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csThis book represents a concise summary of non-relativistic quan-tum mechanics on the level suitable for university students of phys-ics. It covers, perhaps even slightly exceeds, a one-year course of about 50 lectures, requiring basic knowledge of calculus, algebra, classical mechanics and a bit of motivation for the quantum adven-ture. The exposition is succinct, with minimal narration, but with a maximum of explicit and hierarchically structured mathematical derivations.

The text covers all essential topics of university courses of quantum mechanics – from general mathematical formalism to specific applications. The formulation of quantum theory is ac-companied by illustrations of the general concepts of elementary quantum systems. Some subtleties of mathematical foundations are overviewed, but the formalism is used in an accessible, intui-tive way. Besides the traditional topics of non-relativistic quantum mechanics, such as single-particle dynamics, symmetries, semiclas-sical and perturbative approximations, density-matrix formalism, scattering theory, theory of angular momentum, description of many-particle systems – the course also touches upon some mod-ern issues, including quantum entanglement, decoherence, mea-surement, nonlocality, and quantum information. Historical context and chronology of basic achievements is outlined in brief remarks.

The book is intended for beginners as a supplement to lec-tures, however, it may also be used by more advanced students as a compact and comprehensible overview of elementary quantum theory.

I enjoyed reading this book. What I found particularly interesting was the style of the presentation, the original and excellent selec-tion of topics, and the numerous brief historical remarks. The text is succinct but not superficial: the deeper one immerses in reading, one finds even more inspiring remarks. The reader is allerted to the subtleties of the mathematical formulation of quantum mechanics, without getting lost in unnecessary formalism.

Prof. Jean-Paul Blaizot (IPhT, Paris)

Pavel Cejnar

A Condensed Courseof Quantum Mechanics

KA R O L I N U M

A Condensed Course of Quantum Mechanics_mont.indd 1 8/29/13 10:54 AM

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A Condensed Course of Quantum Mechanics

Pavel Cejnar

Reviewed by:

Prof. Jiří Hořejší (Prague)

Prof. Jean-Paul Blaizot (Paris)

Cover Jan Šerých

Typesetting DTP Karolinum Press

First edition

© Charles University in Prague, 2013

© Pavel Cejnar, 2013

ISBN 978-80-246-2321-4

ISBN 978-80-246-2349-8 (pdf)

Ukázka knihy z internetového knihkupectví www.kosmas.cz

Charles University in Prague

Karolinum Press 2013

http://www.cupress.cuni.cz

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Contents

Preface 1

Rough guide to notation 3

INTRODUCTION 5

1. FORMALISM ! 2. SIMPLE SYSTEMS 10

1.1 Space of quantum states . . . . . . . . . . . . . . . . . . . . . . . . 10

Hilbert space. Rigged Hilbert space . . . . . . . . . . . . . . . . . . . 10

Dirac notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Sum & product of spaces . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1 Examples of quantum Hilbert spaces . . . . . . . . . . . . . . . . 15

Single structureless particle with spin 0 or 12 . . . . . . . . . . . . . . 15

2 distinguishable/indistinguishable particles. Bosons & fermions . . . 17

Ensembles of N ≥ 2 particles . . . . . . . . . . . . . . . . . . . . . . 19

1.2 Representation of observables . . . . . . . . . . . . . . . . . . . . 21

Observables as Hermitian operators. Basic properties . . . . . . . . . 21

Eigenvalues & eigenvectors in finite & infinite dimension . . . . . . . 23

Discrete & continuous spectrum. Spectral decomposition . . . . . . . 25

2.2 Examples of quantum operators . . . . . . . . . . . . . . . . . . . 27

Spin-12 operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Coordinate & momentum . . . . . . . . . . . . . . . . . . . . . . . . 29

