Chun Wa Wong Department of Physics and Astronomy, University of California, Los Angeles, CA 90095-1547∗ (Dated: September 21, 2006) Review notes prepared for students of an undergraduate course in quantum mechanics at UCLA, Fall, 2005 – Spring, 2006. Copyright c 2006 Chun Wa Wong
2 2 I. The Rise of Quantum Mechanics (eG = e /4πǫ0 in Gaussian units). Hence U 1.1 Physics & quantum physics: E = T + U = = T. (5) 2 − Physics = objective description of recurrent physi- cal phenomena Bohr’s two quantum postulates (L quantization and Quantum physics = “quantized” systems with both quantum jump on photon emission): wave and particle properties L = mva = n~, n =1, 2, ..., − 1.2 Scientific discoveries: hν = Eni Enf . (6) Facts Theory/Postulates Facts → → For ground state: 1.3 Planck’s black body radiation: U e2 E = 1 = G = 13.6 eV Failure of classical physics: 1 2 − 2a − ~2 1 = T1 = 2 , Emissivity = uc, λmaxT = const, − −2ma 4 ~2 ˚ energy density = u = ρν E , a = 2 =0.53A (Bohr radius), (7) h i meG E exp = kBT, (1) h i 6 For excited states: (ρ = density of states in frequency space). ν E r = n2a, E = 1 . (8) Planck hypothesis of energy quantization (En = n n n2 nhν) explains observation: 1.6 Compton scattering: nx hν ∞ ne hν E = n=0 − = , (2) Energy-momentum conservation for a photon of lab h i ∞ e nx ehν/kB T 1 Pn=0 − − energy E = hc/λ scattered from an electron (of mass m) initially at rest in the lab (before=LHS, P where x = hν/(kBT ). after=RHS; 1=photon, 2= recoiling e):
1.4 Einstein’s photoelectric effect: p = p1 + p2, Photoelectrons ejected from a metal by absorbed E + mc2 = E + m2c4 + p2c2. (9) light of frequency ν have maximum energy: 1 2 q Photon scattered into lab angle θ has longer wave- E = hν W (work function) (3) − length λ1: that depends on the frequency ν of light and not λ λ = λ (1 cos θ), where 1 − e − its intensity. h hc λe = = 2 =2.4 pm (10) 1.5 Bohr’s atomic model: mc mc Electron of mass m in a classically stable circular is the electron Compton wavelength. orbit of radius r = a and velocity v around the 1.7 de Broglie’s matter wave: atomic nucleus: Massive matter and massless light satisfy the same mv2 e2 U energy-momentum and momentum-wavelength re- = G , or T = , (4) r r2 − 2 lations E2 = p2c2 + m2c4, h p = ~k = . (11) ∗Electronic address: [email protected] λ 2
Thus moving matter is postulated to be a wave with LDEs (linear DEs) with constant coefficients a motional de Broglie wavelength λ = h/p, leading have exponential solutions: to the diffraction maxima for both matter and light d of eikx = ikeikx, dx 2-sided formula: 2d sin θ = nλ, d iωt iωt e− = iωe− . (14) 1-sided formula: D sin θ = nλ. dt − These DE are called eigenvalue equations be- II. Waves and quantum waves cause the constant ik is called an eigenvalue of the ikx Classical and quantum waves have identical mathemat- differential operator d/dx, while the solution e is ical properties. called its eigenfunction belonging to its eigenvalue ik. 2.1 Waves in physics: The 1DWEq (12) has four distinct (i.e., linearly Waves rise and fall, travel, have coherence in space independent) eigenfunctions:
and in time, interfere, and diffract. i(kx ωt) Ψ(x, t)= e± ± . (15) Two classes of waves:
(a) Inertial waves in massive media: ocean waves, Energy in both classical and quantum waves is car- sound waves, vibrations of violin strings; ried by the intensity or energy flux (b) Noninertial waves in vacuum: EM waves, I Ψ 2. (16) matter waves. ∝ | | Its quadratic dependence on Ψ is responsible for the 2.2 Mathematical description of waves: interference between two traveling waves Ψi = iθi The 1-dimensional wave equation (1DWEq) Aie with real Ai:
2 2 2 2 2 ∂ 1 ∂ Ψ1 +Ψ2 = Ψ1 + Ψ2 + 2Re(Ψ∗Ψ2) Ψ(x, t)=0 (12) | | | | | | 1 ∂x2 − v2 ∂t2 = A2 + A2 +2A A cos(∆s), 1 2 1 2 ∆ω is an equation of state that gives a wave function where ∆s = ∆k x t . (17) − ∆k Ψ(x, t) describing an unfolding event in spacetime. The wave equation is a partial differential equation In the limit ∆k 0, (PDE) because it depends on more than one vari- → able, here the two variables x and t. dω = v = group velocity In contrast, the solution x(t) of an equation of mo- dk g ω tion such as a Newton equation describes how a = v = = wave velocity. (18) single point x(t) on an object evolves in time. Sim- 6 k ilarly, a snapshot of an object is not necessarily a wave. A wave function Ψ(x, t) is needed to describe Examples: Certain waves satisfy the relation ω = p a wave’s coherent structure in both space and time. Ak . Then vg/v = p. Specific examples are p = 2 for transverse vibrations of a thick bar, p = 3/2 However, a wave (any wave) can have particle prop- for short ripples moving under surface tension, and erties if a point x can be defined on it such that one p =1/2 for long waves on deep sea. can describe how its position changes in time. The result (17) is similar in structure to the squared The wave equation (12) describes linear waves sat- length of a sum of two vectors in space that inter- isfying the superposition principle that a sum sect at an angle θ: of solutions is also a solution : 2 2 2 A1 + A2 = A1 + A2 +2A1A2 cos θ, (19) Ψ = a1Ψ1 + a2Ψ2. (13) | | Two waves of the same amplitude traveling in op- Traveling waves: posite directions interfere form a standing wave: (a) f(x vt)= a wave traveling to the right (+x), − Ψ = A[cos(kx ωt) + cos(kx + ωt)] (b) g(x + vt)= a wave traveling to the left (-x), − = 2A cos(kx)cos(ωt) (20) (c) the composite wave a1f(x vt)+ a2g(x + vt) is also a wave. − with factorized space and time dependences. 3
2.3 Matter-wave quantization: is the normalized probablity density of finding the Light wave of momentum p and energy E = pc is wave in “k space”. (We shall not consider more described by the wave function: complicated properties that also depend on the time t or the energy E.) i(kx ωt) i[(p/~)x (E/~)t] e − = e − . (21) Uncertainty relations: A wave of any kind sat- de Broglie and Schr¨odinger suggested that the RHS isfies the uncertainty relation expression holds also for matter waves so that: 1 ∆x∆k , where ~ ≥ 2 ∂ i(p/~)x i(p/~)x i(p/~)x e =pe ˆ = pe , (∆x)2 = (x x¯)2 , x¯ = x , i ∂x h − i h i (∆k)2 = (k k¯)2 , k¯ = k . (28) ∂ i(E/~)t i(E/~)t i(E/~)t i~ e− = Heˆ − = Ee− .(22) h − i h i ∂t The expression for matter waves is usually written The differential operators that appear are called as the Heisenberg uncertainty principle: the momentum operatorp ˆ and the Hamiltonian (or ~ ˆ ∆x∆p . (29) energy) operator H, respectively. ≥ 2 Commutation relations: The rules of differen- tial calculus dictates that a differential operator Spreading wave packets: A Gaussian wave does not commute with its own variable: packet has the spectral amplitude
∂ ∂ ∂ αk2 1 x = 1+ x , or , x = 1; g(k)= e− , with ∆k = . (30) ∂x ∂x ∂x 2√α ∂ ∂ ∂ It gives rise to a wave function at time t =0 of t = 1+ t , or ,t =1. (23) ∂t ∂t ∂t x2/(4α) f(x, t = 0) e− , with ∆x = √α. (31) These results give the fundamental commuta- ∝ tors of wave mechanics Hence the wave packet at time t = 0 has the mini- ~ mal uncertainty product of [ˆp, x]= , [H,tˆ ]= i~. (24) i 1 ∆x∆k = . (32) 2 2.4 Wave packet and the uncertainty principle: For t = 0, however, the uncertainty product could A wave packet is a superposition of waves of the 6 type (21) be greater than the minimal value of 1/2. This hap- pens if the system is dispersive, meaning a nonzero 1 ∞ i[kx ω(k)t] second Taylor coefficient β in the following expan- f(x, t)= g(k)e − dk, (25) √2π sion: Z−∞ dω 1 d2ω with a linear superposition function g(k) sometimes ω(k) ω(k = 0)+ k + k2 called a spectral amplitude. ≈ dk 2 dk2 0 0 Expectation value (= mean value) of a property 1 = ω(0) + kv + k 2β. (33) A(x, t) of the wave: g 2
∞ The resulting approximate wave function at any A(x, t) ρ (x, t)A(x, t)dx, where h i ≡ P time t can then be evaluated to the closed form Z−∞ 2 x2/(4α ) f(x, t) f(x, t) e− t t , with ρP (x, t) = | | (26) ∝ ∞ f(x, t) 2dx iβ | | xt = x vgt, αt = α + t. (34) −∞ − 2 is the normalizedR probablity density of finding the wave at the spacetime point (x, t). Thus the position x of the wave moves in time with the group velocity The expectation value of a time-independent prop- erty B(k) of the wave can be calculated by using dω v = . (35) the spectral amplitude g(k): g dk 0
∞ Its position uncertainty increases in time to B(k) ρ˜P (k)B(k)dk, where h i ≡ Z−∞ 1/2 2 2 g(k) αt βt ρ˜P (k) = | | (27) ∆x(t)= | | = √α 1+ . (36) ∞ g(k) 2dx √α " 2α # −∞ | | R 4
For a massive particle in free space E2 = p2c2 + in the sense that the probability for finding the m2c4. Hence state somewhere in space at time t is
2 2 6 d E m c ∞ β = ~ = ~ =0. (37) P (t) = ρP (x, t)dx dp2 E3 6 Z−∞ ∞ Thus the spatial width of the Gaussian packet of = Ψ(x, t) 2dx =1. (44) | | a massive particle spreads out in both positive and Z−∞ negative times. By choosing P (t) = 1, the wave function is said to be normalized. III. Wave mechanics 3.3 Conservation of Ψ 2: | | Wave mechanics describes physics by using the The probability density can be moved around, but Schr¨odinger wave eq. and its wave functions. cannot be created or destroyed in the absence of explicit creative or destructive physical processes: 3.1 Schr¨odinger wave equation (SchWEq) : First quantization: The classical E-p relation d ∂ dx ∂ 0 = ρP (x, t)= + ρP (x, t) 2 2 2 2 4 dt ∂t dt ∂x E = p c +m c is quantized into a wave equation by replacing E,p by the differential operators ∂ ∂ = ρ (x, t)+ j (x, t). (45) ∂t P ∂x P ∂ E Hˆ = i~ → ∂t This is called the continuity equation for the con- ~ ∂ served probability current density p pˆ = : (38) → i ∂x jP (x, t)= vρP (x, t), (46) where v = dx/dt. ∂ 2 ~ ∂ 2 i~ Ψ(x, t)= c2 + m2c4 Ψ, (39) ∂t i ∂x " # IV. Wave mechanics in one spatial di- where the wave function Ψ(x, t) has to be added for mension the operators to operate on. In this way, a classical Wave mechanics is applied to simple one-dimensional state of motion “spreads out” into a wave structure. problems to illustrate how the SchWEq is solved and how Nonrelativistic limit: Likewise, the NR energy its wave character affects the properties of a system. E = T + V = p2/2m + V (not containing the rest 2 4.1 Solving the Schr¨odinger wave equation energy mc ) is quantized into the SchWEq: (SchWEq) :
∂ 1 ~ ∂ 2 Separation of varables: Since solutions are i~ Ψ(x, t)= + V (x) Ψ, (40) unique up to a superposition, we may look for a ∂t 2m i ∂x " # solution of the factorized form
Commutation relations: A differential operator Ψ(x, t)= ψ(x)φ(t). (47) does not commute with its variable. Hence Then
~ 2 [ˆp, x]= , [H,tˆ ]= i~. (41) φ(t) pˆ + V ψ(x) i ψ(x)Hφˆ (t) 2m = = E. (48) ψ(x) φ(t) φ(t) ψ(x) Group velocity: E2 = p2c2 + m2c4 EdE = c2pdp. Hence ⇒ On simplification, the first expression is clearly a function of t only, while the middle expression is a E dE function of x only. Hence the separation constant vv = = c2. (42) g p dp E must be a constant independent of t or x. The PDE can thus be broken up into two ordinary 3.2 Born’s interpretation of Ψ 2: DEs: | | The squared wave function gives the probabililty ∂ Hφˆ (t)= i~ φ(t) = Eφ(t), density of finding system at spacetime location ∂t (x, t): 1 ~ ∂ 2 + V (x) ψ(x) = Eψ(x), (49) 2 Ψ(x, t) = ρP (x, t) (43) " 2m i ∂x # | | 5
E q2 E 2 V0 q 2 2 k q V0 2 2 x = 0 k Κ 0 x = 0
BCs: Ψ = Ψ 0 x = 0 x = L Ψ ' = Ψ ' - V0
FIG. 2: Wave numbers for a rectangular potential barrier. FIG. 1: Wave numbers and wave functions at a potential step.
