School of Mathematics and Natural Sciences Physics Department Bergische Universitat¨ Wuppertal
A course on
Introductory Quantum Theory
t
x
by
Reinhard Hentschke
Contents
1 Early developments in quantum mechanics1 1.1 Quantization conditions...... 1 1.2 The black body problem...... 4 1.3 Building a general formalism...... 7 1.4 The hydrogen atom...... 18 1.5 Angular momentum algebra...... 27
2 Formal quantum mechanics 33 2.1 Operator averages...... 36 2.2 The uncertainty principle...... 38 2.3 From wave mechanics to matrix mechanics...... 40 2.4 The time evolution operator...... 41 2.5 The density operator...... 43 2.6 Path integration and the density operator...... 45
3 Scattering theory 51 3.1 The Lippmann-Schwinger equation...... 51 3.2 The 1st Born approximation...... 54 3.3 Partial wave expansion...... 55 3.4 Some remarks on formal scattering theory...... 56
4 Solution methods in quantum mechanics 59 4.1 Time-independent perturbations...... 59 4.2 Variation method...... 64 4.3 The quasi classical approximation...... 66 4.4 Time-dependent perturbation theory...... 67 4.5 Interaction of charges with electromagnetic fields...... 75
5 Many particle systems 77 5.1 Two kinds of statistics...... 77 5.2 Constructing N-particle wave functions...... 78 5.3 Electron spin...... 81 5.4 Simplified theory for helium...... 86 5.5 Simplified theory for the hydrogen molecule...... 88 5.6 Quantum mechanics in chemistry...... 90
iii 6 The quantized radiation field and second quantization 101 6.1 Quantum theory of radiation...... 101 6.2 Second quantization...... 104 6.3 Interacting quantum fields:...... 106 6.4 Outlook...... 106
A Constants and units 109
B Mathematical appendix 111 B.1 Conversion to generalized coordinates...... 111 B.2 Spherical harmonics...... 112 B.3 δ-function...... 113 B.4 Fourier series and Fourier transforms...... 114
C The Casimir effect 115
D Problems 117 Preface mal basis for quantum mechanics treating bound state problems; chapter three deals with scatter- I recall a physics student expressing his opinion ing; chapter four introduces perturbation theory to- that ”real physics” begins with quantum mechan- gether with selected numerical techniques; in chap- ics. This opinion, to which I cannot agree, proba- ter five we turn to many particle systems and ex- bly is due to the ”strangeness” in quantum theory, ploit the fact that in nature there are two types which is summarized in an often cited quote by the of particles-fermions and bosons; the final chap- late Richard Feynman 1 expressing his believe that ter, chapter six, presents a short glimpse at the ”nobody really understands quantum mechanics”. quantized radiation field and second quantization. These lecture notes cover introductory quantum Chapters five and six mark the departure of physics theory to an extend that can be presented in a from chemistry as far as university education is con- one semester course. The subject is approached by cerned. Atomic or molecular systems with more looking first at some of the pressing questions by than two electrons are usually reserved for students the end of the 19th century, when classical physics, of quantum chemistry. Solids as well as advanced in the eyes of many, had come close to explaining quantum theory, which leads to the physics of el- all known physical phenomena. We will focus on ementary particles, appear to be the domain of a special question (e.g. the black body problem), physics students. Here, in a limited sense of course, then introduce an idea or concept to answer this I attempt to satisfy both tastes. Because of this the question in simple terms (e.g. energy quantization), lecture does not follow one particular text. How- relate the quantum theoretical answer to classical ever, in preparing these notes I have been guided theory or experiment, and finally progress deeper by the references [5,6,7,8]. Additional textbooks into the mathematical formalism if it provides a on quantum mechanics containing most of the ma- general basis for answering the next question. In terial discussed here include [9, 10, 11, 12, 13]. this spirit we develop quantum theory by adding in The initial version of these notes was prepared a step by step procedure postulates and abstract for a course on introductory quantum mechanics concepts, testing the theory as we go along, i.e. we during the fall/winter term 2002/2003. Subse- will accept abstract and maybe sometimes counter- quently these notes were used and extended during intuitive concepts as long as they lead to verifiable the summers of 2004, 2006, 2007, as well as during predictions. the fall/winter terms 2008/2009 and 2015/16. I Chapter one is devoted to the early developments owe thanks to many students during these years in quantum theory 2; chapter two provides a for- for their valuable comments.
1 Feynman, Richard Phillips, american physicist, *New Wuppertal, February 2016 (revised version of York 11.5.1918, †Los Angeles 15.2.1988; he made many sem- inal contributions to theoretical physics including the theory January 18, 2021) of superfluid helium and, in particular, to the development of quantum electrodynamics. He introduced the Feynman Reinhard Hentschke diagram technique and the concept of path integration. He not only was a brilliant scientist but also famous for his way of teaching physics. In 1965 he was awarded the Nobel prize Bergische Universit¨at in physics together with Julian S. Schwinger and Sin-Itiro School of Mathematics and Natural Sciences Tomonaga. Gauss-Str. 20 2The student should be aware that these lecture notes, for the most part, cover only a few years in the development 42097 Wuppertal of quantum theory, i.e. ”early” is appropriate in a limited e-mail: [email protected] sense only. In 1900 Planck presented his treatment of black http://constanze.materials.uni-wuppertal.de body radiation; Einstein’s interpretation of the photoelec- tric effect appeared in 1905/06; Bohr’s seminal work ”On the constitution of atoms and molecules” was published in 1913. But after 1925/26, when a formal framework (matrix lecture notes, as almost all texts aimed at beginners, do mechanics/wave mechanics) had been suggested to do cal- (partially!) cover this exciting period only. A good histor- culations, the development became extremely rapid! Maybe ical source on important steps in physics in the 19th and not surprising it took only about five years and a number of early 20th century leading up to the developments discussed textbooks on quantum mechanics had appeared (e.g, Weyl, here is A. Sommerfeld’s ”Atombau und Spektrallinien” [4], 1930 [1]; Dirac, 1930 [2]; von Neumann, 1932 [3]). These which first appeared in 1919!
