 4 Here a is the left turning point, the leads to the Wentzel–Kramers–Brillouin boundary of the classically allowed region (WKB) approximation where c is an (Figure 1.7). The additional phase of π/4 integration constant to be determined by is required to join ψ(x) smoothly with normalization of the wave function, the exponentially decaying solution in the classically forbidden region where p(x)    becomes purely imaginary (‘‘tunneling’’). x A ±i ψWKB(x) =  exp p(x )dx With equal justification, we could have |p(x)|  started from the classically forbidden (121) region to the right of the turning point Several features of Eq. (121) are worth not- b and would find ing: the WKB wave function is an asymp- A totic solution of Schrodinger’s¨ equation in ψWKB(x) =  sin |p(x)| the limit of small λ and satisfies all rules of    b QM, including the superposition principle. × 1 + π  p(x )dx (123) All quantities in Eq. (121) can be calculated, x 4 however, purely from classical mechan- ics. In particular, the quantum probability The uniqueness of ψWKB(x)mandatesthe density |ψWKB(x) | 2 can be directly associ- equality of Eqs (122) and (123). The two are ated with the inverse classical momentum equal if the integral over the whole period ∼|p(x) | −1 or, equivalently, the time the of the (from a to b and back to a) particle spends near x.Thisresultwas equals already anticipated for the highly excited     state of the harmonic oscillator (Figure 1.7). b a + p(x)dx = p(x)dx Furthermore, S0(x) is a double-valued func- a b tion of x corresponding to the particle = + 1  (n 2 )2π (124) passing through a given point moving either to the right or to the left with momen- Equation (124) can be recognized as the tum ±p(x). It is the multiple-valuedness of Bohr–Sommerfeld quantization rule of the p(x)orS(x) that is generic to semiclas- ‘‘old’’ quantum theory put forward prior sical mechanics and that poses a funda- to the development of quantum mechan- mental difficulty in generalizing the WKB ics. Quantization rules were successful in approximation to more than one degree of exploring the of hydro- freedom. gen, one of the great puzzles that classical For describing bound states, a linear dynamics could not resolve. The additional ψWKB 1 ×  combination of with both signs of term 2 2π in Eq. (124) was initially the momentum in Eq. (121) is required introduced as an empirical correction. Only in order to form a standing rather than a within the semiclassical limit of QM does 1.9 Semiclassical Mechanics 33

Fig. 1.12 Confinement of trajectories of an integrable Hamiltonian for constant energy E to the surface of a torus, shown here for a = Path A system with two degrees of freedom (N 2) (after Noid and Marcus, 1977).

Path B

its meaning become transparent: it is the is well defined only under special circum- =  phase loss in units of h 2π of a matter stances, that is, when p(r) is smooth and wave upon reflection at the turning points the action S(r , r) = r p · dr is either sin- 0 r0 (in multidimensional systems, at caustics). gle valued or (finite-order) multiple valued These phase corrections are of the form (called a Lagrangian manifold; Gutzwiller, α(π/4) with α = 2 in the present case. The 1990). Einstein pointed out, as early as quantity α is called the Maslov index and it 1917, that the smoothness applies only to is used to count the number of encounters integrable systems, i.e., systems for which with the caustics (in general, the number the number of constants of motion equals of conjugate points) during a full period of the number of degrees of freedom, N.For the classical orbit. such a system, classical trajectories are con- fined to tori in (as shown in 1.9.2 Figure 1.12 for N = 2) and give rise to The EBK Quantization smooth vector fields p(r) (Figure 1.13). The corresponding quantization conditions are The generalization of the WKB approxi- mation to multidimensional systems faces    α fundamental difficulties: the momentum p(r) · dr = 2π n + (i = 1, ..., N) vector p(r) is, in general, not smooth but it is Ci 4 a highly irregular function of the coordinate (126) = vector r and takes on an infinite number where Ci (i 1, ..., N)denoteN topolog- of different values. Take, for example, a ical distinct circuits on the torus, that is, billiard ball moving in two dimensions on circuits that cannot be smoothly deformed a stadium-shaped billiard table. A ball of into each other without leaving the torus a fixed kinetic energy T = p2/2M can pass (for example, the paths A and B in through any given interior point r with Figure 1.12). The conditions of Eq. (126) a momentum vector p pointing in every are called the Einstein–Brillouin–Keller possible direction. Consequently, the gen- (EBK) quantization rules and are the gen- eralization of Eq. (121), eralization of the WKB approximation    (Eq. 124). The amplitude of Eq. (125) is  i r ψ = | |× · given by the projection of the point den- (r) D(r) exp  p(r ) dr r0 sity of the torus in phase space onto the (125) coordinate space. Since the point density is 34 1 Quantum Mechanics