Its probability current density is conserved across where E appears in both ODEs as the energy the boundary: eigenvalue. The first ODE gives the time- ~ dependent factor k 2 j< = 1 B i(E/~)t m − | | φE (t)= e− , (50) ~q = j = A 2. (55) but the second equation is a second-order ODE > m | T | whose solution requires the imposition of two Its reflection and transmission coefficients at the Boundary conditions at a suitable boundary x = boundary are: x0: R = B 2,T =1 R. (56) ψ<(x0) = ψ>(x0), | | − d 4.4 Potential barrier: ψ′ (x0) ψ<(x) = ψ′ (x0). (51) < ≡ dx > A particle of incident energy E higher than the x0 height V of a rectangular potential barrier is par- 0 Note that if the first BC is violated, ψ′(x0) becomes tially reflected and partially transmitted by a potet- infinite and the DE (differential eq.) cannot be sat- nial barrier. The wave functions for x < 0 before isfied. If the second BC is violated, ψ′′(x0) becomes the barrier and for x>L after the barrier remain infinite. The DE also cannot be satisfied, except for the same as Eq.(54), but the presence of the sec- the special case where V (x0) has a compensating ond boundary at x = L causes reflected waves to infinity at x0. be present in the barrier region
4.2 Particle in a box of length L: ikx ikx 0
T HaL 1 -W 0.8 External 0.6 Sample field
0.4 HbL
0.2 -W Scanning EV0 1 2 3 4 5 Sample tip
FIG. 3: Transmission coefficient across a rectangular potential FIG. 5: Tunneling through a finite potential barrier (a) in barrier. cold emission, and (b) in a scanning tunneling microscope.
V the electron neutrino) powers the conversion of 1 Alpha mass into solar energy in the interior of the Sun. emission Fusion The reaction rate depends on the temperature 0.5 as 4. T T The luminosity of a star usually increases as M 3 of rR its mass M, or 24 of its central temperature. Such 1 2 3 an unusually strongT dependence led Bethe in 1938 -0.5 to look for another source of hydrogen burning. He discovered that this alternative route is a certain CNO cycle that involves certain carbon, nitrogen and oxgen isotopes in intermediate steps. FIG. 4: Tunneling across the Coulomb barrier in α decay and in nuclear fusion. Cold emission: Electrons are bound in metals by their work function W . However, they can be in- duced to tunnel out of the metal by applying an 4.5 Applications of quantum tunneling: external electric field to change the potential to Alpha decay: An α particle inside a nucleus feels a triangular penetrable potential, as illustrated in an attractive nuclear potential. As the α particle Fig. 5(a). leaves the nucleus in alpha decay, it sees the repul- In scanning tunneling microscopes, the tunnel- sive Coulomb potential outside the nucleus. These ing is induced by the proximity of the scanning tip, two potentials combine to form a potential barrier as shown in Fig. 5(b). Since the tunneling current that the α particle must tunnel through on its way increases greatly as the thickness of the potential out, as illustrated in Fig. 4. The decay probablility barrier decreases, one has a very sensitive device for is thus fT , the product of the frequency f of the mapping the geometrical shape of a surface down α particle’s hitting the inside wall of the potential to atomic dimensions. barrier and the transmission coefficient T through the potential barrier (that gives the tunneling prob- 4.6 Bound states in a potential well: ability per attempt). If V (x) constant as x , the lower of these Nuclear fusion: is just the time reversed situation two contant→ values is usually→ ±∞ taken to be the zero of emission. Here a charged particle, say a proton, of the energy scale. Two situations can then be is trying to enter a nucleus from the outside by tun- realized: neling through its very high and thick Coulomb po- tential barrier. The transmission coefficient is typi- (a) A system with E > 0 has continuous energy cally very small, but increases very rapidly with the spectrum with wave functions that are not lo- energy E. If a number of atomic nuclei are exposed calized or square-integrable, while to a gas of protons of increasing temperature, the (b) a system with E < 0 has a discrete energy high-energy tail of the Boltzmann energy distribu- spectrum with localized wave functions that tion of the proton gas will allow more and more are square-integrable. The resulting physical protons to get into atomic nuclei to initiate nuclear states are called bound states in a potential fusion reactions. For example, the pp reaction well. p + p d + e+ + ν (59) → The wave numbers associated with a bound state in (where e+ is the positron and ν is a particle called an attractive one-dimensional square-well potential 7
This is how a wave packet is constructed, namely by taking a linear superposition of waves. Also V=0 Κ 2 1 d E − g(k) = ixf(x). (64) F dk − q2 Thus a position operatorx ˆ = i∂/∂k gives the value
-V0 of the position x associated with the wave function x = -a x = a f(x). The FT shows that waves can be described either by wave functions f(x) in space, or by their spectral FIG. 6: Wave numbers associated with a bound state in an attractive square-well potential. amplitudes g(k) in k-space. Functions of one space can be constructed from those in the other space. Hence wave properties are completely specified ei- are summarized in Fig. 6, where ther in ordinary space or in k-space, a “simplifica- tion” that follows from a wave’s coherent structure. ~2κ2 ~2q2 E = E = , E V = . (60) Table I gives some simple examples of FTs. | | − 2m − 2m Dirac δ-function: Since Such an attractive one-dimensional potential has 1 at least one bound state, the ground state of − f(x) = f(x), (65) even parity whose wave function is proportional to F {F{ }} cos qx. This is because such a wave function al- we find (by writing out the two integral transforms ready “bends” over at the boundaries, and can al- on the LHS in two steps) that ways be matched to a decreasing exponential func- tion outside the potential well. ∞ δ(x x′)f(x′)dx′ = f(x), (66) − Z−∞ V. The mathematical structure of quan- where tum mechanics ′ 1 ∞ ik(x x ) Physical states in quantum mechanics are linear waves. δ(x x′)= e − dk (67) − 2π Both quantum states and classical linear waves can be Z−∞ superposed to form other waves, in the same way that ordinary vectors in space can be added to form other vectors. TABLE I: Some Fourier transforms. Property If f(x)is then f(x) is 5.1 Fourier transform (FT): F{ } The superposition principle for linear wave func- f(x) g(k) tions (solutions of linear wave equations) is formal- 2 αx2 1 k ized in the methods of Fourier series and Fourier e− e− 4α transforms (FT). Both methods and their general- √2α a izations are used in quantum mechanics. unit step function 1 2 sin ka − √2π k The FT of a wave function f(x) in one spatial vari- d n n able x is defined by the integral Derivatives ( dx ) f(x) (ik) g(k)
n d n 1 ∞ ikx Derivatives ( ix) f(x) ( ) g(k) f(x) g(k)= f(x)e− dx. (61) − dk F{ }≡ √2π Z−∞ ika Translation f(x a) e− g(k) For example, − αx Attenuation f(x)e− g(k iα) d ip − f(x) = ikg(k) g(k). (62) ~ b F dx → CC Real g∗(k)= g( k) − Thus the momentum operatorp ˆ = (~/i)∂/∂x yields Real and even g∗(k)= g( k)= g(k) the momentum value p = ~k associated with the −
spectral amplitude g(k). Real and odd g∗(k)= g( k)= g(k) − − Fourier inversion formula: a −a a 1 1 ∞ ikx from to − g(k) f(x)= g(k)e dk. (63) b F { }≡ √2π Complex conjugation Z−∞ 8
is called a Dirac δ-function. transposition with the complex conjugation of all Dirac has given a simple intuitive construction of “matrix elements”. his δ-function as a rectangular function of width ǫ The Dirac bracket φ ψ is a scalar product of and height 1/ǫ with a unit area under it: vectors or an inner producth | i of functions
1/ǫ, when x x′ ǫ/2, ψ1 δ(x x′) = lim | − |≤ (68) φ ψ = (φ1∗, φ2∗) = φ1∗ψ1 + φ2∗ψ2, (71) − ǫ 0 0, otherwise. h | i ψ2 → 5.2 Operators and their eigenfunctions: where the complex conjugation on the bra com- ponents is needed to keep the self scalar product When a differential operator operates on its eigen- ψ ψ nonnegative. For two orthonormal vectors function, it gives a function that is proportional orh | states:i to the eigenfunction itself: i j = δ . (72) ~ ∂ h | i ij peˆ ikx = eikx = ~keikx, i ∂x Eigenbras and eigenkets: One goes beyond sim- ple vector spaces by using the eigenfunctions of op- i(x/~)p ∂ ixp/~ ixp/~ xeˆ − = i~ e− = xe− . (69) ∂p erators such as positionx ˆ and momentump ˆ. In fact, we immediately associate their eigenfunctions The proportionality constant is called its eigen- with abstract eigenstates in the following eigen- value. bra or eigenket form: Two functions or eigenfunctions can be combined xˆ x = x x , and x xˆ = x∗ x = x x , into an inner product: | i | i h | h | h | pˆ p = p p , and p pˆ = p∗ p = p p , (73) | i | i h | h | h | ∞ (φ, ψ) φ∗(x)ψ(x)dx ≡ where the eigenvalues x and p are real, as is true for Z−∞ all physical observables, i.e., properties that can be ∞ measured experimentally. The eigenvalues x and p = gφ∗(k)gψ(k)dk. normally range continuously from to . Z−∞ −∞ ∞ Dirac denotes this inner product by the bracket Projection and quantum destructive mea- surement: A simple 2-dimensional vector A can symbol φ ψ . | i h | i be decomposed into its components Ai = i A with 5.3 Dirac notation: respect to the basis i : h | i | i Functions in function space and states of a physi- A = 1 A + 2 A cal system in quantum mechanics are mathemati- | i | i 1 | i 2 cally similar to vectors in space. Dirac emphasizes N = i i A . (74) this similarity using a mathematical notation that | ih | | i i=1 ! works for vectors, functions and quantum states. X A vector (function, state) can be expanded in terms In a similar way, an N-dimensional vector space is of basis vectors (functions, states). Dirac distin- made up of the N one-dimensional subspaces pro- guishes between two kinds of vectors (functions, jected out by the projection operators states) – column vectors (functions, ket states) and row vectors (complex-conjugated functions, i = i = i i (75) P M | ih | bra states), where the name bra and ket comes from the two parts of the word “bracket”: in the sense that
ψ = ket= ψ 1 + ψ 2 i A = Ai i (76) | i 1| i 2| i P | i | i ψ = 1 = column vector leaves only that part of the vector that is in this ψ2 ith subspace. is also called a measurement sym- Pi φ = bra= φ1∗ 1 + φ2∗ 2 bol (hence i). This measurement is not a classi- h | h | h | cal nondestructiveM measurement but a quantum de- = (φ1∗, φ2∗) = row vector structive measurement that destroys all other com- = φ †, (70) | i ponents of the original vector A . The projection | i For simple vectors, the basis states 1 and 2 are collapses the state A into just the part Ai i in | i | i the ith subspace. | i | i just the unit basis vectors usually denoted e1 and e2, respectively. The † symbol is called an adjoint Completeness relations: The original vector A operation. It combines the matrix operation of can be restored by adding up all its projections| i 9