Chapter 1
Early developments in quantum mechanics
1.1 Quantization conditions h = 6.6256 ± 0.0005 · 10−34Js . The year 1900 marks the beginning of what we call ”Quantum Theory”. Until then various physicists Discrete entities were not new in science at that of reputation had failed to derive a valid formula time. Chemists had long accepted the notion that describing the frequency dependence of radiation matter consists of atoms. This idea was formulated intensity emitted from a black body. A black body by Dalton 3 in his book A new system of chemical can be a metal cavity with walls kept at a fixed philosophy, which appeared in two volumes in 1808 temperature T . A small hole drilled into the walls and 1810. Also, Thomson 4 in 1897 had discovered of this oven will allow radiation to escape - the the electron. However, energy was firmly believed black body radiation. This may sound academic. to be continuous. E.g., we all are used to chang- But what makes this problem especially interest- ing the speed of our car and thus its kinetic energy ing is, that all prior calculations were based on continuously. However, like matter, which macro- apparently well founded theories like Maxwell’s 1 scopically appears continuous but microscopically equations and classical statistical mechanics. These must be ’chemically’ discrete in order to explain pillars of physics, however, as we shall see below, chemical reactions and their mass ratios, energy produced nonsense. In order to obtain a correct also reveals its discrete nature on the microscopic formula describing black body radiation the physi- scale. That classical physics had worked to every- cist Max Planck 2 had to introduce a revolutionary body’s satisfaction had to do with the experimental concept. The energy of the radiation field inside inability of that time to probe the microscopic (or the black body is not continuous. It is the sum of atomic) scale. However, in addition to the black discrete amounts of energy of size hν or ¯hω . Here body problem, where a microscopic phenomenon ν is the radiation frequency (ω = 2πν), and h is the could be measured macroscopically, scientists al- now famous Planck’s constant (dt.: Plancksches ready knew of other troubling facts. Chemists had Wirkungsquantum) come up with the periodic table. But how chemi- cal reactions really work or the nature of chemical bounds were beyond comprehension. Spectral lines 1Maxwell, James Clerk, british physicist, *Edinburgh of simple atoms, e.g., hydrogen (Balmer series 5), 13.6.1831, †Cambridge 5.11.1879; he is best known for his theory of electromagnetism, i.e. the Maxwell equations. 3Dalton, John, british scientist, *Eaglesfield (County However, he also made important contributions in other ar- Cumbria) 5. or 6.9.1766, †Manchester 27.7.1844 eas in physics, e.g. kinetic gas theory, and mathematics. 4Thomson, Sir (since 1908) Joseph John, british 2Planck, Max, german physicist, *Kiel 23.4.1858, physicist, *Cheetham Hill (now Manchester) 18.12.1856, †G¨ottingen4.10.1947; his solution to the black body problem †Cambridge 30.8.1940; Nobel prize in physics 1906. laid the foundation to the development of quantum theory. 5Balmer, Johann Jakob, swiss mathematician, *Lausen For this work he received the Nobel prize in physics in 1918. (Kanton Basel-Landschaft) 1.5.1825, †Basel 12.3.1898
1 2 CHAPTER 1. EARLY DEVELOPMENTS IN QUANTUM MECHANICS had been measured but could not be explained. Be- sides Planck other scientist were productive too. J. 1 2 J. Thomson, who had discovered the electron, sug- mev + φ0 =hω ¯ (1.1) gested a new model for the atom in 1904. Accord- 2 ing to him atoms are small spheres of certain mass with φ0 =hω ¯ 0 (= hν0). The constant φ0 is called containing a uniformly distributed positive charge the work function 9. as well as a corresponding number of electrons em- Nevertheless, Bohr’s postulates were merely pos- bedded into this positive background. An improved tulates. To justify postulates, which were in dis- model was suggested by E. Rutherford 6 in 1911 agreement with Maxwell’s beautiful theory of elec- based on his experiments in which he bombarded tromagnetism, Bohr needed to make verifiable pre- thin metal foils with α-particles. His model was es- dictions based on his model. He needed a quanti- sentially a planetary system with a tiny positive nu- zation condition. In 1924 Louis deBroglie 10 sug- cleus at the center circled by electrons. This imme- gested the relation diately caused a problem. According to Maxwell’s theory accelerated charges emit radiation. There- h fore Rutherford’s atoms should lose energy and col- p = (1.2) lapse. Similar to Planck in the case of the black λ 7 body Niels Bohr postulated a solution to this connecting the momentum of a particle (e.g., an problem. In his model of the atom the electrons are electron) to a wavelength λ via Planck’s constant. confined to discrete orbits E1, E2, .... They may DeBroglie’s idea that an electron may behave as however absorb one of Planck’s radiation packages, a wave was proven experimentally by C. Davis- if it supplies the proper amount of energy separat- son and L. Germer in 1927, who observed electron ing two orbits, e.g.hω ¯ = En+1 − En. This package diffraction from a nickel crystal. This introduced also may be emitted, and