Fig. 1.13 Regular trajectory in coordinate space for an integrable system. The vector field p(r)ismultiple 2 valued but smooth (after Noid and Marcus, 1977).

Y 0

−2

−2 0 24 X

uniform in the classical angle variables θ i semiclassical Van Vleck (Van conjugate to the action (Goldstein, 1959), Vleck, 1928), the amplitude is given by KSC(r , r , t − t ) 1 2 2 1

∂θ1∂θ2∂θ3 = |D(r , r , t − t )| |D(r)|∝det (127) 1 2 2 1  ∂x∂y∂z t2 × i − iαπ exp  L(r(t ))dt (129) t1 2 For a system with one degree of freedom (N = 1), Eq. (125) reduces to Eq. (121) since where L istheclassicalLagrangefunction for the particle traveling along the classi- cally allowed path from r1 at time t1 to 1/2 1/2 ∂θ ∂t ∂θ r2 at time t2 and α counts the number of = ∂x ∂x ∂t singular points of the amplitude factor D

1/2 (similar to the Maslov index). If more than M 1 = ω ∝ (128) one path connects r (t )withr (t ), the p |p|1/2 1 1 2 2 coherent sum of terms of the form in Eq. (129) must be taken. The Van Vleck prop- 1.9.3 agator was recognized only 20 years later Gutzwiller Trace Formula (Feynman, 1948) as the semiclassical limit of the full Feynman quantum propagator As already anticipated by Einstein (1917), tori do not exist in many cases of K(r , r , t − t ) 1 2 2 1    practical interest – for example, for the t2 = { } i motion of an electron in a stadium-shaped Dr exp  L(r, t )dt (130) t1 heterostructure (Marcus et al., 1992). WBK or EBK approximations are where {Dr} stands for the path integral, not applicable since the classical motion that is the continuous sum over all of such an electron is chaotic, destroying paths connecting r1 and r2 including all the smooth vector field p(r)asinthe those that are classically forbidden. In billiard discussed above. The semiclassical the semiclassical limit  → 0, the rapid mechanics for this class of problems was oscillations in the phase integral cancel all developed only recently (Gutzwiller, 1967, contributions except for those stemming 1970, 1971). The starting point is the from the classically allowed paths for which 1.10 Conceptual Aspects of Quantum Mechanics 35  the actions Ldt possess an extremum or, orbit when the initial condition is ever so equivalently, for which the phase in Eq. slightly displaced (Gutzwiller, 1990). The (130) is stationary. smooth average density of states, Equation (129) provides a method for   investigating the semiclassical time evo- d V(E) nav(E) = (133) lution of wave packets in time-dependent dE (2π)N problems. The Fourier–Laplace transform of Eq. (129) with respect to t yields the is given by the rate of the change of the semiclassical Green’s function G(r1, r2, E). classical phase-space volume with energy The spectral density of states of a quantum in units of the volume of Planck’s unit system can be shown to be cell (2π)N associated with each . The derivation of Eq. (132) assumes n(E) = δ(E − Ei) two degrees of freedom and makes explicit i use of the fact that the physical system is 1 completely chaotic and integrable regions =− Tr Im G(r1 = r2, E) (131) π are absent. Only (hyperbolic) unstable periodic are assumed to exist, where the trace implies integration over the each with a positive Lyapunov exponent variables r1, r2 with constraint r1 = r2.Note λi > 0 that measures the instability of that each of these eigenenergies causes each periodic orbit. Equation (132) is a sharp peak (resembling the singularity therefore complementary to the EBK δ of the function) in n(E). Calculation of quantization formula (Eq. 126). This n(E) therefore allows the determination of formula of ‘‘periodic orbit quantization’’ is eigenenergies of quantum systems. It is an asymptotic series, that is, the sum over possible to evaluate Eq. (131) in the semi- all periodic orbits is, in general, divergent. classical limit by using the semiclassical For several systems, the sum over the first Green’s function and evaluating the trace few orbits has been shown to reproduce in the limit  → 0 by the stationary-phase the position of energy levels quite well. method. The result is the trace formula The mathematical and physical properties 1 of the Gutzwiller trace formula are still n(E) = n (E) + Im av iπ a topic of active research in quantum mechanics. × Ti 2sinh(Tiλi/2) periodic orbits   Si π 1.10 × exp i − αi (132)  2 Conceptual Aspects of Quantum Mechanics