Ai i . Hence an identity operator can be con- Dirac δ-function: The equivalence structed| i for this linear vector space ∞ ψ(x) = δ(x x′)ψ(x′)dx′ N − 1 = i i (77) Z−∞ | ih | ∞ i=1 = x ψ = x x′ x′ ψ dx′ (82) X h | i h | ih | i Z−∞ for discrete basis states that satisfy the orthonor- shows that mality relation i j = δ . h | i ij completeness closure x x′ = δ(x x′). (83) The idea of or holds also for h | i − eigenstates of operators such asx ˆ andp ˆ that have It immediately follows that the Dirac δ-function continuous eigenvalues: can be represented in infinitely many ways. For
∞ example, 1 = x x dx | ih | N N Z−∞ x x′ = x i i x′ = φ (x)φ∗(x′). (84) ∞ h | i h | ih | i i i = p p dp. (78) i=1 i=1 | ih | X X Z−∞ The resulting basis states x or p are said to be Quantum measurement and wave-function continuous basis states. | i | i collapse: Electrons emitted from a source pass through a single slit and fall on a distant observing Representations: A discrete representation is a screen on the other side of the slit: As a coherent specification of a vector (or state) A by its com- each | i matter wave, electron forms a diffraction pat- ponents Ai in a discrete basis. A continuous rep- tern on the observing screen in the direction y of resentation uses the continuous basis states x to the diffraction pattern. (Diffraction = spreading of give the components A(x)= x A in the expansion| i h | i a wave into the geometrical shadow regions.) A detector of acceptance ∆y placed at position ∞ A = dx x A(x). (79) y = d on the observing screen detects electrons | i | i Z−∞ from signals of their passage through the detec- tor. If ψ(y′) = y′ ψ is the wave function of self- Wave functions: The (position) wave function of diffracting electronsh | i normalized in some way, the a state or wave ψ is just its component ψ(x) = part that enters the detector is x ψ in the position| i representation. Its compo- nenth | iψ(p) = p ψ in the momentum representa- d+∆y/2 h | i ∆ψ(d)= dy′ y′ y′ ψ . (85) tion is called its momentum wave function. A wave d ∆y/2 | ih | i function can be expanded into an arbitrary discrete Z − representation of basis states i The probability of an electron entering the detector | i is N d+∆y/2 x ψ ψ(x) = x i i ψ 2 h | i≡ h | ih | i ∆P (d)= dy′ ψ(y′) . (86) i=1 d ∆y/2 | | X Z − N Note that the electron described by the wave func- = φi(x)ψi, (80) i=1 tion ψ(y) can be anywhere on the screen before the X detection. However, once it is known to have gone where φi(x)= x i is the wave function of the basis into the detector, e.g., by the detector signal it gen- state i . Incidentally,h | i N must be infinite to match erates, its wave function has collapsed into ∆ψ(d). the number| i of points on the straight line. This is how a quantum measurement can modify the state itself. A wave function can also be expanded in another continuous representation, say the momentum rep- 5.4 Operators and their matrix elements: resentation: Matrix mechanics
∞ Operators as matrices: Operators can be visu- x ψ = ψ(x) = dp x p p ψ h | i h | ih | i alized as matrices because they satisfy matrix op- Z−∞ erations. Hence quantum mechanics is also called 1 ∞ dp ~ ˆ = ei(p/ )xψ(p). (81) matrix mechanics. For example, an operator A √2π ~ acting on a state always gives another state: Z−∞ This is just the inverse Fourier transform relation. Aˆ ψ = c φ (87) | i | i 10
in much the same way that a square matrix multi- Note how the operatorp ˆ becomes a diagonal differ- plying a column vector always gives another column ential operator in the x-representation, and how it vector. It can happen occasionally that the result comes out to the right of the Dirac bracket to avoid is actually proportional to the original state ψ operating on it. The Hamiltonian operator too has | i interesting diagonal representations: Aˆ a = a a . (88) | i | i E E′ E′ = δ(E E′)E′, if continuous E Hˆ E′ = h | i − These special states are called the eigenstates of h | | i n n′ En′ = δnn′ En′ , if discrete. Aˆ. The proportionality constants a are called their h | i (95) eigenvalues. Note how the eigenvalue a is used to label the eigenstate a itself. This operator equa- tion is called an eigenvalue| i equation. Eigenvalue eq. as wave eq.: Matrix mechanics Noncommuting operators: Operators, like ma- can be shown to be equivalent to wave mechanics by writing the operator eigenvalue equation Hˆ Ψ = trices, do not generally commute. For example, | Ei E ΨE as a Schr¨odinger wave equation: ~ | i [ˆp, xˆ]= . (89) ˆ i E x ΨE = x H ΨE h | i h | | i 2 ∞ pˆ In matrix mechanics, commutators like this specify = dx′ x + V (ˆx) x′ Ψ (x′) h | 2m i E the fundamental wave nature of the matter waves Z−∞ 2 described by these operators. It is the starting ∞ 1 ~ d point from which other results can be derived. In = dx′δ(x x′) + V (x′) − 2m i dx′ contrast, the wave property of matter is described Z−∞ " # in wave mechanics by postulating the quantization Ψ (x′) × E rule 2 1 ~ d = + V (x) Ψ (x). (96) ~ ∂ 2m i dx E pˆ = , (90) " # i ∂x and the wave function ψ(x) on whichp ˆ operates. 5.5 Hermitian operators and physical observ- Operator matrix elements: Operators like ma- ables: trices have matrix elements. The matrix elements Hermitian operator: is one for which of a matrix or operator Mˆ can be expressed conve- T niently in the Dirac notation as Hˆ † = Hˆ ∗ = Hˆ (itself). (97)
M = i Mˆ j , (91) ij h | | i It has the important properties that a two-sided representation in terms of a suitable (a) Eigenvalues are real. basis i . A2 2 matrix or operator can thus be written| i in operator× form as (b) Eigenstates can be orthonormalized.
Mˆ = M11 1 1 + M12 1 2 Physical observables can have only real values. | ih | | ih | Hence they are described by Hermitian operators + M21 2 1 + M22 2 2 . (92) | ih | | ih | in quantum mechanics. The ket-bra combinations on the RHS are them- Dirac notation: For any operator Aˆ: self operators or square matrices. The off-diagonal operators are the transition operators ∞ φ Aψˆ = φ†(x)[Aψˆ ](x)dx h | i ψ1 ψ1φ1∗ ψ1φ2∗ Z−∞ ψ φ = (φ∗φ∗)= , (93) ψ 1 2 ψ φ ψ φ ∞ | ih | 2 2 1∗ 2 2∗ = [Aˆ†φ](x) †ψ(x)dx { } Z−∞ while the diagonal operators (with φ = ψ) are pro- = Aˆ†φ ψ . (98) jection operators. h | i An operator in the representation of its own eigen- The Hermiticity of an operator Hˆ is confirmed by states is a diagonal matrix. Thus the equality φ Hˆ ψ = Hφˆ ψ of the two distinct integrals involved.h | | i Noteh that| i if φ(x) is a column x xˆ x′ = x x′ x′ = x′δ(x x′) h | | i h | i − wave function (called a spinor), the transposition ~ d ~ d part of the adjoint operator will come into play to x pˆ x′ = x x′ = δ(x x′) .(94) h | | i h | i i dx − i dx change it into a row spinor. ′ ′ 11
5.6 One-dimensional harmonic oscillator: Oscillator quanta: The quantum number n (qu. The 1DHO problem can be solved elegantly in ma- no. = any number used to label a quantum state) trix mechanics. First write its Hamiltonian gives the number of oscillator energy quanta, each of energy ~ω, that can be emitted from that state. pˆ2 1 Thusa ˆ increases the number of energy quanta in Hˆ = + mω2xˆ2 (99) † 2m 2 the state by 1, and is called a creation operator, while the destruction operatora ˆ decreases n by in the compact form 1. Physical processes that causes the absorption Hˆ 1 or emission of energy quanta can be expressed in hˆ = = qˆ2 +ˆy2 , terms of these operators. ~ω 2 xˆ The ground state which by definition cannot emit wherey ˆ = , x0 any energy has the qu. no. n = 0. It has a ~ x 1 d zero-point energy ω/2 that comes from the wave qˆ = 0 pˆ = . (100) ~ i dy spreading of its wave function from the classical point x = 0 of stable equilibrium. ~ Here the oscillator length x0 = /mω is defined Wave functions: The impossibility of energy by the equation p emission from the ground state is stated by the op- ~ 2 2 erator equation ω = mω x0. (101) aˆ 0 =0. (109) The dimensionless position and momentum opera- | i tors do not commute: Its y-representaion gives the first-order differential [ˆq, yˆ]=1/i. (102) equation for the ground-state (GS) wave function: ˆ The dimensionless Hamiltonian h can be factorized: d y + φ0(y)=0. (110) ˆ 1 dy h =a ˆ†aˆ + 2 , wherea ˆ = 1 (ˆy + iqˆ) √2 The resulting normalized GS wave function is 1 aˆ† = √ (ˆy iqˆ) (103) 2 − y2/2 1 φ0(y)= α0e− , where α0 = . (111) are called the step-down and step-up operators, re- π1/4 spectively. They do not commute Excited states can be created by adding energy a,ˆ aˆ† =1. (104) quanta to the GS:
n Number operator and its eigenstates: The (ˆa†) number operator Nˆ =ˆa aˆ is Hermitian. It has real n = 0 . (112) † | i √n! | i eigenvalues n that can be used to label its eigen- states Consequently, its wave function can be constructed Nˆ n = n n . (105) by successive differentiations of the GS wave func- | i | i tion: It then follows that n is also an energy eigenstate: n | i (ˆa†) ˆ 1 ~ φn(y) = y 0 H n = (n + 2 ) ω n . (106) h | √n! | i | i | i n α0 d y2/2 The eigenvalue n of the number operator is a non- = y e− . (113) √2nn! − dy negative number or integer 0, 1, 2, ... This im- portant quantization property follows from non- negative nature of its Hamiltonian (meaning that Operators in the number representation: In ˆ the number representation n , the only nonzero H 0) and from the nonzero commutators | i h i≥ matrix elements ofa ˆ anda ˆ† lie along the diagonal [N,ˆ aˆ†]=ˆa†, [N,ˆ aˆ]= a.ˆ (107) one line away from the main diagonal of the matrix: − The commutators show thata ˆ† anda ˆ change the 0 √1 0 . 0 0 0 . state n as follows: 0 0 √2 . √1 0 0 . aˆ = , aˆ† = . (114) | i 0 0 0 . 0 √2 0 . aˆ† n = √n +1 n +1 , aˆ n = √n n 1 . (108) ...... | i | i | i | − i 12
Hence the dimensionless position and momentum Ehrenfest’s theorem: The expectation value of operators are the square matrices a physical observable satisfies a classical equation of motion. The expectation value of an operator 0 √1 0 . Aˆ(t) in the time-dependent state 1 1 √1 0 √2 . yˆ = (ˆa +ˆa†)= , √2 √2 0 √2 0 . iHt/ˆ ~ Ψ(t) = e− Ψ(0) is (121) . . .. | i | i 0 √1 0 . ˆ ˆ 1 1 √1 0 √2 . A(t) t Ψ(t) A(t) Ψ(t) qˆ = (ˆa aˆ†)= − . h i ≡ h | | i √ − √ 0 √2 0 . iHt/ˆ ~ ˆ iHt/ˆ ~ i 2 i 2 − = Ψ(0) e A(t)e− Ψ(0) . (122) . . .. h | | i (115) A time differentiation of the three time factors gen- erates three terms: 5.7 Uncertainty principle: ˆ The quintessential wave property described by the d ∂A(t) i Aˆ(t) t = t + [H,ˆ Aˆ(t)] t. (123) Heisenberg uncertainty principle takes on a sur- dth i h ∂t i ~h i prisingly elegant form in matrix mechanics: If the Examples: square uncertainty of the expectation value of a physical observable Aˆ in the quantum state ψ is | i d i 1 defined as xˆ t = [H,ˆ xˆ] t = pˆ t; dth i ~h i mh i (∆A)2 = (Aˆ A¯)2 , (116) d i dVˆ h − iψ pˆ = [H,ˆ pˆ] = . (124) dth it ~h it −h dxˆ it then the product of uncertainties of two physical observables Aˆ and Bˆ in the same quantum state satisfies the inequality VI. Quantum mechanics in three spatial
1 dimensions (∆A)(∆B) [A,ˆ Bˆ] ψ . (117) ≥ 2 h i Different directions in space are independent of one
another in quantum mechanics in the sense that Simultaneous eigenstates: a,b exist for com- | i muting operators Aˆ and Bˆ: [ˆx , xˆ ]=[ˆp , pˆ ]=[ˆx , pˆ ]=0, if i = j. (125) i j i j i j 6 Aˆ a,b = a a,b , Bˆ a,b = b a,b . (118) | i | i | i | i QM problems in three-dimensional space can be solved Then ∆A =0=∆B. by using methods of both wave and matrix mechanics.