the electron returns to its the concept of particle-wave-dualism. DeBroglie’s previous orbit. However, there always exists an or- relation immediately allows to write down an ex- bit whose energy is lowest, and therefore the atom pression for the orbital energy of the electron in cannot collapse. These postulates were in accord 8 the hydrogen atom, which for instance explains the with Einstein’s earlier explanation of the photo- Balmer series 11. electric effect. A metal surface exposed to electro- From the Kepler problem in classical mechanics magnetic radiation (e.g., visible light) emits elec- 1 ¯ ¯ ¯ ¯ we borrow the relation − 2 U = K, where U and K trons if the radiation exceeds a threshold frequency are the average potential and kinetic energy respec- ν0. Below ν0 no electrons are emitted, independent tively 12. For a simple circular orbit of radius a we of the radiation’s intensity! The kinetic energy of 1 2 therefore have the emitted electrons, 2 mev , is described by 9The photoelectric effect can be used to measure ¯h. 10Broglie, Louis-Victor, french physicist, *Dieppe 6Rutherford, Ernest, Lord Rutherford of Nelson (since 15.8.1892, †Louveciennes (D’epartement Yvelines) 1931), british physicist, *Brightwater (New Zealand) 19.3.1987; 1929 Nobel prize in physics. 11 30.8.1871, †Cambridge 19.10.1937; one of the most outstand- This sounds as if Bohr had to wait until deBroglie had ing experimentalists in the 20. century, concentrating on nu- put forward his idea. But this is not true. We merely choose clear physics. He was awarded the Nobel prize in chemistry this path because relation (1.2) allows to interpret Bohr’s in 1908. quantization conditions via the wave-picture of particles. 12 7Bohr, Niels Henrik David, danish physicist, *Kopen- This is easy to show in the special case of a circular 2 hagen 7.10.1885, †Kopenhagen 18.11.1962; he was the lead- orbit. The constant kinetic energy is K = (µ/2)(rϕ˙) , where ing (father) figure in the early development of quantum µ is the (reduced) mass, r is the orbit’s radius, andϕ ˙ is physics. His model of the atom paved the way to our un- the constant angular velocity. The latter may be related derstanding of the nano-world. In 1922 he was awarded the to the constant potential energy, U = −α/r (with α > 0), ¨ Nobel prize in physics. via Newton’s second law, µ~r = −∇U~ = −~er · ∂U/∂r = 2 8Einstein, Albert, german/american physicist, *Ulm −~erα/r . Using polar coordinates, i.e. ~r = r(cos ϕ, sin ϕ) = 2 14.3.1879, †Princeton (New Jersey) 18.4.1955; perhaps the r~er we obtain ~r˙ = rϕ˙(− sin ϕ, cos ϕ) and ~r¨ = −rϕ˙ ~er. We one most outstanding scientists of all times. His work had find the relation (rϕ˙)2 = α/(µr) and therefore K = −U/2. profound influence on all areas in physics and science in gen- This relation, as we have mentioned, holds for general Kepler eral, including cosmology, quantum physics, and many-body orbits if K and U are replaced by their averages over one theory. He was awarded the Nobel prize in physics in 1921. complete orbit (cf. [14] exercise 24). 1.1. QUANTIZATION CONDITIONS 3
But Bohr’s theory must pass another test. When we increase the orbit of the electron to macro- p2 1 e2 = , (1.3) scopic dimensions in what we call a Gedankenex- 2µ 2 a periment it should obey Maxwell’s equations and thus emit radiation according to this classical the- where µ = m m / (m + m ) is the reduced mass, e n e n ory. According to classical mechanics we have and −e is the electron charge 13. Requiring that the | L~ |≡ L = µa2φ˙ = µa2ω . Here L~ is the an- circumference of the orbit corresponds to an inte- cl gular momentum vector, and φ is the angle of the ger number of deBroglie wavelengths, a standing position vector ~r (| ~r |≡ a) with a fixed axis in the electron wave, we find the quantization condition two-body problem. Condition (1.4) immediately implies 2πa = nλ (n = 1, 2,...) . (1.4) L = n¯h . (1.8) Putting (1.2), (1.3) and (1.4) together we obtain Combining this with the above equation as well as (1.5) we obtain for the classical orbital frequency 1 µe2 1 = . (1.5) a ¯h2 n2 µe4 ω = . (1.9) ¯ ¯ 1 ¯ cl 3 3 With E = K + U = 2 U follows ¯h n
For large n this should be equal to ωqt obtained via µe4 1 1 a transition from orbit n + 1 to orbit n: En = − = −13.6 eV (1.6) 2¯h2 n2 n2 4 ! 14. The quantity n is called quantum number. µe 1 1 ωqt = − − Thus, an electromagnetic energy packet or pho- 2¯h3 (n + 1)2 n2 ton emitted during the transition from an excited ! µe4 1 1 state (n = m) to the ground state (n = 1) should = − − 1 2¯h3 n2 1 2 be given by 1 + n µe4 1 1 ≈ − 3 2 1 − 2 − 1 1 2¯h n n ¯hω = −13.6 eV − 1 . (1.7) m→1 2 µe4 m = . ¯h3n3 This is an agreement with the experiment! 15 This is an example of the important correspon- 13 −1 Here we have omitted the factor (4πo) multiplying dence principle, which describes the classical limit the potential energy, i.e. we occasionally toggle between the of quantum theory. MKSA and the Gauß unit systems. Here we may look at the classical limit from an- 14If a charge q is accelerated due to the force F = qE, where E is the electric field strength, between two parallel other angle. For large n we always require plates of a capacitor, then the energy ∆E it gains is given by ∆E n → ω . (1.10) ¯h∆n cl ∆E = F d = qEd = −dq∆φ/d = −q∆φ .