The sum extends over all unstable classical QM has proved to be an extraordinarily periodic orbits, each characterized by the successful theory and, so far, has never period Ti, the instability (or ‘‘Lyapunov’’) been in contradiction to experimental exponent λi,andtheactionSi,allofwhich . The Schrodinger¨ equation are functions of the energy E.Anorbit has cleared every test carried out so far, is unstable if its Lyapunov exponent is including a precision test of its linearity: positive. The term λi measures the rate of theupperboundforcorrectionsdueto exponential growth of the distance from the nonlinear terms in the wave functions, 36 1 Quantum Mechanics

ψ(x)n (n > 1), in the Schrodinger¨ equation of the system. The point to be emphasized stands currently at one part in 1027 is that it is not our incomplete knowledge (Majumder et al., 1990). Even so, the of an independently existing objective real- conceptual and philosophical aspects of ity, but the fate of the cat itself, that can QM have remained a topic of lively only be resolved by the measurement. Such discussion stimulated, to a large part, a measurement is therefore, quite literally, by the counterintuitive consequences of a ‘‘demolition’’ measurement leading to QM that apparently defy common-sense the ‘‘collapse’’ of the wave function to one interpretation. The debate, which can be of the two possible states. In hypotheti- traced to the origins of QM and Einstein’s cal thought experiments for macroscopic objections summarized in his famous systems such as Schrodinger’s¨ cat prob- quote ‘‘Gott wurfelt¨ nicht’’ (‘‘God does not lem, the average over a huge number play dice’’), is centered around two aspects: (∼1023) of uncontrolled degrees of free- the superposition principle, Eq. (15), and dom will, for all practical purposes, destroy the probability interpretation of the wave any well-definded phase relation or coher- function, Eq. (39). ence between the ‘‘states’’ dead and alive. In contrast, the of QM can be put to 1.10.1 a rigorous test for microscopic systems, for Quantum Mechanics and Physical Reality example, by performing correlation mea- surements on two particles emitted from The counterintuitive aspects of the super- the same source. Examples along these position principle are highlighted by the lines were, for example, put forward by of ‘‘Schrodinger’s¨ cat.’’ Einstein, Podolsky and Rosen (1935) to The coherent superposition of two states, challenge the purely probabilistic interpre- each of which is presumably an eigen- tation of QM. state of an extremely complex many-body Consider the emission of two elec- Schrodinger¨ equation, describes the state trons from, for example, a of the cat, being either dead, |d ,oralive, (Figure 1.14). The spin of each electron |a : i = 1,2 can be in a coherent superposition state of spin up |+ and spin down |− , |ψcat =cd|d +ca|a (134) |ψ =α |+ + β |− (136) 2 2 i i i with |cd | +|ca | = 1. Only by way of measurement, that is, by projection onto relative to one particular Cartesian axis, the states |a or |d , for example, the z axis. One possible 2 2 spin state of the (ignoring Pa =| a|ψcat | , Pd =| d|ψcat | (135) here the antisymmetrization requirement for fermions) would be an uncorrelated canwefindoutaboutthefateofthecat. product state, While the meaning of two sharply different outcomes of Eq. (135), ‘‘dead’’ or ‘‘alive,’’ is | =|+ |− deeply engraved in our classical , ψ 1 2 (137) the quantum state of a coherent super- position, Eq. (134), defies any classical Already Schrodinger¨ pointed out that interpretation. Yet it is an accessible state quantum theory allows to be alternatively 1.10 Conceptual Aspects of Quantum Mechanics 37