5.8 Classical correspondence: 6.1 Separation of variables in rectangular coor- Bohr’s correspondence principle: Quantum dinates: mechanics reproduces classical mechanics in the If a Hamiltonian Hˆ = Hˆ1 + Hˆ2 + Hˆ3 is separable limit of large quantum numbers. Example: The into terms referring to different dimensions in rect- probability density of quantum state n of the | i angular coordinates, its energy eigenstates can be 1DHO agrees with the classical value: factorized into the product states n = n n n = | 1 2 3i 2 n1 n2 n3 , one in each spatial dimension: lim ψQM(x) ρCM(x). (119) | i| i| i n n → →∞ ˆ H n1n2n3 = (E1 + E2 + E3) n1n2n3 . (126) For a periodic system of period T , the classical | i | i probability density can be defined in terms of the The method works for other operators too: probablity dP (x) of finding the classical point mass within a width dx of the position x during a mo- 2 2 2 2 pˆ p1p2p3 = (p + p + p ) p1p2p3 . (127) ment dt in time t: | i 1 2 3 | i 2 Examples: dP (x) = ρCM(x)dx = dt, or T (a) 3-dim HO: En = ~ω(n+3/2),n = n1 +n2 +n3. CM 2 ρ (x) = , (120) 2 2 2 2 v(x)T (b) Free particle: p = p1 + p2 + p3 in the state p1p2p3 . where v(x) = dx/dt is the velocity at position x. | i See Fig. 4-18, 19 of Bransden/Joachain for a com- (c) Particle in a box of sides Li: The discrete mo- parison between quantum and classical ρs for the mentum spectrum is defined by the quantum 1DHO. numbers ni = kiLi/π =1, 2, .... 13
6.2 Separation in spherical coordinates: The kinetic energy operator in spherical coordi- Kinetic energy: In the spherical coordinates r = nates is 2 (r, θ, φ), pˆ separates into a radial part and an an- ˆ2 gular part: ˆ ˆ ˆ 1 2 L T = Tr + TL = pˆr + 2 , (136) 2m r ! Lˆ2 pˆ2 =p ˆ2 + , r r2 Central potential: Central potentials are spheri- ex ey ez cally symmetric and depend only on the radial dis- where Lˆ = ˆr pˆ = xˆ yˆ zˆ tance from the center of potential (or force): × pˆ pˆ pˆ x y z V = V (r) = f(θ, φ). (137) = Lˆxex + Lˆ yey + Lˆzez (128) 6 is the orbital angular momentum operator. Note For central potentials, the Hamiltonian separates that the position and momentum operators in a in the radial and angular coordinates typical term of Lˆ such as e yˆpˆ commute because x z ˆ ˆ ˆ ˆ they refer to different directions in space. However, H = (Tr + V )+ TL. (138) Lˆ is quite a different type of operator from ˆr or pˆ, As a result, the wave function of definite energy E because its components Lˆ do not commute with i factorizes as follows: one another:
ˆ ˆ ~ˆ ψEℓm(r)= r Eℓm = REℓ(r)Yℓm(θ, φ). (139) [Li, Lj]= i Lk, (i, j, k in RH order). (129) h | i On the other hand, The radial factor satisfies the Schr¨odinger equation
ˆ2 ˆ 1 2 1 2 [L , Li]=0, for any component i. (130) pˆ + ~ ℓ(ℓ + 1) + V (r) R (r) 2m r r2 Eℓ ˆ 2 ˆ The two commuting operators L , Lz can be used = EREℓ(r), (140) to define the simultaneous eigenstate ℓm : | i where Lˆ 2 ℓm = ~2ℓ(ℓ + 1) ℓm , | i | i 1 d d ˆ ~ pˆ2 = ~2 2 = ~2 r2 Lz ℓm = m ℓm , (131) r − ∇r − r2 dr dr | i | i 1 d2 where ℓ and m are called the orbital (angular mo- = ~2 r. (141) mentum) and magnetic quantum numbers, respec- − r dr2 tively. This eigenstate describes completely an an- gular state in spherical coordinates. The resulting Note that the wave function REℓ(r) depends on angular wave function ℓ as well as the energy eigenvalue E. An en- ergy eigenstate in three-dimensional space has only θφ ℓm = Y (θ, φ) (132) three state labels. h | i ℓm Laplace equation: of electrostatics is also im- is called a spherical harmonic. portant in QM because its solutions describe the ˆ In the position represention, the Lz equation can quantum states of a static or resting free particle be written as the differential equation (V =0, p = 0 and E = 0):
~ ∂ ∇2 Y (θ, φ)= mY (θ, φ). (133) ψ(r)=0. (142) i ∂φ ℓm ℓm The solutions regular at the origin r = 0 are the The solutions in the variable φ are solid spherical harmonics (or harmonic polynomi- imφ als) Yℓm(θ, φ)= f(θ)e . (134) (r)= rℓY (θ, φ). (143) These solutions are periodic in φ with a period of Yℓm ℓm 2π: Solutions of the Laplace equations are sometimes called harmonic functions. eim(φ+2π) = eimφ. (135) The solutions in rectangular coordinates are sim- Hence m must be an integer (0, 1, 2, ...). It will pler. They are all rectangular polynomials with turn out that ℓ is a nonnegative± integer± bounded terms of the form xn1 yn2 zn3 having the same de- from below by m . gree n = n + n + n . All rectangular polynomials | | 1 2 3 14
Spherical coordinates: The energy eigenstates TABLE II: Rectangular polynomials and solid spherical har- are monics. 2ν n Rect. Poly. Harmonic? Number ℓm? nℓm (aˆ†) ℓ=n 2ν,m(aˆ†) 000 , ν =0, 1, ...(150) Y | i∝ Y − | i All the states shown in Table II with x replaced by 01 Y 1 00 Y the operatora ˆ1†, etc., are valid 3DHO states. The 1 z Y 3 1m=0 degeneracy of these states is thus Y x iy Y m = 1 n ± ± 1 2 2 2 D(n)= (n + 1)(n +2)= (2ℓ + 1). (151) 2 2z x y Y 5 2 =0 2 m ℓ=even or odd z(x − iy)Y− Y m = 1 X ± 2 ± (x iy) Y m = 2 6.4 Plane wave in free space: x2 ±+ y2 + z2 N 1— ± Three-dimensional plane waves have the product wave functions ik r i(k x+k y+k z) can be written in the spherical form with definite e · = e x y z ℓ,m: ∞ = Aℓ0REℓ(r)Yℓ0(θ, φ) ℓ=0 n harmonic, if ℓ = n X r Yℓm(θ, φ)= , (144) non harmonic, if ℓ the componentsa ˆi† of the vector operator aˆ† can between the electron and the proton. The degen- create a 3DHO state. Their energy eigenvalues are eracy of states is then n 1 E = (n + 3 )~ω, where − n 2 d(n)= (2ℓ +1)= n2. (155) n = n1 + n2 + n3 =0, 1, 2... (149) Xℓ=0 15 Wave functions: are TABLE III: States of the hydrogen atom. The radial wave ψ (r)= R (r)Y (θ, φ). (156) functions are expressed in terms of the dimensionsless distance nℓm nℓ ℓm r˜ = r/a. The radial functions Rnℓ are functions of n En ℓ m Spect.Symbol Rnℓ(˜r) ~2κ2 r˜ ρ = κr, where = E. (157) 1 E1 0 0 1s e− 2m − r/˜ 2 2 E1/4 1 0, 1 2pre ˜ − Then ± r/˜ 2 0 0 2s (1 r/˜ 2)e− − u (ρ) nℓ ℓ ρ 2 ℓ r/n˜ Rnℓ(ρ)= = ρ e− v(ρ) (158) n E1/n ℓmax = n 1 n[ℓmax]r ˜ max e− ρ ℓ− n[ℓ] can be written in two other forms with each func- tion satisfying a different differential equation: 7.1 Angular momentum: ℓ(ℓ + 1) 2m e2 2R (r)= G + κ2 R , The orbital angular momentum operator ∇r nℓ r2 − ~2 r nℓ ex ey ez ℓ(ℓ + 1) ρ0 Lˆ = ˆr pˆ = xˆ yˆ zˆ u′′ (ρ)= +1 u , nℓ 2 nℓ × pˆ pˆ pˆ ρ − ρ x y z = Lˆxex + Lˆ yey + Lˆzez (163) ρv′′(ρ)+2(ℓ +1 ρ)v′ + (ρ 2ℓ 2)v =0. (159) − 0 − − is very different from xˆ and pˆ because its compo- where ρ0 = 2/κa and v′(ρ) = dv(ρ)/dρ. All solu- nents do not commute with one another: tions have ˆ ˆ ~ˆ ˆ ˆ ~ˆ E [Lx, Ly]= i Lz, or L L = i L. (164) ρ =2n, giving E = 1 , (160) × 0 n n2 However, every component Lˆi commutes with in agreement with the Bohr model. ˆ2 ˆ2 ˆ2 ˆ2 L = Lx + Ly + Lz. (165) Polynomials vν (ρ): The factor v(ρ) in Eq.(158) is a polynomial of ρ: The angular momentum states ℓm are the simul- taneous eigenstate of the two commuting| i operators (a) v (ρ) = 1 gives solutions with ℓ = n 1, ˆ2 ˆ 0 − L , Lz: (b) v1(ρ)= ρ c(a constant) gives solutions with − Lˆ2 ℓm = ~2ℓ(ℓ + 1) ℓm , ℓ = n 2, | i | i − Lˆ ℓm = ~m ℓm , (166) (c) polynomials of degree ν, namely z| i | i ν The nature of these states can be studied by using j vν (ρ)= cj ρ , (161) the j=0 X Ladder operators: for angular momentum states give solutions with ℓ = n 1 ν. Lˆ = Lˆx iLˆy. (167) − − ± ± Hydrogen states: The properties of the states of These operators have the structure of the hydrogen atom are summarized in Table III. iφ The spectroscopy notation used is: x iy = r sin θe± (168) ± [ℓ]= s, p, d, f, ... for ℓ =0, 1, 2, 3, ... (162) that in wave functions have the magnetic quan- tum numbers m = 1. Hence Lˆ can be called ± ± The spectroscopic symbol for a quantum state is the m = 1 components of Lˆ in spherical coor- 2s+1 ± [ℓ]j , giving the spin degeneracy 2s + 1, the dinates. Operators and states are both matrices, orbital angular momentum [ℓ] and the total spin though states are not square matrices. They can quantum number j. be multiplied together. In spherical coordinates, the m value of a product of functions or matrices is the sum of the m values of the individual func- VII. Spin and statistics tions or matrices. Hence we expect that Lˆ ℓm The study of orbital angular momentum states leads to be a state of magnetic quantum number m±| 1,i us to intrinsic spins and quantum statistics. respectively. ± 16 Before demonstrating this result, let us first note 7.2 Addition of angular momenta: however that the ladder operators, like their con- ˆ ˆ Product states: Two angular momentum opera- stituent rectangular components of Lx, Ly, com- tors Lˆ and Sˆ are independent if [Lˆ , Sˆ ] = 0. The ˆ2 i j mute with L itself. Hence they do not change the simultaneous eigenstates of the commuting opera- orbital quantum number ℓ when acting on the state 2 2 tors Lˆ , Lˆz, Sˆ , Sˆz are the product states ℓm . However, they too do not commute with Lˆ : | i z ˆ ˆ ~ˆ ℓmℓsms = ℓmℓ sms . (178) [Lz, L ]= L . (169) | i | i| i ± ± ± These commutators can be used to show explicitly They have the degeneracy that Lˆ+ is the step-up operator for m, increasing its value by 1, while the step-down operator Lˆ d(ℓ,s)=(2ℓ + 1)(2s + 1). (179) decreases m by 1: − Coupled states: Two independent angular mo- φ = Lˆ ℓm = Aℓm ℓm 1 . (170) | ±i ±| i | ± i menta Lˆ and Sˆ can be added to the total angular Aℓm can be found by calculating the normalization momentum: φ φ using h ±| ±i ˆ ˆ ˆ ˆ 2 ˆ2 ˆ2 ˆ2 ˆ ˆ ~ˆ J = L + S. (180) L Lz = Lx + Ly = L L Lz. (171) − ∓ ± ± 2 2 2 The result is The four operators Lˆ , Sˆ , Jˆ , Jˆz can be shown to commute among themselves. Hence their simulta- Lˆ ℓm = ~ ℓ(ℓ + 1) m(m 1) ℓm 1 . (172) ±| i − ± | ± i neous eigenstates ℓsjm exist: | i Possible ℓmp values: m2 is bounded because Lˆ2 ℓsjm = ~2ℓ(ℓ + 1) ℓsjm , Lˆ2 < Lˆ2 , or m2 <ℓ(ℓ + 1). (173) | i | i h zi h i Sˆ2 ℓsjm = ~2s(s + 1) ℓsjm , | i | i Hence m m m . From Eq.(172): 2 2 min ≤ ≤ max Jˆ ℓsjm = ~ j(j + 1) ℓsjm , ˆ | i | i L+ ℓmmax = 0 mmax = ℓ, Jˆ ℓsjm = ~m ℓsjm . (181) | i ⇒ z| i | i Lˆ ℓmmin = 0 mmin = ℓ. (174) −| i ⇒ − So the total number of steps N on the ladder is Clebsch-Gordan coefficients: The two distinct ways (product or coupled) of specifying the states mmax mmin =2ℓ = N 1. (175) of two angular momenta are related by a change of − − representation: Assuming that ladders with any number of steps exist in nature, one finds N 1 ℓsjm = ℓmℓsms ℓmℓsms ℓsjm , | i | ih | | i Odd N = 2ℓ +1 ℓ = − = integer, m m ! ⇒ 2 Xℓ s N 1 Even N = 2s +1 s = − = half integer. ⇒ 2 ℓmℓsms = ℓsjm ℓsjm ℓmℓsms , (182) | i | ih | | i (176) j X The orbital quantum numbers ℓs are integers be- where each sum involves only one summation in- cause they are also the integer degrees of the har- dex because mℓ + ms = m. The transformation monic polynomials . The half-integer s’s are Yℓm (Clebsch-Gordan) coefficients are chosen real, and the spin quantum numbers of an intrinsic spin an- therefore gular momentum Sˆ. The word “spin” is also used for any angular momentum. ℓsjm ℓmℓsms = ℓmℓsms ℓsjm . (183) 2 h | i h | i Angular-momentum representations: Lˆ , Lˆz and therefore also L ˆ , Lˆx, Lˆy are N N matrices ± × Examples: Two spin 1/2 states can be coupled to for those states with N steps in their ladder. a total spin of S =1 or 0: Example: For spin s = 1/2, the ladder has only two steps, with spin states “up” and “down”. 