Here d is the plate-to-plate separation and ∆φ is the voltage In the case of Eq. (1.6) for example we have drop from one plate to the other. If q = −1.602 · 10−19 C, the charge of an electron, and ∆φ = 1 V , then ∆E equals 1 4 eV . Notice: 1 eV = 96.5 kJ/mol. ∆En dEn µe 1 (1.9) ≈ = =hw ¯ cl . 15This particular series of frequencies is called Lyman- ∆n dn ¯h2 n3 series (1906). Balmer’s series corresponds tohω ¯ m→2 = −13.53eV 1 − 1 . m2 4 Integrating Eq. (1.10) we obtain 4 CHAPTER 1. EARLY DEVELOPMENTS IN QUANTUM MECHANICS
I Z En Z n ∂ 1 dE 0 T (E) = pdq . = dn = n . (1.11) ∂E h 0 ν (E) 0 Note that T (E) = 1/ν(E) is the period of the orbit This equation may be cast in another useful form, corresponding to the energy E. Finally we inte- i.e. grate this equation from 0 to E, which yields the desired equality I pdq = hn , (1.12) Z E Z E 0 I 0 0 dE T (E )dE = 0 = pdq . (1.13) where the loop integral covers the area in classical 0 0 ν (E ) phase space bounded by a trajectory corresponding This is a result of classical mechanics. We test to the energy E 16. Eq. (1.13) using the classical harmonic oscillator, We show Eq. (1.12) utilizing the classical action E = p2/(2m) + (k/2)x2, as an example. Because (dt.: Wirkung), S = S (q, t), as a function of the the oscillator’s period is T = 2πpm/k we have (generalized) coordinate q and time t 17. It is useful to introduce a Legendre transformation to a new ˜ ˜ Z E Z E r function, S = S (q, H), where H is the Hamilton 0 0 0 m 18 T (E )dE = T dE = 2π E. (1.14) function , via 0 0 k Now we evaluate the right side of Eq. (1.13), i.e. dS˜ = d (Ht) + dS ∂S ∂S √ s = tdH + Hdt + dq + dt I Z 2E/k k ∂q ∂t pdq = 2 √ dx 2m E − x2 |{z} − 2E/k 2 |{z} =−H =p r √ 2E Z 1 p = pdq + tdH = 2 2mE dz 1 − z2 k −1 Setting H = E = const we integrate the last equa- | {z } =π/2 tion between q and qo to obtain rm = 2π E, (1.15) k Z q S˜(q, E) − S˜(q ,E) = pdq 0 . o where p2E/k is the amplitude of the oscillator. qo Obviously the two results, (1.14) and (1.15), do Now we take the derivative with respect to E: agree.
∂S˜(q, E) ∂S˜(q ,E) ∂ Z q − o = pdq 0 . 1.2 The black body problem ∂E ∂E ∂E q | {z } | {z } o =t =to At this point we want to consider the black body problem explicitly to figure out what goes wrong as 19 Here t and to correspond to q = q(t) and qo = q(to). well as how to correct classical theory . Extending the integration over a full period of the 19Historically, this system has been looked upon from (periodic) motion yields two, practically identical but conceptually different, points of view: (i) The earlier one, the one we take here, views the 16 ~ H Notice that by using pφ =| L | we obtain pφdφ = radiation field as a collection of independent oscillators. (ii) H Ldφ = nh. This is the generalized form of Eq. (1.8). Later, after the photon had fully established itself as an ele- 17cf. the section on Hamilton-Jacobi theory in Ref. [14]. mentary particle, the radiation field was treated as a gas of 18Hamilton, Sir (since 1835) William Rowan, irish math- Bosons. But this is a matter of statistical mechanics, which ematician and physicist, *Dublin 4.8.1805, †Dunsink (near we mention only in order to prevent the reader from getting Dublin) 2.9.1865. confused. 1.2. THE BLACK BODY PROBLEM 5
A black body cavity contains electromagnetic ra- diation. Its classical energy density is given by 1 ∗ q~ = √ c~ + c (1.22) k,α 4πc k,α ~k,α ~ 2 ~ 2 iω E + H p = −√ c − c∗ (1.23) W = (1.16) ~k,α ~k,α ~k,α 8π 4πc the final form [15] (Eq. (2.49)), where E~ and H~ are the electric and the magnetic field strengths, respectively. The X 1 total field energy E in the cavity is given by E = p2 + ω2q2 . (1.24) 2 ~k,α ~k,α ~k,α Z This means that the electromagnetic field energy E = dV W , (1.17) P V inside the cavity corresponds to ~k,α uncoupled one-dimensional (1D) harmonic oscillators. where V is the volume of the cavity. We may use At this point electrodynamics has done its job, a gauge, where E~ = − 1 ∂A~ and thus Eq. (1.17) c ∂t and we now turn to statistical mechanics. In sta- becomes tistical mechanics you will learn that the average energy contained in a system consisting of a fixed !2 number of particles (e.g., oscillators) inside a fixed 1 Z 1 ∂A~ 2 E = + ∇~ × A~ dV (1.18) volume, which is in thermal equilibrium with its 8π c ∂t surroundings (here the cavity walls), is given by