z z

y

x ++ e He e L L x 1 2 Fig. 1.14 Two-electron emission from helium with the electrons being spin-analyzed in two Stern–Gerlach-type filters with different orientations (along the x and z axis, respectively). The difference in length, L1 < L2, makes the spin selection in the left detector happen first (inducing a ‘‘’’) and in the right detector later. √ in an entangled (or quantum-correlated) α =−β = 1/ 2, superposition state, 1 |ψ =√ (|+ 1|− 2 −|− 1|+ 2) (139) 2 |ψ =α|+ 1|− 2 + β|− 1|+ 2 (138) is rotationally symmetric and equally applies when we inquire about the spin In fact, the electron pair in the (singlet) orientation along the x rather than along = ground state of helium with total spin S 0 the z axis. If we postselect the spin-up α = forms such√ an entangled pair with direction (+)alongthex axis for the first − = β 1/ 2. The hallmark of entangled particle, the second particle will have spin states of the form Eq. (138) is that they down (−)alongthesameaxis.Ifwe, cannot be represented as a factorized instead, measured for the second parti- product in the form of Eq. (137). cle the relative to the z axis The consequences of such quantum cor- asbefore(Figure1.14),wewouldfind relations are profound: when measuring both spin up and down with 50% prob- + the spin of particle 1 to be up ( ), we ability, and thus a vanishing average spin. know with certainty that particle 2 has Different projection protocols for the first − itsspindown( ). The ‘‘collapse’’ of the spin imply entirely different results for entangled wave function when measur- thesecond.Theemergenceofaspecific ing particle 1 particle 2 to switch ‘‘physical reality’’ out of a specific mea- its spin from a completely undetermined surement context is often referred to as state to a well-defined value. We therefore and is unknown in obtain information about the orientation classical physics. It has therefore been of spin 2 without actually performing a questioned whether the quantum mechan- measurement on this particle. This corre- icaldescriptionofastatesuchasinEq. lation, often referred to as the nonlocality (138) can be considered a complete pic- of QM, is a consequence of the probabilis- ture of physical reality (Einstein, Podolsky tic description and superposition of wave and Rosen, 1935). In particular, it was functions in QM. The of a defi- conjectured that particles may have addi- nite physical property or physical reality by tional properties that are underlying the ‘‘postselection’’ through the measurement quantum-mechanical picture but which are of an entangled state goes even further. For hidden from our view. Such ‘‘hidden vari- example, the entangled state Eq. (138) with ables’’ would ensure that physical systems 38 1 Quantum Mechanics