1 1 SM = 11 = , 1 1 1 1 = , 2 2 ↑↑ 2 2 ↓↓ ˆ2 | i | 1 i | − i | i Then the spin operators are 2 2 matrices: S = 1 1 10 = + ; (3/4)~2diag(1, 1). The Pauli× spin matrices are | 2 2 i √2 |↑↓ ↓↑i ˆ ~ 1 1 00 = 1 , (184) the components of σˆ =2S/ : | 2 2 i √2 |↑↓−↓↑i 1 0 0 1 0 i σ = , σ = , σ = . (177) where ( ) denotes the spin up (down) magnetic z 0 1 x 1 0 y i −0 ↑ ↓ − state of the spin 1/2 system. 17 Vector model: The length J of the vector sum J = L + S satisfies a triangle inequality: TABLE IV: electronic structure of the elements. The quan- tum numbers of some atomic ground states are also given as the spectroscopic symbol 2S+1[L] . L S J L + S, but J | − | ≤ ≤ Z Element Spatial Configuration Ground State jmin = ℓ s j jmax = ℓ + s, (185) 2 1 H 1s S1/2 | − | ≤ ≤ 2 1 2 He (1s) S0 for quantum angular momenta involves the quan- tum numbers. 2 Construction of total angular momentum 3 Li (He)2s S1/2 2 1 states: The jth ladder of states has 2j + 1 steps. 4 Be (He)(2s) S0 The topmost state m = ℓ + s of the longest ladder 2 2 5 B (He)(2s) 2p P1/2 jmax = ℓ + s is simply 6 C (He)(2s)2 (2p)2 2 3 ℓs ℓ + sℓ + s = ℓℓss . (186) 7 N (He)(2s) (2p) | i | i 8 O (He)(2s)2 (2p)4 2 5 2 9 F (He)(2s) (2p) P3/2 Then step down to find the remaining states on the 2 6 1 ladder: 10 Ne (He)(2s) (2p) S0 Jˆ ℓsjm = ~ j(j + 1) m(m 1) ℓsjm 1 −| i − − | − i If β = α is the same s.p. state: = (Lˆp + Sˆ ) ℓmℓsms − − | i mℓms Bosons: Ψ (1, 2) = ψ (1)ψ (2) =0, but X αα α α 6 ℓmℓsms ℓsjm , (187) × h | i Fermions : Ψαα(1, 2) = 0. (191) where the RHS is a sum over two terms involving These important results are called: Lˆ ℓmℓ and Sˆ sms , respectively. −| i −| i Bose-Einstein condensation: The ground state The topmost state of the next ladder with j = of a system of identical bosons is one where every ℓ + s 1 is the remaining or unconstructed state boson is in the lowest energy single-particle state. with m− = ℓ + s 1. It is constructed by orthonor- Pauli exclusion principle: No two identical malization from−ℓsℓ + sℓ + s 1 , using the prod- fermions can populate the same single-particle uct representation.| By convention,− i its overall phase state. is taken to be the Wigner (aka Condon-Shortley) phases. The other states on the ladder are found Many identical particles: The wave function of by stepping down. This construction method works N identical bosons is symmetric under any per- for all the remaining j ladders. mutation of the N particle labels. For identical fermions, the wave function changes sign under an 7.3 Identical particles in quantum mechanics: odd permutation of the N particle labels, but re- Spin-statistics theorem: Identical particles of mains unchanged under an even permutation. integer (half integer) spins satisfy the Bose-Einstein 7.4 The Periodic Table: (Fermi-Dirac) statistics. They are called bosons (fermions). Electronic structure of the elements: is sum- marized in Table IV: Two identical particles are indistinguishable in their probability densities Helium atom: Its two electrons with spins point- ing and can populate the same 1s spatial state. ↑ ↓ ρ(2, 1) = ρ(1, 2), (188) They complete the principal qu. no. n = 1 shell. The shell n can accomodate 2n2 electrons. but their wave functions may differ by a negative To get the lowest energy, both electrons in the He sign ground state (GS) occupy the lowest s.p. spatial state nℓm = 100. The total wave function bosons Ψ(2, 1) = Ψ(1, 2) for . (189) ± fermions Ψ (1, 2) = ψ (r )ψ (r )χ (1, 2) 0 100 1 100 2 SM=00 = Ψ (2, 1) (192) Boson (Fermion) wave functions are symmetric (an- − 0 tisymmetric) in the particle labels. must be antisymmetric in the two fermion labels 1 Examples: If α, β are single-particle (s.p.) states: and 2. Hence the spin “function” χ (not a “state”) for the total intrinsic spin must be the antisymmet- Ψ (1, 2) [ψ (1)ψ (2) ψ (2)ψ (1)] . (190) ric function with S = 0. αβ ∝ α β ± α β 18 The atomic Hamiltonian The corresponding states nx can be labeled by the qu. no. n . They form| a discretei basis for wave Ze2 Ze2 e2 x Hˆ (1, 2) = G G + G (193) functions in the representative 1D box. − r1 − r2 r12 Three-dimensional space: The normalized 3D gives the GS energy (with Z = 2) wave functions in the representative cube are 3 1 ik r 2π E0(He) Z ER + E12 109eV+30eV. (194) r n = e · , where k = n , (198) ≈− ≈− h | i L3/2 i L i where ER = 13.6 eV is the Rydberg energy. and n = n1n2n3. Closed subshells: The ms value of the spin Fermi gas: For identical fermions, each spatial state sm changes when the quantization axiz e | si z state can be populated at most singly (the spin changes direction. However, if all ms states are degeneracy being treated separately). Then n is filled by 2s + 1 particles, they will remain filled called a (vector) occupation number. A free parti- when the arbitrarily chosen direction ez changes. 2 2 2 4 cle of mass m has energy Ek = (~k) c + m c . Hence the total MS is 0, and this unique state of So the lowest energy state of a free Fermi gas of 2s + 1 particles has S = 0. Similarly, the complete N fermions in the cube is one wherep the states of filling of the 2ℓ + 1 states of different mℓ gives a lowest k or occupation number n = n are popu- unique spherical symmetric spatial system of 2ℓ +1 lated. With spatial isotropy (spherical| symmetry),| particles that has L = ML = 0. This is why the 1 the occupation pattern is that of a sphere in n or ground states of He, Be and Ne are all S0 states. k space of radius k = kF : Spin-orbit interaction: The atomic Hamilto- 3 nian contains a relativistic term ALSLˆ Sˆ, ALS > 3 L 3 · N = dN = d n = 2π d k 0. It gives a negative (attractive) energy of Z Z Z (A /2)(ℓ+1) for the s.p. state with the total an- kF − LS dN L 3 4π 3 gular momentum j = ℓ 1/2. For the j = ℓ +1/2 = dk dk = 2π 3 kF . (199) − Z0 state on the other hand, the energy is repulsive, namely (ALS/2)ℓ. This explains why the B ground Here state in the spatial 2p configuration has the quan- 3 2 dN = L 4πk2 (200) tum numbers P1/2 of the lowest-energy single elec- dk 2π tron state outside the closed n = 1 shell and the density of states closed 2s subshell. is called the in k-space. The radius of the Fermi sphere in k-space The F ground state is one electron short of the closed n = 2 shell. The lowest energy is obtained by N k = (6π2ρ )1/3, where ρ = , (201) putting the missing electron or hole in the higher F # # L3 j = 3/2 state to give a 2P ground state, i.e., 3/2 is called the Fermi momentum. It does not depend leaving a higher state unpopulated by populating on the size of the cube. For electrons with their spin all the lower-energy states in the 2p subshell. degeneracy d(s) = 2, the total electron number is 7.5 Free Fermi gas: Ne =2N. Box normalization: For a particle extending over The energy per particle in a free Fermi gas is all space, the plane-wave wave functions eik r are · dN not square-integrable. This feature causes awk- E dk Ekdk = Ek gas = wardness including the appearance of Dirac δ- N h i dN R dk dk functions. The problem can be circumvented by (3/5)ǫ , if nonrelativistic; subdividing space into identical cubes of sides L = F R (202) (3/4)~ckF , if very relativistic. satisfying a periodic boundary condition (BC). The wave functions in each identical cube can then be The nonrelativistic energy normalized. ~2k2 One-dimensional space: The periodic BC ǫ = F (203) F 2m ikx(x+L) ikxx φkx (x + L)= e = φkx (x)= e (195) at the surface k = kF of the Fermi sphere is called gives the discrete spectrum the (NR) Fermi energy. Chandrasekhar limit: A white dwarf star con- k L =2πn , with n =0, 1, 2, ... (196) x x x ± ± tains Z protons, Z electrons and A Z neutrons. − and the normalized wave functions The main contributions to its energy are the repul- sive kinetic energy of the highly relativistic elec- φ (x)= 1 eikxx. (197) kx √L tron gas and the attractive gravitational potential 19 energy of its mass M. Both energies are inversely The equation is solved by equating terms of the proportional to R, the radius of the star, but they same power of λ on the two sides of the equation: have different M dependences: 0 ˆ (0) (0) (0) (0) λ : H0ψn = En ψn ψn = φn, > 0, unbound, ⇒ M 4/3 M 2 1 (1) E = b c =0, critical, λ : En = Hnn′ , Tot R − R < 0, collapse. H′ ψ(1) = kn φ , n − (0) (0) k (204) k=n Ek En X6 − ˆ The white dwarf will collapse gravitationally if its where Hkn′ = k H′ n . (210) h | | i mass exceeds the critical mass (1) b 3/2 8.2 Calculation of En : M = . (205) c c Many perturbations are made up of sums of fac- torable terms of the type VIII. Approximation methods for time- ˆ ˆ independent problems H′ = Q(r)A (211) Few problems in quantum mechanics are exactly solv- involving a radial factor Q(r) and an angle-spin fac- able. We first consider some approximation methods for tor Aˆ for a system with intrinsic spin S. Then Hamiltonians that are time-independent. E(1) = Q Aˆ , 8.1 Time-independent perturbation theory: n nℓh iα ∞ 2 2 matrices: can be diagonalized exactly. The where Q = R (r) 2 Q(r)r2dr, × nℓ | nℓ | Hermitian matrix Z0 α = ℓmℓsms or ℓsjm. (212) 0 b H = , (206) b a Examples: where both a and b are real, has the eigenvalues and orthonormal eigenvectors Lˆ Sˆ = 1 Jˆ2 Lˆ2 Sˆ2 , h · i 2 h − − iℓsjm 1 ˆ ˆ ~ E = (a ∆E), Lz +2Sz ℓmℓsms = (mℓ +2ms) , 1,2 2 ∓ h i 2 ~m b Sˆ = Jˆ Sˆ . (213) ∆E = a2 +4b2 a +2 ; z ℓsjm ~2 ℓsjm ≈ a h i j(j + 1)h · i pcos θ sin θ ψ = , ψ = , 1 sin θ 2 −cos θ 8.3 Variational method: E b Variational principle: For any wave function ψ: tan θ = 1 . (207) b ≈−a ψ Hˆ ψ Many problems of physical interest involving only Eψ = Hˆ ψ h | | i E0(GS). (214) h i ≡ ψ ψ ≥ two dominant states can be solved approximately h | i by this method. The approximate wave function ψ gives a higher Perturbation theory: Let the Hamiltonian Hˆ = energy when it contains excited-state components Hˆ0 + Hˆ ′ contains a main part Hˆ0 whose energy of higher energies. (0) eigenvalues En , eigenfunctions φn or eigenstates Variational wave function: is any wave func- n are already known. If a complicated but weak tion carrying a number of parameters b that can be | i ˆ perturbation H′ is now added to the system, the varied in order to minimize the energy eigenvalue equation ˆ ˆ ˆ ψ(b) H ψ(b) (H0 + λH′)ψn = Enψn (208) E(b)= h | | i . (215) ψ(b) ψ(b) can be solved by a systematic expansion in powers h | i of λ that can be set back to its numerical value of The energy minimum E(b0) where dE(b)/db =0 is 1 at the end of the expansion. the best variational estimate because