(cf. [15] (2.18) and (2.19) as well as (2.15)). The P −βEm Eme ∂ X vector potential in turn is expressed as a Fourier hEi = m = − ln e−βEm ,(1.25) P e−βEm ∂β series m m −1 where β = kBT . T is the temperature, and kB is Boltzmann’s constant. The sum P is over all en- ~ 1 X n −iωt m A (~r, t) = √ c~k,α~u~k,α(~r)e ergy values accessible to the system (cf. [16] section V ~k,α 2.1). o +c∗ ~u∗ (~r)eiωt , (1.19) We interrupt briefly to explain the difference be- ~k,α ~k,α tween hEi as compared to E used above. Statis- tical mechanics considers the electromagnetic field where k = 2π n , k = 2π n , k = 2π n (n , n , x L x y L y z L z x y inside the cavity as in thermal equilibrium with the n = ±1, ±2,...), V = L3, and z cavity walls kept at fixed temperature T . In this sense energy may be exchanged between the field (α) i~k·~r and the walls, i.e. the field’s energy fluctuates. ~u~ (~r) = ~ e . (1.20) k,α In classical physics energy is continuous, and Here and in the following the asterisk ∗ indicates the summation in Eq. (1.25) becomes an inte- the complex conjugate. ~(α) is real unit vector in gration. From classical mechanics we know that a the α-direction of the plane perpendicular to the harmonic oscillator at constant energy is described momentum vector ~k. Inserting Eq. (1.19) into Eq. by a trajectory in phase space. Changing energy (1.18) we obtain 20 means moving to another trajectory. Summing over all possible energy states (microstates) of the system, with their proper statistical weight, there- 2 X w ∗ fore means integrating over the entire phase space. E = c~ c~ (1.21) 2πc2 k,α k,α Thus, for a single oscillator we have ~k,α and via X Z ∞ ∼ dpdq . 20this is a homework problem m −∞ 6 CHAPTER 1. EARLY DEVELOPMENTS IN QUANTUM MECHANICS
We neglect a constant factor (the phase space den- (ω = pk/m). This is exactly what Planck had sity), because it drops out of the calculation of hEi assumed in 1900. which now, for a single oscillator, becomes Again we consider a single 1D oscillator. This time energy is discrete, and the summation remains Z ∞ a summation. The right side of Eq. (1.25) now ∂ −β(p2+ω2q2)/2 − ln dpdqe = kBT. becomes ∂β −∞
Because the oscillators are independent we obtain ∞ ∂ X ¯hω for the entire cavity − ln e−βhωn¯ = ∂β eβhω¯ − 1 n=0 X hEi = k T 1 . (1.26) B 21. For the entire cavity we obtain ~k,α The problem is now reduced to counting the num- X ¯hω ber of modes P ! hEi = (1.31) ~k,α eβhω¯ − 1 It is useful to transform the summation into an ~k,α integration. The general prescription for this is and therefore Z Z ∞ X 3 2 → ρ~ d k = 4πρ~ dkk . (1.27) k k ∞ 3 0 hEi (1.27) ¯h Z ω ~k = 2 3 dω βhω¯ V π c 0 e − 1 ~ 2 4 ρ~k is the density in k-space given by (cf. [16]; Eq. (3.13)) π (k T ) = B . 15 (¯hc)3 V ρ = . (1.28) ~k (2π)3 This in fact is the correct result in agreement with the experiment 22! Notice that in the (classical) This follows because the ~k-vectors lie on a cubic limit ¯h → 0 the integrand becomes ∼ ω3(1 + −1 grid, whose gridpoints are given by (kx, ky, kz). β¯hω − 1) and we recover the classical result of The lattice constant of this grid is 2π/L. The com- Eq. (1.29). bination of Eqs. (1.26) and (1.28) finally gives In the next section we learn that the above equa- tion for the energy levels of a harmonic oscillator hEi Z ∞ ω2 is not quite correct. It neglects the so called zero- = k T dω = ∞ . (1.29) point energy. However, here the zero-point energy V B π2c3 0 is a constant which does not contribute to the ra- This means that the energy density of the black diation detected outside the black body cavity, be- body radiation field is infinity. Clearly, this result cause photons correspond to transitions from and cannot be correct. And it is very disturbing that to excited oscillator states only 23. we have obtained it via correct application of elec- 21Using P∞ zn = (1 − z)−1 for z < 1. trodynamics and classical statistical mechanics! n=0 22It is possible to drill a hole into one of the container walls We now make a second attempt at solving this and observe the photon current density I(T ) = chEi/V . problem - using the quantization condition (1.12)! This is Stefan’s law (I(T ) ∝ T 4). Notice that the T 4- We conclude that the energy of the radiation field dependence may be argued on classical grounds - not, how- written in the form of Eq. (1.24) should be quan- ever, the coefficient [18] (section 12.1). In 1989 the cosmic background explorer or COBE-satellite tized as follows: For a single 1D harmonic oscillator measured the intensity spectrum of the cosmic background we have radiation over a wide range of frequencies finding perfect agreement with Planck’s distribution (cf. Ref. [16]; section 3.1). 1 I (1.15) E 23This does not mean that zero-point energy is not inter- n = pdx = n (1.30) h ¯hω esting as we demonstrate in appendix C. 1.3. BUILDING A GENERAL FORMALISM 7