Hamiltonian of free particle & particle in potential . . . . . . . . . . . 30

Orbital angular momentum. Isotropic Hamiltonians . . . . . . . . . . 33

Hamiltonian of a particle in electromagnetic field . . . . . . . . . . . 37

1.3 Compatible and incompatible observables . . . . . . . . . . . . . 39

Compatible observables. Complete set . . . . . . . . . . . . . . . . . 39

Incompatible observables. Uncertainty relation . . . . . . . . . . . . . 41

Analogy with Poisson brackets . . . . . . . . . . . . . . . . . . . . . . 42

Equivalent representations . . . . . . . . . . . . . . . . . . . . . . . . 43

2.3 Examples of commuting & noncommuting operators . . . . . . 44

Coordinate, momentum & associated representations . . . . . . . . . 44

Angular momentum components . . . . . . . . . . . . . . . . . . . . . 47

Complete sets of commuting operators for structureless particle . . . 49

1.4 Representation of physical transformations . . . . . . . . . . . . 50

Properties of unitary operators . . . . . . . . . . . . . . . . . . . . . 50

Canonical & symmetry transformations . . . . . . . . . . . . . . . . . 52

Basics of group theory . . . . . . . . . . . . . . . . . . . . . . . . . . 54

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2.4 Fundamental spatio-temporal symmetries . . . . . . . . . . . . . 56

Space translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Space rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Space inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Time translation & reversal. Galilean transformations . . . . . . . . . 62

Symmetry & degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . 64

1.5 Unitary evolution of quantum systems . . . . . . . . . . . . . . . 65

Nonstationary Schrodinger equation. Flow. Continuity equation. . . . 65

Conservation laws & symmetries . . . . . . . . . . . . . . . . . . . . . 67

Energy × time uncertainty. (Non)exponential decay . . . . . . . . . . 68

Hamiltonians depending on time. Dyson series . . . . . . . . . . . . . 71

Schrodinger, Heisenberg & Dirac description . . . . . . . . . . . . . . 73

Green operator. Single-particle propagator . . . . . . . . . . . . . . . 74

2.5 Examples of quantum evolution . . . . . . . . . . . . . . . . . . . 76

Two-level system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Free particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Coherent states in harmonic oscillator . . . . . . . . . . . . . . . . . . 79

Spin in rotating magnetic field . . . . . . . . . . . . . . . . . . . . . . 81

1.6 Quantum measurement . . . . . . . . . . . . . . . . . . . . . . . . 83

State vector reduction & consequences . . . . . . . . . . . . . . . . . 83

EPR situation. Interpretation problems . . . . . . . . . . . . . . . . . 85

2.6 Implications & applications of quantum measurement . . . . . 89

Paradoxes of quantum measurement . . . . . . . . . . . . . . . . . . . 89

Applications of quantum measurement . . . . . . . . . . . . . . . . . 91

Hidden variables. Bell inequalities. Nonlocality . . . . . . . . . . . . 92

1.7 Quantum statistical physics . . . . . . . . . . . . . . . . . . . . . . 94

Pure and mixed states. Density operator . . . . . . . . . . . . . . . . 95

Entropy. Canonical ensemble . . . . . . . . . . . . . . . . . . . . . . 96

Wigner distribution function . . . . . . . . . . . . . . . . . . . . . . . 98

Density operator for open systems . . . . . . . . . . . . . . . . . . . . 99

Evolution of density operator: closed & open systems . . . . . . . . . 101

2.7 Examples of statistical description . . . . . . . . . . . . . . . . . 104

Harmonic oscillator at nonzero temperature . . . . . . . . . . . . . . 104

Coherent superposition vs. statistical mixture . . . . . . . . . . . . . 105

Density operator and decoherence for a two-state system . . . . . . . 106

3. QUANTUM-CLASSICAL CORRESPONDENCE 108

3.1 Classical limit of quantum mechanics . . . . . . . . . . . . . . . . 108

The limit ~→ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Ehrenfest theorem. Role of decoherence . . . . . . . . . . . . . . . . . 109

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3.2 WKB approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Classical Hamilton-Jacobi theory . . . . . . . . . . . . . . . . . . . . 112

WKB equations & interpretation . . . . . . . . . . . . . . . . . . . . 114

Quasiclassical approximation . . . . . . . . . . . . . . . . . . . . . . . 115

3.3 Feynman integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Formulation of quantum mechanics in terms of trajectories . . . . . . 118