it is closest to the true ground-state (GS) energy E0. This λ expansion is done in both En and ψn: He atom: The ee repulsion between the two elec- ˆ ˆ (0) (1) (H0 + λH′) ψn + λψn + ... trons in the He atom can be eliminated approxi- mately by reducing the actual nuclear charge Z =2 (0) (1) (0) (1) = En + λEn + ... ψn + λψn + ... . (209) to the effective charge z 27/16. ≈ 20 Hartree mean-field screening: In an atom with where ψ are the time-independent eigenstates of | ii Z electrons, the mutual ee interactions can be re- Hˆ . Then it evolves at time t into placed by a mean field experienced by each electron iHt/ˆ ~ characterized by an effective nuclear charge of Ψ(t) = e− ψ = d1 Ψ1(t) + d2 Ψ2(t) | i | ~i | i ~ | i = d e iE1t/ ψ + d e iE2t/ ψ . (221) Z, for r 0 1 − 1 2 − 2 z(r)= (216) | i | i 1, for r → → ∞ Unperturbed basis: It is often useful to express Ψ(t) in terms of the time-dependent unperturbed | i 8.4 WKB approximation: basis states Since any complex number can be written in the iE(0) t/~ 1 iφ Φ (t) = e− 1 , amplitude-phase form Ae , one can look for a so- | 1 i 0 lution of the 1D Schr¨odinger wave equation in this iE(0) t/~ 0 amplitude-phase form. The WKB approximation Φ (t) = e− 2 (222) | 2 i 1 is a semi-classical result used between the two clas- sical turning points x1 and x2 of the potential V (x) (0) (0) where the local wave number of energies E1 = 0 and E2 = a: 2m Ψ(t) = c1(t) Φ1(t) + c2(t) Φ2(t) . (223) k(x)= [E V (x)] (217) | i | i | i ~2 − 2 r The probabilities ci(t) can be found by using the energy eigenstates| | is real. It is obtained by ignoring the A′′(x) = d2A(x)/dx2 term. Then cos θ sin θ ψ1 = , ψ2 = − (224) iφ(x) sin θ cos θ ψ(x) = A(x)e C x ′ ′ i x k(x )dx in Eq.(221): ψW KB(x) = e± 1 . (218) ≈ k(x) R 2 i∆ωt 2 c1(t) = d1 cos θ d2e− sin θ , p | | − WKB quantization: At a turning point where 2 i∆ωt 2 c (t) = d sin θ + d e− cos θ , (225) the potential has a finite slope, the wave func- | 2 | 1 2 tion can penetrate into the classically forbidden re- ~ where ∆ω = (E 2 E1)/ . These probabilities are gion where the local wave number becomes purely in general functions− of t. Indeed, their time depen- imaginary k iκ. Under this analytic continua- → dences betray the fact that the φi are not energy tion, the amplitude factor becomes complex, A eigenstates, for otherwise they will be stationary iπ/4 ∝ 1/√k e− /√κ, thereby contributing an addi- states with time-indepedent probabilites. tional→ phase of π/4. The resulting WKB-modified The expansion (223) is particularly interesting Bohr-Sommerfeld− quantization condition is when the basis states φ are themselves physically | ii x2 m observable, as happens in k(x)dx = n π, n =1, 2, ..., (219) − 4 Neutrino oscillations: Neutrinos come in three Zx1 distinct varieties or “flavors”: νe, νµ, ντ , associated where m is the number of turning points where the with the electron e, muon µ and tau τ, respec- potential is not an wall. The resulting WKB ∞ tively. Although they are physically distinct from energies are often quite good. They even agree one another, they are not mass eigenstates (energy with the exact energies in the case of the one- eigenstates at zero momentum). The mass eigen- dimensional harmonic oscillator potentials. states are linear combinations of these neutrino ba- sis states of different flavors. For this reason, the IX. Approximation methods for time- flavor probabilities are not constants of motion, but dependent problems instead they oscillate in time. Time-dependent problems are more complicated partly Solar neutrino oscillations: The neutrinos ini- because one has to keep track of the time evolution of a tially produced in the solar interior where masses state, but mostly because the Hamiltonian itself might are converted to energy are νe = φ1, and not be time-dependent. νµ = φ2, i.e., c2(t = 0) = d1 sin θ + d2 cos θ = 0, or d = cos θ, d = sin θ. Allowing only oscil- 1 2 − 9.1 The time-dependent two-state problem: lation into νµ for simplicity, the probabilities for is exactly solvable for the time-independent Hamil- finding νe and νµ at a later time t are tonian (206). Let a state be prepared initially as c (t) 2 = 1 sin2 2θ sin2(∆ωt/2), | 1 | − ψ Ψ(t = 0) = d ψ + d ψ , (220) c (t) 2 = sin2 2θ sin2(∆ωt/2). (226) | i ≡ | i 1| 1i 2| 2i | 2 | 21 The neutrinos may have very small masses, so that (TD) perturbation Hˆ ′. Unfortunately, the TD ex- pansion coefficients cn(t) must be obtained more m2c3 laboriously by actually solving the TDSchEq E p c + i . (227) i ≈ i 2p i ∂ i~ Ψ(t) = (Hˆ0 + λHˆ ′) Ψ(t) . (234) The the probability for finding νe of energy E at a ∂t| i | i distance L = ct from the Sun is The resulting differential equations (DEs) of mo- 2 2 2 2 4 tion for ck(t), c1(t) = 1 sin 2θ sin [A(∆m c )L/E]; | |2 −2 2 ∆m = m m , ~ iωbkt | 1 − 2| i c˙b(t)= λ Hbk′ e ck(t), (235) 1 2 1 k A = =1.27 GeV(eV)− km− . (228) X 4~c are a set of coupled first-order DEs that can be Experimental data for solar neutrinos of energies (0) (0) ~ solved easily in a PC. Here ωbk = (Eb Ek )/ . broadly distributed around a mean value of E = − Systematic TD perturbation theory: is con- 3 MeV give cerned with the Taylor expansion in λ (eventually 2 4 5 2 2 set to 1): (∆m )c =7.3 10− (eV) , tan θ =0.41. (229) × (0) (1) cb(t)= cb (t)+ λcb + ... (236) 9.2 Time-dependent perturbation theory: ˆ Eq.(235) can then be separated term by term into Expansion in exact eigenstates: Suppose H = (n) 6 separate equations for cb : Hˆ (t), and its energy eigenvalues Ej and eigenstates ψ are known. Then any arbitrary quantum state | j i λ0 terms : i~c˙(0)(t) = 0 c(0)(t) = const, at time t can be expanded in terms of these energy b ⇒ b (1) (0) 1 ~ iωbk t eigenstates: λ terms : i c˙b (t) = Hbk′ e ck , ... (237) k N X iE t/~ Ψ(t) = d Ψ (t) = d e− j ψ . (230) These first-order DEs in time can be integrated di- | i j | j i j | j i j=1 j rectly, making it easy to calculate first-order wave X X functions in TD perturbation theory. Expansion in basis states: This same state can Sinusoidal perturbation: Let the sinusoidal per- also be expanded in terms of a known set of basis turbation iE(0) states Φn(t) = e− n φn that are eigenstates of | i | i (0) Hˆ ′ = Vˆ cos ωt, 0 t t0 only, (238) a known Hamiltonian Hˆ0 of energies En : ≤ ≤ be turned on for a finite period of time. After the iE(0) t/~ Ψ(t) = cn(t) Φn(t) = cn(t)e− n φn (231). perturbation has ceased, a state initially in φ | i | i | i | ai n n will find itself in the state φ , b = a, with the X X b probability | i 6 It happens occasionally that both ψ and φ states | i | i are physically observable. Then it is of interest to 2 V 2 t2 c(1)(t t ) ba j2(z) 0 , (239) express one observational probabilities in terms of b ≥ 0 ≈ 2 0 ~2 the other expansion coefficients. For example, 1 2 where z = 2 (ωba ω)t0, N ± 2 iEj t/~ j0(z) = sin(z)/z, cn(t) = Unj dj e− , (232) | | (0) (0) ~ j=1 ωba = [Eb Ea ]/ . (240) X − Thus most of the excitations are close to the two is a generalization of what is done for neutrino os- (0) (0) states φ with energies E = Ea ~ω. cillations to higher dimensions. Here U, with | bi b ± Golden rule: The transition rate during the in- Unj = φn ψj , (233) teraction lasting a time t from a state a to a group h | i 0 of states b is thus is an N N unitary matrix if both sets of states are × normalized. U is then made up of the eigenstates 1 ∞ (1) 2 ~ Wba cb ρbd( ωba) ψj stored columnwise. ≡ t0 | | | i Z−∞ 2 TD perturbation: The expansion (231) can also 2π Vba ˆ ˆ ˆ ρb, (241) be used if H = H0 + H′ contains a time-dependent ≈ ~ 2 22 where ρb = dNb/dEb is the density of final states. The new features include the appearance of a ge- geo Example: The Bohr transition of an atomic elec- ometrical phase ϕn and of transitions to other states k = n. tron from a higher atomic state b to a lower state 6 a with the emission of a photon of energy ~ω = Impulsive perturbation: An impulsive pertur- (0) (0) Eb Ea is caused by the electric dipole interac- bation contains a δ-function in time: tion − Hˆ ′(x, t)= Aˆ(x)δ(t). (249) Hˆ ′ = Vˆ cos(ωt). (242) The TDSchEQ can be solved exactly to show that Here Vˆ = ǫ Dˆ E0 is the electric-dipole interaction − · ˆ the δ-function makes the state discontinuous at t = between the electric dipole operator D = qˆr of the 0: atomic electron and the electric field E = ǫE0 of the emitted photon. The latter can be obtained 1 Aˆ − Aˆ from the photon energy density in a cube of side L: Ψ(0+) = 1 ~ 1+ ~ Ψ(0 ) , (250) | i − 2i ! 2i ! | − i ~ω ǫ E2 = 0 0 , (243) L3 2 X. Applications where ǫ0 is the permittivity of free space. When Hˆ ′ is used in the Golden Rule, the resulting transi- In this chapter, we see how quantum problems of great tion rate is the Einstein coefficient for spontaneous physical interest have been solved exactly or approxi- photon emission by the atom: mately. 4 ω3 1 A = W = D 2, (244) 10.1 Solving special matrix eigenvalue problems: ba 3 c3~ 4πǫ | ba| 0 Special matrices of arbitrarily large dimensions of 9.3 Sudden, adiabatic and impulsive perturba- great physical interest can sometimes be solved an- tions: alytically by symmetry considerations. In sudden perturbation, the Hamiltonian To begin, it is easy to verify by inspection the eigen- changes suddenly by a finite amount to the final values and eigenvectors of the Pauli matrix σx: value Hˆa. If the change is so sudden that the state 0 1 1 1 ψ at t = 0 has not changed, the state at times = , | i 1 0 1 1 t> 0 is just 0 1 1 1 iHˆ t/~ Ψ(t) = e− a ψ = . (251) | i | i 1 0 1 − 1 iE(a)t/~ − − = e− n φ φ ψ | nih n| i n Indeed, three simple rules often go a long way in X finding the eigenvectors and eigenvalues of certain = Φ (t) φ ψ . (245) | n ih n| i Hermitian matrices: n X (a) The vector v = (1, 1, ..., 1), where all compo- Adiabatic perturbation: When a time- 1 nents are 1, is an eigenvector of a matrix where dependent Hamiltonian Hˆ (t) changes very slowly the matrix elements of each row is a permu- in time, we expect the instantaneous eigenvalues (t) (t) tation of those of the first row. Its eigenvalue En (t) and eigenstates ψn defined by the eigen- | i E1 is just the sum of the matrix elements of value equation any row. ˆ (t) (t) (t) H(t) ψn = En (t) ψn (246) (b) Eigenvectors of a Hermitian matrix can be | i | i made othogonal to one another. to give a good approximation. That is (c) The sum of eigenvalues is equal to the trace of (t) En(t) En (t), (247) the matrix. ≈ a result known as the adiabatic theorem. Example 1: The vector v1 = (1, 1, 1) is an eigen- If one watches the system for a sufficiently long vector of the 3 3 matrix ((0,1,1),(1,0,1),(1,1,0)), × time, however, sooner or later additional features with eigenvalue E1 = 2. The three vectors v2 = will appear. They can be described conveniently (2, 1, 1), v3 = ( 1, 2, 1), v4 = ( 1, 1, 2) are (t) − − − − − − by using the time-dependent basis ψ : all orthogonal to v1, but not linearly independent | k i ′ of one another. Each has the eigenvalue -1. The ′ ′ (i/~) t E(t )(t )dt (t) 0 k orthogonal eigenvectors can be taken to be v1, v2, Ψ(t) = ck(t)e− ψk . (248) | i R | i and v5 = (v3 v4)/3=(0, 1, 1). Xk − − 23 Example 2: The vectors v1, v2 and v5 are also Hence Hˆ ′ has nonzero matrix elements only be- the orthogonal eigenvectors of the Hermitian ma- tween two opposite-parity states i, j with the same trix ((1,1,1),(1,1,1),(1,1,1)), with the eigenvalues 3, magnetic quantum number: mi = mj . 