1.3 Building a general formal- ideas are due to Heisenberg, Jordan, and Born 24 25 26 ism ): We seek a system of 2k matrices q , . . . , q , . . . , p , . . . p , which correspond to the 1 k 1 k Using energy quantization we are able to explain generalized coordinates and momenta of the me- the crude structure of the spectral lines of the hy- chanical analog. The key requirement is that these drogen atom, and we have solved the black body matrices must obey the commutator relations problem. But there remains much more to ex- h i plain: What happens if a second electron is added, q , q ≡ q q − q q = 0 , (1.33) how can we calculate intensities of spectral lines m n m n n m etc. Clearly, we need a general theory for electrons, atoms, molecules, radiation and their interaction. h i p , p ≡ p p − p p = 0 (1.34) A first step may be the following thought: Look- m n m n n m ing into our mathematical tool chest, we may ask and whether quantization can be formulated in terms of eigenvalue problems either through the use of ma- h i trices or differential equations. The latter are es- p , q ≡ p q − q p = −i¯hδmn . (1.35) pecially appealing if we think of particles as waves m n m n n m - as standing waves possessing a discrete spectrum Even though these relations seemingly appear out of frequencies. What we may aim for at this point of the blue, we remind the reader of the close is an eigenvalue differential equation for a parti- correspondence between these so called commu- cle (e.g., an electron) in an external potential, U, tators and the Poisson brackets in Hamiltonian yielding discrete energy levels like mechanics ({qm, qn} = 0, {pm, pn} = 0 and {pm, qn} = δmn [14]). Secondly the matrix 4 2 H q , . . . , q , . . . , p , . . . , p , which is the analog µe 1 e 1 k 1 k En = − for U = − 2¯h2 n2 r of the Hamilton function H, becomes diagonal. As an example we consider the 1D harmonic os- in the case of the Coulomb potential or p2 1 2 2 cillator, i.e. H = 2m + 2 mω x . We make a bold guess and write down the matrices 1 2 En =hωn ¯ for U = kx 2 0 1√ 0 0 ··· r 1√ 0 2√ 0 ··· ¯h 0 2 0 3 0 in the case of the harmonic oscillator. Consequently √ x = 0 0 3 0 (1.36) we seek an equation of the form 2mω ...... 0 . and H | ni = En | ni , (1.32)
0 −1 0 0 ··· where En are energy eigenvalues, and H is an √ r 1√ 0 − 2√ 0 ··· ¯hmω 0 2 0 − 3 0 operator, the so called Hamilton operator, corre- √ p = i 0 0 3 0 , (1.37) sponding to the classical Hamilton function H. 2 ...... But what is | ni? If we think in terms of matrix . . 0 . theory, then | ni is an eigenvector corresponding 24Heisenberg, Werner Karl, german physicist, *W¨urzburg to the eigenvalue En. In the following we will talk 5.12.1901, †M¨unchen 1.2.1976; one of the major contributors about energy eigenstates, and | ni will be called a to the development of quantum theory, of which his uncer- tainty principle remains a central and characteristic piece. state vector (dt.: Zustandsvektor). He received the Nobel prize in physics in 1932. 25Jordan, Ernst Pascual, german physicist, *Hannover Matrix mechanics: 18.10.1902, †Hamburg 31.7.1980. 26Born, Max, german physicist, *Breslau 11.12.1882, †G¨ottingen5.1.1970; a leading figure in quantum theory in The prescription for a quantum theory based pre-war Germany. He shared the 1954 Nobel prize in Physics on matrix algebra is as follows (The following with W. Bothe. 8 CHAPTER 1. EARLY DEVELOPMENTS IN QUANTUM MECHANICS which indeed obey the relations (1.33) to (1.35). the ideas of wave mechanics which we explore next. Inserting (1.36) and (1.37) into Wave mechanics: 1 mω2 H = p2 + x2 (1.38) We return to the notion that particles may be- 2m 2 have as waves. A plane wave traveling in the posi- yields tive x-direction may be described via hx i 1 0 0 0 ··· ψ (x, t) ∼ cos 2π − νt , (1.41) 2 λ 0 3 0 0 ··· 2 0 0 5 0 ··· 2 where λ is the wavelength, and ν is the frequency. H =hω ¯ 0 0 0 7 . (1.39) 2 . . . . We may also write this as ...... ψ (x, t) ∼ cos (kx − ωt) . (1.42) Thus, we obtain the desired eigenvalues En for If we think about particles, we think about local- . . ized objects. However, our ψ (x, t) thus far is not 0 localized. Therefore we construct a wave packet by | ni = 1 , (1.40) writing 0 . . Z ∞ where 1 stands for the nth element. ψ (x, t) = dkw (k) cos (kx − ωt) (1.43) There is an interesting difference to our previous 0 result, Eq. (1.30), based on the original quantiza- with tion condition. The lowest attainable energy, the 1 27 ground state energy, now is ¯hω ! However, the 2 2 2 w (k) = e−b k . (1.44) increments are still equal tohω ¯ . We recall that the original quantization condition, H pdq = nh, re- Setting t = 0 for the moment we obtain for sulted from (1.10), which is based on large n. Our r new quantum mechanical theory however produces π 2 2 ψ (x, t = 0) = e−x /4b . (1.45) the proper ground state energy as we shall see. 4b2 Notice also that there is a basic problem inher- We see that by adding up a sufficient number of ent in the above prescription for