Application to the Aharonov-Bohm effect . . . . . . . . . . . . . . . . 119

Application to the density of states . . . . . . . . . . . . . . . . . . . 120

4. ANGULAR MOMENTUM 123

4.1 General features of angular momentum . . . . . . . . . . . . . . 123

Eigenvalues and ladder operators . . . . . . . . . . . . . . . . . . . . 123

Addition of two angular momenta . . . . . . . . . . . . . . . . . . . . 125

Addition of three angular momenta . . . . . . . . . . . . . . . . . . . 128

4.2 Irreducible tensor operators . . . . . . . . . . . . . . . . . . . . . 129

Euler angles. Wigner functions. Rotation group irreps . . . . . . . . . 129

Spherical tensors. Wigner-Eckart theorem . . . . . . . . . . . . . . . 130

5. APPROXIMATION TECHNIQUES 133

5.1 Variational method . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Dynamical & stationary variational principle. Ritz method . . . . . . 133

5.2 Stationary perturbation method . . . . . . . . . . . . . . . . . . . 136

General setup & equations . . . . . . . . . . . . . . . . . . . . . . . . 136

Nondegenerate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Degenerate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Application in atomic physics . . . . . . . . . . . . . . . . . . . . . . 141

Application to level dynamics . . . . . . . . . . . . . . . . . . . . . . 145

Driven systems. Adiabatic approximation . . . . . . . . . . . . . . . 147

5.3 Nonstationary perturbation method . . . . . . . . . . . . . . . . 149

General formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Step perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Exponential & periodic perturbations . . . . . . . . . . . . . . . . . . 154

Application to stimulated electromagnetic transitions . . . . . . . . . 155

6. SCATTERING THEORY 157

6.1 Elementary description of elastic scattering . . . . . . . . . . . . 158

Scattering by fixed potential. Cross section . . . . . . . . . . . . . . . 158

Two-body problem. Center-of-mass system . . . . . . . . . . . . . . . 159

Effect of particle indistinguishability in cross section . . . . . . . . . . 160

6.2 Perturbative approach the scattering problem . . . . . . . . . . 161

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Lippmann-Schwinger equation . . . . . . . . . . . . . . . . . . . . . . 161Born series for scattering amplitude . . . . . . . . . . . . . . . . . . . 164

6.3 Method of partial waves . . . . . . . . . . . . . . . . . . . . . . . . 166Expression of elastic scattering in terms of spherical waves . . . . . . 166Inclusion of inelastic scattering . . . . . . . . . . . . . . . . . . . . . 172Low-energy & resonance scattering . . . . . . . . . . . . . . . . . . . 174

7. MANY-BODY SYSTEMS 1757.1 Formalism of particle creation/annihilation operators . . . . . 176

Hilbert space of bosons & fermions . . . . . . . . . . . . . . . . . . . 176Bosonic & fermionic creation/annihilation operators . . . . . . . . . . 177Operators in bosonic & fermionic N -particle spaces . . . . . . . . . . 181Quantization of electromagnetic field . . . . . . . . . . . . . . . . . . 186

7.2 Many-body techniques . . . . . . . . . . . . . . . . . . . . . . . . . 189Fermionic mean field & Hartree-Fock method . . . . . . . . . . . . . 189Bosonic condensates & Hartree-Bose method . . . . . . . . . . . . . . 192Pairing & BCS method . . . . . . . . . . . . . . . . . . . . . . . . . . 193Quantum gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

Ukázka knihy z internetového knihkupectví www.kosmas.cz

1

Preface

This book was conceived as a collection of notes to my two-semester lecture onquantum mechanics for third-year students of physics at the Faculty of Mathematicsand Physics of the Charles University in Prague. It was created in 2011-12.

At first, I just wanted to write down the most important facts, formulas andderivations in a compact form. The information flew in a succinct, “staccato” style,organized in larger and smaller bits (the and I items), rarely interrupted bywordy explanations. I enjoyed the thick, homogeneous mathematical form of thenotes. Calculations, calculations, calculations. . . I thought of a horrified historian orsociologist who finds no oasis of words. This is how we, tough guys, speak!