0, and 0, respectively. This is a special case of the Example: In the n = 2 shell of the hydrogen atom, general result that two matrices M and M have 1 2 there are four states: nℓm = 200; 210, 21 1. Hˆ ′ common eigenvectors if connects only the two opposite-parity m =± 0 states. For these two states alone, it takes the form of a M2 = M1 + aI, (252) 2 2 matrix b((01), (1, 0)) whose eigenstates are × 1 where I is the identity matrix. Since the parame- E1,2 = b : ψ1,2 = (φ200 φ210) . (256) ter a can be complex, we see that non-Hermitian ∓ √2 ± matrices can also have orthogonal eigenvectors. Their energies E1,2 thus move away from the undis- turbed energy of the two states with m = 1 by an Example 3: The arbitrarily large N N matrix ± whose matrix elements are all 1’s has× one “collec- amount proportional to b, which is proportional to . [The state ℓm has parity ( 1)ℓ.] tive” eigenvector v1 = (1, 1, ..., 1) with eigenvalue E | i − N. The remaining eigenvalues are all 0. The or- 10.3 Aharonov-Bohm effect: thogonal eigenvectors can be chosen to be v2 = Gauge transformation: In classical electromag- (N 1, 1, 1, ..., 1), v3 = (0,N 2, 1, ..., 1), − − − − − − − netism (EM) the electric and magnetic fields are ..., vN = (0, 0, ..., 0, 1, 1). − uniquely defined, but the EM potential (the scalar BCS theory of superconductivity: can be un- potential ϕ and vector potential A) are not unique. derstood conceptually by using a large dimensional They can change by a gauge transformation N N matrix Hamiltonian with the same small × negative matrix element ǫ everywhere. All eigen- A A′ = A + ∇Λ, − → states have energy 0, except the collective eigen- ∂Λ ϕ ϕ′ = ϕ , (257) state v1 = (1, 1, ..., 1) that has energy ∆ = ǫN. → − ∂t The energy gap ∆ that separates the ground− state from the excited states makes it hard for the sys- where Λ is any scalar field, without changing the tem to be excited. If the ground state contains EM fields current-carrying electrons, then at sufficiently low B = ∇ A, temperatures (below a certain critical temperature × ∂A Tc), these electrons cannot lose energy by inelastic E = ∇ϕ . (258) collisions with the crystal lattice in the conductor. − − ∂t When this happens, both dissipation and resistiv- Relativistic electrodynamics: The basic dy- ity vanish, and the medium becomes superconduct- namical variable in classical mechanics is the 4- ing. (BCS = Bardeen, Cooper and Schrieffer who momentum (p,iE/c). To add EM, one needs an discovered this fundamental theory of superconduc- EM 4-vector. There is a unique qualifying candi- tivity.) date: the EM 4-potential (A,iϕ/c). Hence electro- 10.2 Stark effect in an external electric field: dynamics for a particle of mass m and charge q can be built up from the sum (p qA,i(E qϕ)/c). A charge q moved a distance z against a constant This combination transforms− like a 4-vector− un- external electric field E = e has the (dipole) in- E z der Lorentz transformations, thus guaranteeing the teraction energy correct result in different inertial frames. As a re- sult, the EM 4-potential plays a more fundamental E′ = q E.dr = q z. (253) role in quantum mechanics than the EM fields. − − E Z The energy-momentum relation for relativistic elec- The resulting perturbing Hamiltonian is trodynamics is then 2 2 2 2 4 Hˆ ′ = Dˆ , where Dˆ = qˆr (254) (E qϕ) = (p qA) c + m c . (259) −E z − − is called the electric dipole operator of the charge. One can find from this the nonrelativistic kinetic This operator is odd in parity, changing sign when energy TNR: z z. Hence all first-order energies vanish: T E mc2 →− ≡ − (1) 1 2 E = n Hˆ ′ n =0. (255) TNR = qϕ + (p qA) . (260) n h | | i ≈ 2m − Since z = r cos θ is independent of the azimuthal Quantum electrodynamics: We are now in a angle φ, Dˆ z has magnetic quantum number m = 0. position to quantize the mechanical terms in NR 24 electrodynamics, leaving the EM potential unquan- after one complete circuit. Hence its Aharonov- tized for simplicity: Bohm phase can only be an integral multiple of 2π: 1 2 qΦ Hˆ = (pˆ qA) + Vˆ + qϕ. (261) g =2πn = , or Φ = nΦ0. (268) 2m − ~ As a result, the magnetic flux enclosed by c can Aharonov-Bohm phase: The EM interaction re- only exist in integral multiples of a quantized unit sides primarily in the phase (or gauge) of quantum flux Φ0 = h/q. Experimental measurement deter- wave functions. For the special case where ϕ = 0, mines that q = 2e in superconductors, thus show- this important result can be demonstrated readily ing that the super− current is carried by pairs of by showing that the wave function ψ in the pres- electrons, now called Cooper pairs. ence of the vector potential A differs from the wave Gauge interactions: Other fundamental inter- function ψ′ without A by an A-dependent phase g: actions also work through the phase of quantum ig ψ(r)= e ψ′(r). (262) wave functions. Hence they are called gauge inter- actions. Indeed, the information contained in the If true, the phase factor eig serves the purpose of phase of a wave function is usually more important removing the A dependence from the Schr¨odinger than that contained in its amplitude. A picture of equation when moved to the left of a “gauged” fac- Snow White can be made to appear in our eye as tor: a coherent wave. One can take the phase informa- tion from Snow White’s wave function and use it ~ ~ ig ig with the amplitude part of the wave function from ∇ qA e ψ′(r)= e ∇ ψ′(r). (263) i − i a picture of Happy, one of the seven dwarfs. A composite wave like this has been constructed the- Since oretically. Believe it or not, one sees Snow White in the resulting hybrid picture, and not the dwarf. ~ ~ ig ig ∇ e ψ′(r)= e ~(∇g)+ ∇ ψ′(r), (264) i i 10.4 Magnetic resonance: Rabi measured the magnetic moments of atomic g has to satisfy the differential equation states by finding the Zeeman splitting of the en- ergies of different magnetic substates by resonance ~∇g = qA, or ~dg = qA dr. (265) · matching. Atomic magnetic moments are tradition- ally expressed as gµ in units of the Bohr magneton Hence B µB = e~/2m, the theoretically value g being called q r the Land`e g factor. The Hamiltonian involved is g = ~ A(r′) dr′, (266) r · ˆ ˆ ˆ Z 0 H = H0 + H′, (269) where the line integral is in general path-dependent. where Hˆ0 = Hˆ1 + HˆLS + Hˆ2 contains the usual The integral is then said to be non-integrable. This NR Hamiltonian Hˆ1, the relativistic spin-orbit term non-integrability can be made explicit by integrat- HˆLS, and the Zeeman Hamiltonian ing around a closed circuit c once and simplifying the result with the help of Stokes’s theorem: Hˆ = Mˆ B 2 − · z gµBBz q = Jˆz = ω0Jˆz (270) g = A(r′) dr′ ~ ~ · Ic q q that gives the splitting between the magnetic sub- = (∇ A) dS = Φ, (267) ~ × · ~ states of the same j. Finally a small oscillatory ZS magnetic field is applied in the x-direction to give where the total magnetic flux Φ across the surface a perturbing Hamiltonian S enclosed by the circuit c depends on the magnetic ˆ ˆ ~ field ∇ A on and inside c. Such a dependence on H′ = ωxJx cos(ωt), where ωx = gµBBx. (271) × properties of the system outside the path c shows This perturbation causes transitions between the why quantum mechanics in particular, and waves magnetic substates. Any quantum state more generally, describe nonlocal phenomena that are spread out in space. Ψ(t) = c (t) Φ (t) | i m | m i Magnetic flux quantization: Suppose the path m X iω t c is located in a region free of any EM field, then = c (t)e− m φ , (272) m | mi the wave function must return to its original value m X 25 (0) where ωm = Em /~, can be expressed in term of the is the differential solid angle in a suitable inertial unperturbed states Φ (t) . The time-dependent frame. | m i expansion coefficient cm(t) then satisfies the TD- Example: Two beams of the same cross sectional SchEq area A collide head-on where their paths cross each other. Each beam is made up of bunches of Ni(i = ~ iωmkt i c˙m(t)= Hmk′ e ck(t), (273) 1, 2) particles per bunch. The target can be taken k X to be one bunch of N2 particles in beam 2. Then the effective incident flux is I = fN1/A, where f is where ωmk = ωm ωk. − the frequency of collision between bunches. Example: The problem can be solved approxi- Rutherford cross section: For the scattering of mately for cm(t) for the atomic n[ℓ]j = 2p1/2 con- figuration that has only two magnetic substates φ α-particles of charge Z1 = 2 from an atomic nu- (0) ± cleus of charge Z2, Rutherford found from classical of energies E = ~ω0/2. For example, if the sys- ± ± mechanics that in the CM frame tem is initially in the upper state (c+ =1,c = 0), − then the probability of finding the system in the dσ d 2 1 upper and lower states at a later time t are respec- = , dΩ 4 sin4(θ/2) tively: 2 Z1Z2eG 2 2 2 2 where d = , (277) P++ = c+ = cos (ωRt/2) + sin χ sin (ωRt/2), TCM | |2 2 2 P + = c = cos χ sin (ωRt/2), where (274) − | −| is the distance of closest approach (or turning point) for head-on collision at the NR kinetic en- ω = (∆ω)2 + (ω /2)2, ∆ω = ω ω, ergy T in the CM frame. Note that for backscat- R x 0 − CM ∆ω ωx/2 tering θ = π, the Rutherford differential cross sec- sin χ = p , cos χ = . (275) 2 ω ω tion d /16 gives a direct measurement of the posi- R R tion d of the classical turning point. Magnetic resonance: As the applied frequency Total cross section: is proportional to the total ω passes the Zeeman frequency ω0, the minimum reaction rate: population of the φ+ state falls sharply to 0 at ω0 dσ d and then rises sharply back up to 1 again, thus σ = dσ = dΩ= R = R. (278) ≈ dΩ L L giving a clear signal for the measurement of ω0. Z Z Z The third frequency ωx controls the period τ of oscillation of the population between the two mag- 11.2 Quantum theory of scattering in a nutshell: netic substates. At full resonance (ω = ω0), one finds τ =4π/ωx. Partial-wave expansion: of a plane wave into spherical waves of good angular momentum ℓ around the origin of coordinates has the Rayleigh XI. Scattering theory form: Information about the dynamical properties of micro- ∞ ik r ℓ scopic systems like atoms, nuclei and particles can be ψk(r) = e · = i (2ℓ + 1)Pℓ(cos θ)jℓ(kr), obtained by scattering beams of projectiles from targets ℓ=0 containing them, as first demonstrated by Rutherford, X ∞ i Geiger and Marsden who elucidated atomic structure by ℓ →∞ i (2ℓ + 1)Pℓ(cos θ) scattering α particles from a gold foil. r ∼ 2k Xℓ=0 e i(kr ℓπ/2) ei(kr ℓπ/2) 11.1 