constructing the plane waves, putting in many more waves around quantum analog of a mechanical system. The clas- k = 0 (long wavelengths), we obtain an object, i.e. sical generalized coordinates and momenta may be a wave packet localized around x = 0. The exact interchanged without altering the Hamilton func- form of w (k) does not matter. Here we choose a tion. The corresponding operators or matrices, Gaussian shape because it is simpler to integrate. according to (1.35), may (in general) not be in- You may try this out yourself using the Mathemat- terchanged without altering H. Fortunately we ica program shown in Fig. 1.1, e.g. you may sub- do not run into this problem often, because for 2 stitute e−|k| instead of e−k . H = K (p , . . . , p ) + U (q , . . . , q ) there is no such 1 k 1 k As Fig. 1.2 shows, we may put t > 0 and ob- problem. serve the wave packet travelling along x. Here we We do not want to investigate the origin of choose ω = c k, where c = 1 is a constant velocity the relations (1.33) to (1.35), which is heuristic p p identical for all plane waves in the packet 28. This in any case, because quantum mechanics cannot representation of a free particle moving in space be derived from mechanics. We rather want appears acceptable. to suggest a second recipe for constructing a However we have not used the correct dispersion quantum theory. Later, however, we will return to relation (here: ω ∝ k). From Eq. (1.2) we have matrix mechanics and show how it is connected to 27This energy is also called zero-point energy. 28Exact solution: ψ (x, t) = p π exp − (x − t)2 /4b2. 4b2 1.3. BUILDING A GENERAL FORMALISM 9
1.0
0.8
0.6 1 0.4 0.75 0.5 2 0.25 0.2 0 1.5 1 x -4 t -4 -2 2 4 -2 0.5 -0.2 0 x 2 1.0 4 0
0.8
0.6
0.4
0.2 1 0.75 0.5 2 x 0.25 -4 -2 2 4 0 1.5 - 0.2 1 w k_ := Exp -k^2 -4 t -2 Plot Sum w k Cos 2 k x , k, 0, 1, .01 100, @ D @ D Π 0 0.5 x, -5, 5 , PlotRange ® -.2, 1 , AxesLabel ® x, " " , x TextStyle ® FontSize ® 12 , PlotStyle ® Black 2 @ @ @ D @ H L D 8
¯hk2 The operators ~r ≡ ~r and ~p ≡ −i¯h∇~ again satisfy ω = , (1.46) 2m the commutator relations (1.33), (1.34), and (1.35), i.e. i.e. ω ∝ k2. Fig. 1.2 illustrates what happens h¯ to the wave packet (setting 2m = 1). It loses its localization as t increases! [xm, xn] = 0 (1.51) Nevertheless, let us proceed and write more gen- erally h i p , p = 0 (1.52) m n Z ψ (~r, t) = d3pw (~p) ei(~p·~r−Et)/h¯ . (1.47) all space h i p , x = −i¯hδmn . (1.53) Here we use complex waves, which is equivalent to m n ~p·~r Et using cos h¯ − h¯ as long as we apply linear op- More precisely erations to ψ (~r, t). Next we apply the operators ~ 2 h i h i ∇ and ∂t to Eq. (1.47): p , x ≡ p , x ψ (~r, t) m n m n = −i¯h (∂ x ψ − x ∂ ψ) Z 2 xm n n xm 2 3 p i (~p·~r−Et) ∇~ ψ (~r, t) = d p − w (~p) e h¯ . = −i¯hδmnψ (~r, t) . (1.54) ¯h2 Notice also that we may satisfy the classical rela- and p2 tion E = 2m +U (~r, t) (with E = const) by formally adding a term U (~r, t) ψ (~r, t) to (1.48). The result Z 3 i i (~p·~r−Et) is Schr¨odinger’sequation for a single particle mov- ∂ ψ (~r, t) = d p − E w (~p) e h¯ . t ¯h ing in a potential U (~r, t) 29:
h¯2 Multiplication of the upper relation by − 2m and 2 ¯h 2 the lower relation by i¯h yields, if both relations are − ∇~ + U (~r, t) − i¯h∂t ψ (~r, t) = 0 . (1.55) combined, 2m This equation, which yet has to prove itself, is the ¯h2 starting point for all calculations in wave mechan- − ∇~ 2 − i¯h∂ ψ (~r, t) = 0 , (1.48) ics. 2m t Notice that we can insert the Ansatz where we have used E = p2/2m! This differential i equation looks very much like a diffusion equation − h¯ Et ih¯ ψ (~r, t) = ψ (~r) e (1.56) with a complex diffusion constant 2m . It describes the free particle. into Eq. (1.55. This then yields the time-inde- Notice that analogous to matrix quantum me- pendent or stationary Schr¨odingerequation chanics, where we have replaced the classical coor- dinates and momenta by q and p , we now define m m ¯h2 the operators − ∇~ 2 + U (~r) − E ψ (~r) = 0 . (1.57) 2m
29Schr¨odinger, Erwin, austrian physicist, *Vienna E → i¯h∂t (1.49) 12.8.1887, †Vienna 4.1.1961; he shared the 1993 Nobel prize in physics with Paul A. M. Dirac for his groundbreaking and contributions to quantum theory. 1.3. BUILDING A GENERAL FORMALISM 11
the problem, i.e.h ¯2/(2ma2), is proportional to a−2. U(x) ∞ ∞ Here a is the wall-to-wall separation, but generally speaking it is the linear dimension of the space to which the particle is confined. This remains true in three dimensions. In the case of an atom, a typi- cally is one to several 10−10m. For a particle con- fined to the nucleus a is on the order of 10−14m. This means that typical energies of chemical re- actions involving the electrons (responsible for the x size of the atom) are around 8 orders of magnitude (!) less than the typical energy of nuclear reactions. 0 a In the two regions q ≤ 0 and q ≥ 1 it is reasonable to assume that ψ(q) must vanish in order for (1.60) to remain true. In the region 0 < q < 1 we have Figure 1.3: Infinite rectangular potential. 2 −∂q − ψ (q) = 0 . (1.62) A 1D particle between infinite walls: A solution is