However, I discovered that the dense form of the notes was hardly digestible evenfor tough guys. I had to add some words. To create a “storyteller” who wraps thebare formulas into some minimal amount of phrases. His voice, though still ratherlaconic, may help to provide the proper motivation and clarify the relevant context.I also formed a system of specific “environments” to facilitate the navigation. Inparticular: Among crowds of calculations there appears a hierarchy of highlightedformulas:∗

important essential 1 essential 2 crucial

Assumptions or foundational concepts, irreducible to other statements/concepts,appear in boxes:† Answer to ultimate question of life, universe & everything = 42

Here and there come some historical notes:‡ J 2013: Condensed Course issuedHandmade schemes (drawn on a whiteboard) illustrate some basic notions.

In this way, the notes have turned into a more serious thing. They almost becamea textbook ! The one distinguished from many others by expanded mathematicalderivations (they are mostly given really step by step) and reduced verbal stuffing(just necessary comments in between calculations). This makes the book particularlywell suited for conservation purposes—acquired knowledge needs to be stored in acondensed, dense enough form, having a compact, nearly tabular structure.

However, as follows from what has been said, this book cannot be considered astandard textbook. It may hardly be read with ease and fluency of some more epictreatises. One rather needs to proceed cautiously as a detective, who has to preciselyfix all objects on the stage (all symbols, relations etc.) before making any small stepforward. This book can be used as a teaching tool, but preferably together with an

∗Such formulas are highly recommended to memorize! Although all students of physics & mathematics seem toshare a deep contempt for any kind of memorization, I have to stress that all results cannot be rederived in reasonabletime limits. There is no escape from saving the key formulas to the memory and using them as quickly reachablestarting points for further calculations.†However, these assumptions do not constitute a closed system of axioms in the strict mathematical sense.‡I believe that knowledge of history is an important part of understanding. The concepts do not levitate in vacuum

but grow from the roots formed by concrete circumstances of their creation. If overlooking these roots, one maymisunderstand the concepts.

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oral course or a more talkative textbook on quantum mechanics. Below I list someof my favorite candidates for additional guiding texts [1–8].

I have to stress that the notes cover only some parts of non-relativistic quan-tum mechanics. The selection of topics is partly fixed by the settled presentationof the field, and partly results from my personal orientation. The strategy is tointroduce the complete general formalism along with its exemplary applications tosimple systems (this takes approx. one semester) and then (in the second semester)to proceed to some more specialized problems. Relativistic quantum mechanics istotally absent here; it is postponed as a prelude for the quantum field theory course.

Quantum mechanics is a complex subject. It obligates one to have the skills of amathematician as well as the thinking of a philosopher. Indeed, the mathematicalbasis of quantum physics is rather abstract and it is not obvious how to connect itwith the observed “reality”. No physical theory but quantum mechanics needs sucha sophisticated PR department. We will touch the interpretation issues here, butonly very slightly. Those who want to cultivate their opinion (but not to disappearfrom the intelligible world) are forwarded to the classic [9]. The life saving trick inthis terra incognita is to tune mind to the joy of thinking rather than to the demandof final answers. The concluding part of the theory may still be missing.

Before we start I should not forget to thank all the brave testers—the first men,mostly students, who have been subject to the influence of this book at its variousstages of preparation. They were clever enough to discover a lot of mistakes. Besure that the remaining mistakes are due to their generous decision to leave somefish for the successors.

In Prague, January 2013

Recommended textbooks:[1]

J.J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1985, 1994)A modified edition of the same book:[

2]

J.J. Sakurai, J.J. Napolitano, Modern Quantum Mechanics (Addison-Wesley, 2011)[3]

G. Auletta, M. Fortunato, G. Parisi, Quantum Mechanics (Cambridge University Press, 2009)[4]

L.E. Ballantine, Quantum Mechanics. A Modern Development (World Scientific, Singapore, 1998)[5]