have only properties that are ‘‘real,’’ that of ‘‘nonlocality’’ are untenable (Hasegawa is, independent of a measurement and et al., 2006; Groblacher¨ et al., 2007). that the result of any measurement is only dependent on ‘‘local’’ system prop- 1.10.2 erties rather than on a remote detection of another observable. The knowledge of ‘‘hidden variables’’ would allow one to The most fundamental unit of informa- remove the probabilistic aspects of QM. tion in classical information theory is the TheroleofprobabilityinQMcouldthen so-called bit, which can take on only one be the same as in : of two mutually exclusive values (typically a convenient way to quantify our incom- labeled as ‘‘0’’ and ‘‘1’’). In conventional plete knowledge of the state of a system. computers, all numbers and operations The hidden-variable hypothesis, or alterna- with them are encoded in this binary sys- tem. The corresponding quantum analog tively, the quantum correlation built into of the bit – named –isgivenbya entangled states, Eq. (138), has been probed two-state system with the two states |+ by testing the violation of Bell’s inequality and |− from Eq. (136), now labeled as |0 (Bell, 1969; Clauser et al., 1969). Experi- and |1 . What sets the qubit apart from the ments with correlated pairs of particles, in conventional classical bit is precisely the particular photon pairs – for example, by ability to be in a coherent superposition of Aspect and coworkers (1982) – have fal- both basis states, sified the hidden-variable hypothesis by showing that entangled states can violate |Q =α|0 +β|1 (140) Bell’s inequality while being in full agree- ment with the of QM. More with |α | 2+|β | 2 = 1. In quantum informa- recent experiments have succeeded in mea- tion theory, the intrinsic indeterminacy of suring entangled photons at a distance of the qubit’s value before a measurement is several hundred meters apart and with exploited as a resource rather than treated light polarizers (used as photon analyz- as a deficiency. ers), which switch their axes so rapidly One potential application of is that during the photons’ time of flight subsumed under the name ‘‘quantum the polarizers have no chance to exchange computation’’. The basic building block information on their respective orientation. of a so-called quantum computer would (The ‘‘collapse’’ of the wave function by be a number of N entangled qubits measuring one of the two photons occurs that can be in any superposition of instantaneously for both photons, but can- 2N basis states. An example for N = not be used to transmit any information 2 is just the maximally entangled pair and therefore does not violate the relativis- in Eq. (139). While a collection of N tic relations.) Even under such classical bits can only be in a single stringent conditions, a violation of Bell’s one of the 2N possible states at a inequality by more than 30 standard devia- given time, N qubitscanbeinany tions was observed experimentally (Weihs superposition state and thus can clearly et al., 1998). Also so-called noncontex- have a much higher information content. tual hidden-variable theories and realis- Owing to the superposition principle, any tic models that assume a certain type of the (unitary) operations of the 1.10 Conceptual Aspects of Quantum Mechanics 39 quantum computer performed on the A truly random sequence of numbers can initial state could process all the involved be generated, for example, by repeatedly 2N basis states simultaneously. Several measuring the spin orientation (or photon algorithms have been suggested (Shor, polarization) of an unpolarized spin, a 1994; Grover, 1996) for how this inherently situation that may be realized by first quantum feature of ‘‘parallel processing’’ projecting the first spin of an entangled could be used to solve fundamental pair along a perpendicular direction (see computational problems much faster, i.e., theexampleofFig.1.14above). with much fewer operations than with the Interception of a message shared with corresponding classical computations. the recipient through an entangled pair The manipulation of quantum states by a third party would require a mea- within the framework of quantum com- surement, which leads to a wave function putation and quantum information pro- collapse and to a detectable reduction in cessing involves a set of rules as, for quantum correlations. Using the princi- example, the so-called no cloning theorem ples of QM for the generation and trans- (Wootters and Zurek, 1982), which states mission of a secure key – a the impossibility to create identical copies technique that is referred to as quantum of an arbitrary unknown quantum state. – has been successfully imple- Although only imperfect ‘‘clones’’ of a mented for setups involving very large quantum state are possible, a transfer of distances (144 km) between the sender and a quantum state from one system to a recipient (Ursin et al., 2007). second, possibly distant, system is allowed according to QM. Such a ‘‘teleportation’’ of 1.10.3 a quantum state involves the destruction Decoherence and Measurement Process of the original state during the teleporta- tion procedure and is typically performed One fundamental obstacle in practically through an intermediate entangled state implementing quantum computation and of three particles (Bennett et al., 1993; quantum information protocols is the chal- Bouwmeester et al., 1997). lenge to keep the coherent superposition The laws of QM can also be used for of a quantum state, in particular of entan- secure information transfer. In classical gled pairs, ‘‘alive’’ for extended periods cryptography, secret messages are enci- of time. Any unintentional or intentional phered by way of a code that the recipient of interaction of the system inevitably and a message must know in order to decipher irreversibly converts the quantum super- a message’s content. To prevent the break- position into a classical probability distri- ing of the code by a third unauthorized bution and destroys the operating principle party, the key for the encoding algorithm is of a quantum information device. The preferably defined as a random sequence measurement process and the destruction of numbers, which is generated anew for of , frequently called decoher- each message transmitted. ence, are closely intertwined. In the early Challenges to classical cryptography are days of quantum theory, the measure- the need of a truly random key and the risk ment device that caused the ‘‘collapse’’ of key corruption through the interception of the wave function was considered to by a third party. Both of theses challenges be a ‘‘classical’’ apparatus decoupled from can be met by intrinsic quantum features. the quantum dynamics to be observed 40 1 Quantum Mechanics