The first example, to which we apply our new √ equation in the form of (1.57), is a particle trapped ψ (q) = ψo sin( q) . (1.63) between two barriers, one at x = 0 and the other Notice that ψ (0) = 1 and thus this ψ ties contin- one at x = a as shown in Fig. 1.3. This is a one- uously on to ψ (q) = 0 in region q ≤ 0. Why we dimensional (1D) problem. In this case Eq. (1.57) should require the two solutions to be the same at becomes q = 0 is not clear at this moment, but it becomes clear below when we discuss the physical meaning ¯h2 of ψ. In order to also tie (1.63) continuously to − ∂ 2 + U (x) − E ψ (x) = 0 . (1.58) 2m x ψ (q) = 0 in region q ≥ 1, we must require
Before we do anything to solve this problem, we √ transform Eq. (1.58) into its dimensionless form. = πn (n = 1, 2,...) . (1.64) This is done by introducing the new dimensionless We exclude n = 0, because it would mean ψ (q) coordinate vanishes everywhere, which should mean that there is no particle at all. Thus, the two ad hoc boundary q = x/a (1.59) conditions yield the discrete energy levels and subsequently dividing the above equation ¯h2 through ¯h2/(2ma2): E = π2n2 . (1.65) n 2ma2 available to our particle on the basis of the (sta- 2 −∂q + u (q) − ψ (q) = 0 , (1.60) tionary) Schr¨odingerequation. Can we test this result? Well, our problem is where somewhat artificial, but there is a resemblance of our particle to an ideal classical gas particle (or ¯h2 0 0 < q < 1 molecule). We know that the average thermal en- E = and u = . (1.61) 2ma2 ∞ otherwise ergy per particle in an ideal gas is
This is not merely cosmetics. There is valuable 3 information here. Notice that the typical energy of hEi = kBT, (1.66) 2 12 CHAPTER 1. EARLY DEVELOPMENTS IN QUANTUM MECHANICS
i.e. kBT/2 contributed by each momentum compo- i.e. the 1D harmonic oscillator. Eq. (1.57) in this nent. In the present 1D case we therefore expect case becomes exactly this. Combination of Eqs. (1.25) and (1.65) yields 2 ¯h 2 1 2 2 − ∂x + mω x − E ψ (x) = 0 . (1.70) ∞ 2m 2 2 X 2 hEi = kBTot ∂t ln exp[−n /t] , (1.67) The solution is n=1
h¯2 2 1/4 where t = T/To and To = ( 2ma2 π )/kB. If we mω 1 ψn (x) = √ (1.71) consider the limit t 1, i.e. the limit of low tem- π¯h n!2n perature, it is sufficient to only keep the n = 1-term rmω h mω i in the sum and our result becomes ×H x exp − x2 n ¯h 2¯h with hEit1 = kBTo . (1.68)
If we assume t 1 instead, i.e. high temperatures, n 2 d 2 then we may write H (q) = (−1)n eq e−q . (1.72) n dqn Z ∞ 2 −n2/t kBT The Hn (q) are Hermite polynomials (H0 (q) = 1, hEit1 = kBTot ∂t ln dne = .(1.69) H (q) = 2q,...) 30. Each index n corresponds to 0 2 1 an energy eigenvalue This is indeed the expected classical result, which follows in the limit of high temperature or, alterna- tively, in the limit h → 0. In the low temperature 1 En =hω ¯ n + n = 0, 1, 2,... (1.74) limit we find that our solution deviates from the 2 classical result and hEi approaches a finite value. The crossover from classical to quantum behavior This means that Schr¨odinger’s equation agrees with (1.39)! Notice in particular that E = 1 ¯hω. Notice apparently happens when T approaches T from 0 2 o also that the discrete eigenvalues result from the above. But what is the value of To? If we as- sume that our particle is an Argon atom and that requirement ψn(±∞) = 0. This is illustrated in −16 Fig. 1.4 for the case of n = 0, which shows the a = 1cm, then we find To ∼ 10 K! Remark: This is the reason why in Statistical Mechanics numerical solution for three different energies, only the translational contribution to the energy of an one of which corresponds to the ’possible’ energy E = 1 ¯hω. However, what is the meaning of the so atomic or molecular gas can always be calculated 0 2 based on classical mechanics. called wave function ψn (x)? Even though we have shown that our solution 30Substitution of x = p¯h/ (mω)q into Eq. (1.70) yields produces the correct result in the classical limit, this does not prove that Schr¨odinger’sequation 2 2 ∂q − q + 2 ψ(q) = 0 yields correct results in cases when experiments 2 deviate from the classical expectations. Thus, we with E =hω ¯ . The Ansatz ψ(q) = constHn(q) exp[−q /2] transforms this equation to its final form known in mathe- must study more examples. matics as Hermite’s differential equation:
The meaning of ψ (~r, t): 2 ∂q − 2q∂q + 2 − 1 Hn(q) = 0 (1.73)