The scattering cross sections: − − − (279) × r − r The differential cross section This is the time-independent wave function of a sta- dσ 1 d = R (276) tionary scattering state. The time-dependent wave dΩ L dΩ iωt function carries an additional time factor e− , 2 is the angular distribution d /dΩ of the reaction where ω = ~k /2m. The additional time factor ikr rate per unit luminosity L.RL itself is the product shows that the wave function e− /r describes an R ingoing INT , where I is the flux or current density of the spherical wave collapsing towards the ori- ikr incident beam (number of incoming particles per gin, while the wave function e /r describes an outgoing spherical wave expanding out from the ori- second per cross sectional area of the beam, or N˙ i/a gin. for a beam of uniform cross section a) and NT is the number of target particles illuminated by this Scattered wave: When a target particle is placed beam. L depends on both beam and target. dΩ at the origin of coordinates, the radial part jℓ of the 26 wave function (279) will be replaced by the func- Total cross section: is the result integrated over tion Rℓ(r) from the SchEq with a potential in it. At the solid angle dΩ: large distances r, the ingoing waves are still collaps- ing towards the origin, and therefore do not know if dσ σtot = dΩ= f ∗(θ)f(θ)dΩ a potential is present there. In elastic scatterings, dΩ Z Z an outgoing spherical wave has the same normal- = σtot,ℓ , (286) ization as before, and can be changed at most by a ℓ phase, here taken to be 2δℓ, that resides in a phase X 2iδ factor Sℓ = e ℓ called an S-matrix element: where the two sums over ℓ, one from each f, has been simplified to only one sum by using the or- ∞ ℓ i thogonality relation for Legendre polynomilas ψ(r) i (2ℓ + 1)Pℓ(cos θ) r→∞∼ 2k ℓ=0 1 4π X ′ ′ i(kr ℓπ/2) i(kr ℓπ/2) Pℓ(cos θ)Pℓ (cos θ)dΩ= δℓℓ , and (287) e− − e − 1 2ℓ +1 Sℓ Z− × r − r 4π 2 σtot,ℓ = 2 (2ℓ + 1) sin δℓ. (288) = ψk(r)+ ψsc(r) where (280) k Optical theorem: If δℓ is real, one finds that ∞ ℓ 2 ψsc(r) →∞ i (2ℓ + 1)Pℓ(cos θ) sin δℓ r ∼ Imf(θ =0) = (2ℓ + 1)Imfℓ = (2ℓ + 1) ℓ=0 k X ℓ ℓ i(kr ℓπ/2) S 1 e − X X ℓ − k × 2ik r = σtot. (289) 4π eikr = f(θ) . (281) This theorem holds even when the phase shift δℓ = r αℓ + iβℓ becomes complex. is the scattered wave function, the additional wave function generated by the potential. Containing only outgoing spherical waves, it can be written 11.3 Breit-Wigner resonance: asymptotically (r ) in terms of an angle- The partial-wave total cross section σℓ = → ∞ 2 dependent quantity called the scattering amplitude σℓ, max sin θℓ reaches a maximum value of σℓ, max = 2 (4π/k )(2ℓ + 1) whenever δℓ(k)= π/2. This maxi- ∞ mum value is called the unitarity limit. f(θ) = (2ℓ + 1)fℓPℓ(cos θ), ℓ=0 If δℓ(E) rises through δℓ(E = ER)= π/2 sifficiently X rapidly, a Taylor expansion about E = ER needs Sℓ 1 1 where f = − = eiδℓ sin δ (282) to be taken only to the terms ℓ 2ik k ℓ π dδ if the scattering is elastic, meaning that the phase δ + (E E ) ℓ . (290) ℓ ≈ 2 − R dE shift δℓ is real. ER Γ/2 Flux: or probability current density is Then tan δℓ , where ≈ −E E − R ~ Γ 1 j = (ψ∗∇ψ ψ∇ψ∗) jk + jsc, (283) = > 0. (291) 2im − ≈ 2 (dδℓ/dE)ER ~ where jk = k/m is the flux of the plane wave, and Resonance: In the neighborhood of a resonance, the partial-wave total cross section has the simple ~k f(θ) 2 j = er | | (284) E-dependence: sc m r2 2 tan δℓ is the flux of the scattered wave at large distances. σℓ = σℓ, max 2 1 + tan δℓ Differential cross section: A detector of area (Γ/2)2 2 dA = r dΩ then measures the differential cross sec- = σℓ, max . (292) (E E )2 + (Γ/2)2 tion − R This function has a sharp maximum at E = E and dσ 1 d 1 dA R falls rapidly to half its maximal value at E E = = R = ~ (jsc er) R dΩ Lk dΩ k/m · dΩ Γ/2. Hence Γ is called the resonance width− (or = f(θ) 2. (285) ±full width at half maximum, FWHM). | | 27 Decaying state: The partial-wave scattering am- The photon in question cannot propagate freely. It plitude near resonance has the simple form can only exist momentarily around its point source (the point charge), and is then said to be a virtual 1 1 1 f = eiδℓ sin δ = photon. In Feynman’s diagrammatic language, the ℓ k ℓ k cot δ i ℓ − Coulomb interaction arises from the emission of a 1 Γ/2 virtual photon by a source (the q here) and its ab- = − . (293) k (E ER)+ iΓ/2 sorption by a test charge at distance r from it, or − vice versa. At the complex energy E = E iΓ/2, the time- R The virtual particle in Yukawa’s theory of inter- dependent probability density decays− exponentially actions is a boson, otherwise it cannot be emitted 2 i(E iΓ/2)t/~ 2 Ψ(r,t) = ψ(r)e− R− because of the conservation of fermion number. A | | | | 2 Γt/~ virtual boson that is massive satisfies the Yukawa = ψ e− . (294) | | equation (∇2 µ2)φ(r)= 4πgδ(r), (299) − − 11.4 Yukawa’s theory of interactions: where the unit of charge is now denoted g, and µ = ~ Yukawa showed in 1935 that both the Coulomb mc/ is the inverse reduced Compton wavelength. The solution of this DE that vanishes at r = can interaction mediated by the exchange of massless ∞ photons between charges, and the strong or nuclear be shown to be the Yukawa potential interaction mediated by the exchange of massive g µr φ(r)= e− . (300) bosons called mesons, can be derived from quan- r tum mechanics. Nuclear forces: were known in 1937 to have a Klein-Gordon equation: The Einstein energy- 15 2 2 2 2 4 range of 1/µ 1.4 fm. (fm = 10− m.) This momentum relation E = p c +m c can be quan- fact allowed Yukawa≈ to predict that these forces tized into the Klein-Gordon equation in free space: are caused by the exchange of bosons of mass ~ 2 Hˆ 2Φ(r,t)= pˆ2c2 + m2c4 Φ(r,t), (295) m = µ /c 140 MeV/c . Such strongly inter- acting bosons,≈ now called π mesons or pions, were where Hˆ = i~∂/∂t and pˆ = (~/i)∇. The result- discovered in 1947. iEt/~ ing wave functions Φ oscillates in time as e− . These free-space solutions are said to be on the energy shell, because the E-p relation is satisfied 11.5 Scattering at low energies: on a spherical shell in p-space. Impact parameter: In classical mechanics, a par- Static solutions of the KG equation: The ticle impacting at a transverse distance b (the im- KG equation does not have time-independent so- pact parameter) from the center of a square-well lutions in free space, because these solutions have potential of range R will not “see” the potential if E = 0 and therefore cannot satisfy Einstein’s b > R. If the incident momentum in the center- E-p relation. Yukawa showed that there are of-mass frame is p, the interaction vanishes when the angular momentum ℓ = r p exceeds the value static (hence energy-nonconserving) solutions near ~ × a point charge q located at the origin that satisfy ℓmax = Max(kR/ ), where Max(z) is the largest the inhomogenious DE: integer in z. The wave spreading in quantum mechanics makes 2 2 2 4 2 pˆ c + m c φ(r)=4π(~c) qδ(r), (296) the connection between ℓ and b less sharply defined, but it remains true that the scattering phase shift For m = 0, the Yukawa equation simplifies to the δℓ vanishes sharply when ℓ>ℓmax. Consequently, Poisson equation ℓmax 2 ∇ φ(r)= 4πqδ(r). (297) f(θ) (2ℓ + 1)fℓPℓ(cos θ), (301) ≈ − ℓ=0 One can show with the help of Gauss’s theorem X (or Gauss’s law in electrostatics) that the resulting Hence at sufficiently low energies, the S-wave (ℓ = static wave function of a massless photon around a 0) term dominates: point charge that vanishes at r = is sin2 δ 4π ∞ σ 4π 0 = . (302) q tot k2 2 2 2 φ(r)= . (298) ≈ k + k cot δ0 r The fact that the leading Taylor term for ℓ =0 is This is just the Coulomb potential around the point independent of k comes from the properties of the charge q (in Gaussian units, or q = qSI /√4πǫ0). scattering wave function. 28 Effective-range expansion: 11.7 Born approximation for two-particle scatter- ings: 1 1 1 2 k cot δ0 = = + r0k + ..., (303) The Born (or first Born) approximation for the −a(k) −a 2 0 scattering of two particles is a first-order TDPT where a0 is called the scattering length and r0 is that can be obtained from the called the effective range. Golden rule: A weak two-body interaction iωt iωt Hˆ ′ = Veˆ − + Vˆ †e (311) 11.6 Phase shifts for finite-range potentials: causes the scattering from an initial two-particle Radial wave function: The wave function in the state i to a final two-particle state f within the solid presence of a potential V is 2 angle d Ωf around the final relative momentum kf . The transition rate is given, in first-order TDPT, ∞ u (r) ψ(r)= iℓ(2ℓ + 1)P (cos θ) ℓ , (304) by the Golden Rule: ℓ kr ℓ=0 X dw 2π ρ (E) fi = f Vˆ i 2 f where the radial wave function uℓ satisfies the ra- dΩ ~ |h | | i| 4π dial wave equation µ (~k ) = f f L3 f Vˆ i 2 . (312) 2~4 d2u 2µ ℓ(ℓ + 1) (2π) |h | | i| ℓ + (E V )u u =0. (305) 2 ~2 ℓ 2 ℓ dr − − r Here µj , kj are the reduced mass and relative mo- where µ is the reduced mass in the center-of-mass mentum, respectively, in the state j (= i,f) of the system. two-body system in a cube of side L, and the den- sity of final state factor is Attractive S-wave square-well potential: For ℓ = 0 in an attractive square-well potential of depth 3 2 ρf (E) L k dk V , the radial wave equation is = . (313) 0 4π 2π dE 2 d uin 2 For r < R : + κ u =0, The scattering is elastic if kf = ki, and inelastic if dr2 in kf , ki. u = sin(κr); (306) in The resulting differential scattering cross sec- tion 2 d uout 2 For r > R : + k u =0, dσfi 1 dwfi dr2 out (314) dΩ ≡ Jinc dΩ uout = sin(kr + δ0); (307) is just the transition rate per unit incident flux (or 2mE 2mV current density), which is k2 = , κ2 = 0 , κ2 = k2 + κ2. (308) ~2 0 ~2 0 v ~k J = i = i (315) inc L3 µ L3 The phase shift δ0 is determined by matching the i inside and outside logarithmic derivatives in = L in a cube of side L. Note that Jinc has the expected out at r = R, where = (du/dr)/u, to give 2 1 L L dimension of m− s− . The final result for the Born 1 1 approximation can be written compactly as tan κR = tan(kR + δ0), or κ k dσ µ µ k fi = i f f V˜ (q) 2, (316) 1 k 2~4 δ = tan− tan κR kR. (309) dΩf (2π) ki | | 0 κ − where q = kf ki, and Levinson’s theorem: If there are m bound states − inside the potential, 3 iq r 3 V˜ (q) L f Vˆ i = e− · V (r)d r (317) ≡ h | | i Z δℓ(k =0)= mπ, (310) is obtained by using the plane-wave wave functions except for an S-wave bound state at E = 0, which in the relative coordinate r normalized to one par- contributes only π/2, half of the normal contribu- ticle in a cube of side L: tion. 1 ik r 1 ik r r i = e i· , f r = e− f · . (318) h | i L3/2 h | i L3/2 29 Thus the cross section in the Born approximation can be calculated directly from the Fourier trans- form V˜ (q) of the interaction potential V (r). Examples of V˜ (q) are those from the δ-shell and the Yukawa potentials: 4πR V (r) = A δ(r R) V˜ (q)= A sin(qR); 0 − ⇒ 0 q e αr 4π V (r) − V˜ (q) . (319) ∝ r ⇒ ∝ α2 + q2