A. Peres, Quantum Theory: Concepts and Methods (Kluwer, 1995)[6]

A. Bohm, Quantum Mechanics: Foundations and Applications (Springer, 1979, 1993)[7]

W. Greiner Quantum Mechanics: An Introduction (Springer, 1989),W. Greiner, Quantum Mechanics: Special Chapters (Springer, 1998)W. Greiner, B Muller, Quantum Mechanics: Symmetries (Springer, 1989)[

8]

E. Merzbacher, Quantum Mechanics (Wiley, 1998)

Further reading:[9]

J.S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, 1987)

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Rough guide to notation (succinct and incomplete)

symbol meaning

Spaces, state vectors & wavefunctions

H, H, H Gelfand’s hierarchy of spaces (rigged Hilbert space)`2, L2(R3), Cd specific separable or finite Hilbert spaces|ψ〉, 〈ψ′|; 〈ψ′|ψ〉 “ket” & “bra” forms of state vectors; scalar product

||ψ|| =√〈ψ|ψ〉 vector norm

α|ψ〉+β|ψ′〉 superposition≡ linear combination of state vectors (α, β∈C)|φi〉,|Φij〉≡|φ1i〉1|φ2j〉2 general basis vector in H; separable basis vector in H1 ⊗H2

|ψ〉1|ψ′〉2 general separable vector in H1 ⊗H2

|a〉, |ai〉, |a(k)i 〉 eigenvector of A with eigenvalue a or ai (degeneracy index k)

|Ei〉, |E〉 energy eigenvectors| ↑〉, | ↓〉 up & down projection states of spin s=1

2|lml〉|sms〉 , |jmj〉 states with (orbital

spin ,total) ang. momentum ( ls , j), projection m•ψ(~x,ms︸︷︷︸) ≡ Ψ(~x) single-particle wavefunction in single/multicomponent forms

Ψ(ξ1 . . . ξN) N -particle wavefunction

Rnl(r) = unl(r)r radial wavefunction

Span|ψ1〉...|ψn〉 linear space spanned by the given vectorsN , dH normalization coefficient & dimension of space H

Observables & operators

O, O†, O−1 operator, its Hermitian conjugate & inverse

I, U identity operator & unitary operator

Pa, Π(a1,a2) projectors to discrete & continuous eigenvalue subspaces

||A|| operator norm

A1 ⊗ A2 tensor product of operators acting in H1 ⊗H2

H, T , V ; H ′ Hamiltonian, its kinetic & potential terms; perturbation~∇, ∆ gradient & Laplace operator (or also an interval, gap...)

~x, ~p, P coordinate, momentum vectors & spatial parity operator

~L, ~S; ~J , J± orbital, spin & total angular momentum, shift operators for J3

~σ the triplet of Pauli matrices

T∆o |o〉→|o+∆o〉 eigenvector shift operator for general operator O

Gi, CG generator & Casimir operator of a group Gb, b†; a, a†; c, c† annihilation, creation operators for bosons, fermions, or both

N , Nk total number of particles & number of particles in kthstate

R~nφ, R(αβγ) rotation operator inH & rotation matrix in 3D (Euler angles)

U(t), U(t1,t0) evolution operator for times t0t→ t1

T , T time reversal operator & time ordering of operator product

G(t), G(~xt|~x0t0) Green operator & propagator

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OS, OH(t), OD(t) Schrodinger, Heisenberg, Dirac representations of operator

[Aλ1×Bλ1]λµ tensor coupling of spherical tensor operators Aλ1µ1, Bλ1

µ2

[A, B],A, B commutator & anticommutator of operatorsA,B Poisson bracket of classical observables A,B

Tr O, Tr1O trace of operator/matrix, partial trace over H1 in H1 ⊗H2

Det O, Def(O) determinant of matrix/operator, definition domain of operator

Statistics, probabilities & densitiespψ(a) probability to measure value a of observable A in state |ψ〉〈A〉ψ, 〈a〉c average of A-distribution in |ψ〉, average of a for a parameter c〈〈A2〉〉ψ≡∆2

ψA dispersion of A-distribution in |ψ〉 ≡ squared uncertainty

pc(a|b) conditional probability of a given b (depending on parameter c)