(Bohr, 1928). The notion of open quan- The expectation value of any physical tum systems has superseded this view observable represented by an operator AS and provides a widely accepted unified acting only on the states in the Hilbert description of measurement and decoher- space of the system S gives in the presence ence within the realm of quantum theory. of entangled environmental degrees of The starting point here is the fundamental freedom, insightthatthequantumsystem(S)tobe

observed is never completely isolated from AS = ψ|AS|ψ the environment (E), which itself is quan- 2 =|α| +|AS|+ S tum rather than classical. The Hamiltonian S +|β|2 −| |− describing both system and environment S AS S = is given by H = HS+HE+HS−E,contain- Tr(ρAS) (143) ing the Hamiltonian of the system (HS) =||2|+ + | + | |2|− −| and its environment (HE), as well as the with ρ α SS β SS . interaction between the two (HS−E). A It is the coherent entanglement with the state describing both system and envi- environment, Eq. (142), that leads to (par- ronment, |ψ , even when at the initial tial) decoherence within the system (S)and time t = 0 prepared in a factorizable state, to a statistical mixture described by the | = =| i | j ψ(t 0) ψS ψE ,willevolveinthe density operator introduced above (Eq. 40) course of time under the influence of the with probabilities P+ =|α|2 and P− =|β|2. interaction HS−E into an entangled, non- The ubiquitous coupling of a physical factorizeable state system to the environment provides the natural setting for the transition from | = | i | j ψ(t) aij ψS ψE (t) (141) the quantum of superpo- i,j sitions to the definiteness of the classical measurement. The classical limit induced | i | j Here, ψS (and ψE ) are basis states of by decoherence is conceptually different the system (of the environment), which from the classical limit of quantum sys- often, but not always, are eigenstates of tems of large quantum numbers, which is the Hamilton operator of the system HS the focus of semiclassical mechanics (see (of the environment HE). ‘‘Tracing out’’ Section 1.9). In spite of recent progress, it the (unobserved) environment degrees of is generally still considered an open ques- freedom of the entangled state by taking tion if decoherence concepts alone can fully expectation values leads to decoherence explain the quantum-to-classical transition in the (observed) system and mimics and provide definite answers to the widely the effect of a classical measurement discussed questions regarding the ‘‘quan- device. tum .’’ Consider, as an illustrative example, The above considerations demonstrate a situation in which the system and that it is very unlikely for a macroscopic the environment are each represented object such as Schrodinger’s¨ cat to ever be by a two-dimensional Hilbert space and in a superposition state ‘‘dead and alive’’ as described by the following entangled state: in Eq. (134) because it is extremely difficult to isolate the cat from its environment. For all practical purposes, an exchange of | = |+ |− + |− |+ ψ(t) α S E β S E (142) particles and heat between the cat and 1.11 Relativistic Wave Equations 41 its environment will prevent the cat, for 1.11.1 itself, to be in a pure quantum state. The Klein–Gordon Equation It will rather be described by a density matrix with strongly reduced quantum We use the same recipe that led us in correlations. As both the cat and the Section 1.3 to the Schrodinger¨ equation, i.e., we replace the quantities H and p by environment are macroscopic objects with differential operators (Eq. 11), motivated many degrees of freedom, decoherence bythedeBrogliehypothesis.Inthis sets in at an extremely short timescale. case, however, we use the relativistic Recent experiments have been able to relation between energy and momentum sufficiently decouple ‘‘mesoscopic’’ objects (see Chapter 2) instead of the nonrelativistic as superconducting rings (Friedman et al., one, Eq. (9): 2000) from their environment to observe  these systems in superposition states. E − V(r) = M2c4 + p2c2 (144)

Following this recipe, we are led to 1.11  Relativistic Wave Equations d i ψ(r, t) = −2∇2c2 + M2c4 dt When the characteristic speed v of particles  in a physical system becomes comparable +V(r) ψ(r, t) (145) to the , v ≤ c, the Schrodinger¨ equation, which incorporates nonrelativis- Equation (145) has a major drawback: tic quantum dynamics, fails. Generaliza- it treats the space and time derivatives tions to relativistic wave equations face, asymmetrically. The Lorentz invariance however, fundamental difficulties due to of requires, however, the possibility of particle creation and that space and time coordinates be treated destruction in relativistic dynamics (see on equal footing. We thus start with the Chapter 2). For example, if photons have square of Eq. (144) and arrive at   energies in excess of twice the relativis- 2 2 ∂ ticrestenergyofelectrons,Eγ > 2mec , i − V(r) ψ(r, t) ∂t a pair consisting of an electron and its = −2∇2 2 + 2 4 ψ antiparticle, a positron, can be created. ( c M c ) (r, t) (146) Therefore, the interpretation of |ψ(r)|2 as This is the time-dependent Klein– a single-particle probability density is no Gordon equation, applicable to particles longer possible. A consistent resolution of with integer spin. The corresponding sta- the difficulties can be accomplished within tionary Klein–Gordon equation follows by the framework of quantum field theory, replacing i∂/∂t by E. Equation (146) gives which is outside the scope of this chapter. an eigenvalue equation in E2 rather than The relativistic analogs to the Schrodinger¨ in E itself. Consequently, upon taking equation discussed in the following section the square root of the eigenvalue, solu- are only applicable to relativistic systems in tions with both signs, ± p2c2 + M2c4, which particle production and destruction will appear. The physical meaning of are not yet important. negative-energy solutions is discussed 42 1 Quantum Mechanics

below. For a with charge q 1 spin 2 particle interacts with an electro- in the presence of an electromagnetic field, magnetic field as described by minimal both the φ(V = qφ)andthe coupling, the stationary Dirac equation electromagnetic vector potential A must be becomes included by ‘‘minimum coupling’’:   [E − qφ(r)]ψ(r)      −→ − = ∇−  p p qA qA (147) = cα ∇−qA + βMc2 ψ(r) (151) i i