ρ(~x, t), ~j(~x, t) probability density & flow at point ~x, time tρ, Wρ(~x, ~p), Sρ density operator/matrix, Wigner distribution function, entropy%(E) density of energy eigenstates

Functionsjl, nl, h

±l (kr) Bessel, Neumann & Hankel functions

Lji (ρ), Hn(ξ)

associatedgeneralized

Laguerre polynomials & Hermite polynomials

Plm(cosϑ),Ylm(ϑ,ϕ) associated Legendre polynomial, spherical harmonics (sph.angles)

Djm′m(αβγ) Wigner matrix function≡Dj

m′m(R) (Euler angles of rotation R)

δ(x), δε(x); Θ(x) Dirac δ-function, sequence of functionsε→0−→ δ; step function

Z(β), Z(β, µ) (grand)canonical partition funcs. (inv.temperature,chem.potential)S[~x(t)]S(~x,t)

, L(~x, ~x) classical action (functional & function forms), Lagrangian

V (~x), ~A(~x) scalar & vector potentialsSji, Pji,Wji(t) j→ i transition amplitude, probability & rate (time)Fl, Sl, δl(k) partial wave amplitude, S-matrix & phase shift (|wavevector|)f~k(~k ′)≡f~k(ϑ,ϕ) scattering amplitude (direction/angles)

dσdΩ(ϑ,ϕ) differential cross section (σ ≡ integral cross section)

Miscellaneous(1, 2, 3)≡(x, y, z) indices of Cartesian components

~n,

(~nx,~ny,~nz)(~nr,~nϑ,~nϕ)

unit vector,

Cartesianspherical

orthonormal coordinate vectors

δij, εijk Kronecker & Levi-Civita symbols

Cjmj1m1j2m2

Clebsch-Gordan coefficient ≡ 〈j1j2jm|j1m1j2m2〉~, c, e Planck constant, speed of light, elementary chargeM,M; q particle mass & two-particle reduced mass; particle charge~k, ω, λ wavevector, frequency, wavelength (or perturbation parameter)εk, nk energies & occupation numbers of single-particle statesXii∈D,X(c)c∈C discrete/continuous set of objectsMin,Max,SupXii minimum, maximum, supremum of a set of numbers• ; iff blind index denoting objects from a given set; if and only if

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INTRODUCTION

Before sailing out, we encourage the crew to get ready for adventures. Quantummechanics deals with phenomena, which are rather unusual from our common macro-scopic experience. Description of these phenomena makes us sacrifice some principleswhich we used to consider self-evident.

Quantum levelQuantum theory describes objects on the atomic and subatomic scales, but alsolarger objects if they are observed with an extremely high resolution.

I Planck constantThe domain of applicability of quantum mechanics determined with the aid of

a new constant: ~ .= 1.05 · 10−34 J·s .

= 0.66 eV·fs (units of action)

I Consider 2 classical trajectories q1(t) & q2(t) (in a general multidimensionalconfiguration space) which (in the given experimental situation) are on the limitof distinguishability. The difference of actions: ∆S= |S[q1(t)]−S[q2(t)]|

Classical mechanicsQuantum mechanics

applies if the relevant actions satisfy

∆S ~∆S . ~

In particular, if the minimumof action measured with resolution∼ ~ is wide with respect todistinguishable trajectories,quantum description is unavoidable.

J Historical remark1900: Max Planck introduced ~ along with the quanta of electromagnetic radiationto explain the blackbody radiation law1905: Albert Einstein confirmed elmag. quanta in the explanation of photoeffect1913: Niels Bohr introduces a quantum model of atoms (“old quantum mechanics”)

Double slit experiment

According to Richard Feynman & some others, this is the most crucial quantumexperiment that allows one to realize how unusual the quantum world is.

I Arrangement

Emitter E of individual particles, shield with slits A and B, screen SBoth trajectories ~xA(t) and ~xB(t) from ~xE to ~xS minimize the action

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Suppose |SA − SB| . ~

I Regimes of measurement

(a) Interference setup: particle position measured only at the screen⇒ interference pattern with individual particle hits

(b) Which-path setup: prior the screen measurement, the particle positionmeasured immediately after the slits ⇒ no interference pattern

Delayed-choice experiment: The choice of setup (a)/(b) is made after theparticle passed the slits. The same outcome as if the decision was made before.

Paradox: The outcome of the interference setup indicates a wave-like behaviorof particles (passage through both slits simultaneously). The outcome of thewhich-path setup shows a corpuscular behavior (passage through one slit only).The outcome of the delayed-choice experiment invalidates the possibility thatthe particle “changes clothes” according to the setup selected.

J Historical remark1805 (approx.): Thomas Young performed double-slit experiment with light1927: C. Davisson & L. Germer demonstrate interference of electrons on crystals1961: first double-slit experiment with massive particles (electrons)1970’s: double-slit experiments with individual electrons1990’s-present: progress in realizations of which-path setup & delayed-choice exp.

Wavefunction & superposition principle

To explain the outcome of the interference setup of the double-slit experiment,one has to assume that particles possess some wave properties.

I Particle attributed by a wavefunction: ψ(~x, t) ≡√ρ(~x, t) eiϕ(~x,t) ∈ C

Squared modulus |ψ(~x, t)|2 = ρ(~x, t) ≥ 0 is the probability density to

detect the particle at position ~x. Normalization:

∫|ψ(~x, t)|2 d~x = 1 ∀t

Phase ϕ(~x, t) ∈ R has no “classical” interpretation

Ukázka knihy z internetového knihkupectví www.kosmas.cz

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ψ(~x, t) ≡ instantaneous density of the probability amplitude for finding theparticle at various places (particle is inherently a delocalized object!)

I Superposition of wavefunctions

The outcome of the interference setup depends on the fact that waves can besummed up. Consider 2 wavefunctions ψA(~x, t) & ψB(~x, t)∫|ψA|2d~x <∞,

∫|ψB|2d~x <∞ ⇒

∫|αψA+βψB|2d~x <∞ ∀α, β ∈ C

⇒ any linear combination of normalizable wavefunctions is a normalizable wave-function ⇒ these functions form a linear vector space L2(R3)

I Interference phenomenon

Probability density for a superposition of waves is not the sum of densities forindividual waves

Chooseα=|α|eiϕαβ=|β|eiϕβ

such that

∫|αψA+βψB|2 d~x = 1 (with

ψA

ψB

normalized)

⇒∣∣αψA +βψB

∣∣2︸ ︷︷ ︸ραA+βB

= |αψA|2︸ ︷︷ ︸|α|2ρA

+ |βψB|2︸ ︷︷ ︸|β|2ρB

+ 2|αβψAψB| cos(ϕA+ϕα−ϕB−ϕβ)︸ ︷︷ ︸interference terms

I Description of the interference setup in the double slit experiment

1) Initial wavefunction between emission (t=0) and slits (t=tAB): ψ(~x, t)

2) Wf. at t&tAB (right after the slits): ψ(~x, t+AB) ≈ αδA(~x−~xA) + βδB(~x−~xB)with δ•(~x−~x•) ≡ wf. localized on the respective slit (δ•=0 away from the slit)and α, β ≡ coefficients depending on the “experimental details”

3) Wf. at tS=tAB+∆t (just before screen): ψ(~x, tS) ≈ αψA(~x,∆t)+βψB(~x,∆t)with ψ•(~x,∆t) ≡ the wf. developed from δ•(~x−~x•) in time ∆t

⇒ Distribution on screen: ρ(~xS) ≈ |αψA(~xS,∆t) + βψB(~xS,∆t)|2

I Dirac delta function (mathematical intermezzo)

δ(x) ≡ a generalized function (distribution) ≡ limit of a series of ordinary

functions: δ(x) = limε→0

δε(x) with, e.g.: δε(x)≡

1ε for x∈[− ε2 ,+

ε2 ]

0 otherwise

⇒ Support [δ(x)] ≡ x=0 &+∞∫−∞

δ(x) dx = 1

Other limiting realizations of δ-function:

δε(x) = 1π

εε2+x2 (Cauchy or Breit-Wigner form)

δε(x) = 1√2πε2

e−x2

2ε2 (Gaussian form)

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