With this coupling, the Klein–Gordon Dirac permit the probability- equation is invariant under gauge trans- | |2 = density interpretation, that is, ψ(x) formations of the electromagnetic field as 4 | |2 i=1 ψi(x) . The physical meaning of the well as under Lorentz transformations. four components can be easily understood by considering a free particle. The energy 1.11.2 eigenvalues are E =± p2c2 + M2c4.In The Dirac Equation addition to the positive energies we find negative eigenenergies, which are a feature Instead of taking the square of Eq. (144), of relativistic quantum mechanics and Dirac pursued a different approach in which result from the fact that the developing a Lorentz-invariant relativistic square-root operator permits both the wave equation. He ‘‘linearized’’ the square positive and negative sign of E.The root four- can be decomposed into two    2-spinors ψ = (φ1, φ2)oneofwhich c2p2 + M2c4 −→ c α · p + βMc2 represents the positive-energy solution and (148) the other the negative-energy solution. Each component of the two-spinor wave with the subsidiary condition that the function can be understood as a product square of the linearized operator satisfies of a configuration-space wave function ikr e and a spin 1/2eigenstate|ms =   2 ±1/2 . A major success of the Dirac cα · p + βMc2 = c2p2 + M2c4 (149) theory was that the intrinsic degree of an electron spin, which was a mere It turned out that solutions to this equation add-on within the Schrodinger¨ theory, can be found only if all α (i = x, y, z) i is incorporated into the Dirac equation and β are taken as matrices of minimum from the outset. Furthermore, the g dimension 4 × 4. Equation (149) is satisfied factor for the magnetic moment of the if the matrices α , β satisfy the relations i electron as well as the spectrum of the hydrogen atom can be derived from the α α + α α = δ i k k i 2 ik Dirac equation to a very high degree of + = αiβ βαi 0 (150) approximation. The pieces still missing are the quantum electrodynamic (QED) Accordingly, ψ is now a four-component corrections that lead to small energy entity (called a spinor), ψ ={ψi, i = shifts of atomic levels (‘‘’’) 1, ...,4}. The resulting Dirac equation and the deviations of the g factor for 1 describes spin 2 particles. When a charged the anomalous magnetic moment from 1.11 Relativistic Wave Equations 43

Fig. 1.15 Particle–antiparticle production visualized as the excitation of a E negative-energy particle to positive energies leaving behind a hole.

= 2 E 2mec

2, g = 2 × 1.001 16. These corrections can QM, whereas imperfections in the coher- be determined only within quantum field ence are described by a density matrix with theory, the theory of the interactions (partially) classical correlations. Decoher- of charged particles with the quantized ence stands for the loss of coherence as electromagnetic field. induced, for example, by coupling an iso- The negative-energy states, according lated quantum system to its environment. to Dirac, can be interpreted in terms of Correspondence Principle: The princi- antiparticle states. The state of ple that quantum-mechanical and classical empty space is represented by a fully description of physical phenomena should occupied ‘‘sea’’ of negative-energy states. become equivalent in the limit of vanishing While this ‘‘sea’’ itself is not observ- de Broglie wavelength λ → 0, or equiva- able, ‘‘holes’’ created in the sea by excit- lently, of large quantum numbers. ing negative-energy electrons to posi- de Broglie Wave: According to de Broglie, tive energies (Figure 1.15) are observ- the motion of a particle of mass M and able and appear as antiparticles. For v is associated with a matter wave example, a missing electron with charge with wavelength λ = h/(Mv). h is Planck’s −e,energy−E, and momentum p repre- quantum. sents a positron with charge e,energyE, and momentum −p. The negative-energy Delta (δ)Function: A particular case of a states of electrons therefore offer a sim- distribution, frequently used in quantum ple explanation of particle–antiparticle mechanics to formulate the orthogonality in terms of the excita- and normalization conditions of states with tion of a negative-energy particle to a a of eigenvalues. positive-energy state (the particle) leaving Distributions: Also called generalized func- a hole in the negative-energy ‘‘sea’’ (the tions. Linear-continuous functionals oper- antiparticle) behind. ating on a metric . They form the of the metric vector space. Glossary In the context of quantum mechanics, the space is a Hilbert space and the operation of Coherence and Decoherence: In QM, the functional on an element of the Hilbert the term coherence stands for the abil- space is called formation of a . ity of a particle’s state to interfere. Eigenvalue Problem: The solution of Perfectly coherent states (‘‘pure states’’) theeigenvalueequationforthematrix are subject to the superposition principle of A, Aψ = λψ where λ is a number called 44 1 Quantum Mechanics

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