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  • Quantum Mechanics 3:

    the quantum mechanics of many-particle systems

    W.J.P. Beenakker

    Academic year 2017 – 2018

    Contents of the common part of the course:

    1) Occupation-number representation

    2) Quantum statistics (up to § 2.5.3)

    Module 1 (high-energy physics):

    2) Quantum statistics (rest)

    3) Relativistic 1-particle quantum mechanics

    4) Quantization of the electromagnetic field

    5) Many-particle interpretation of the relativistic quantum mechanics

    The following books have been used:

    F. Schwabl, “Advanced Quantum Mechanics”, third edition (Springer, 2005);

    David J. Griffiths, “Introduction to Quantum Mechanics”, second edition

    (Prentice Hall, Pearson Education Ltd, 2005);

    Eugen Merzbacher, “Quantum Mechanics”, third edition (John Wiley & Sons, 2003);

    B.H. Bransden and C.J. Joachain, “Quantum Mechanics”, second edition

    (Prentice Hall, Pearson Education Ltd, 2000).

  • Contents

    1 Occupation-number representation 1

    1.1 Summary on identical particles in quantum mechanics . . . . . . . . . . . . 1

    1.2 Occupation-number representation . . . . . . . . . . . . . . . . . . . . . . 5

    1.2.1 Construction of Fock space . . . . . . . . . . . . . . . . . . . . . . . 5

    1.3 Switching to a continuous 1-particle representation . . . . . . . . . . . . . 12

    1.3.1 Position and momentum representation . . . . . . . . . . . . . . . . 14

    1.4 Additive many-particle quantities and particle conservation . . . . . . . . . 15

    1.4.1 Additive 1-particle quantities . . . . . . . . . . . . . . . . . . . . . 16

    1.4.2 Additive 2-particle quantities . . . . . . . . . . . . . . . . . . . . . 19

    1.5 Heisenberg picture and second quantization . . . . . . . . . . . . . . . . . 20

    1.6 Examples and applications: bosonic systems . . . . . . . . . . . . . . . . . 23

    1.6.1 The linear harmonic oscillator as identical-particle system . . . . . 23

    1.6.2 Forced oscillators: coherent states and quasi particles . . . . . . . . 26

    1.6.3 Superfluidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

    1.6.4 Superfluidity for weakly repulsive spin-0 bosons (part 1) . . . . . . 33

    1.6.5 Intermezzo: the Bogolyubov transformation for bosons . . . . . . . 38

    1.6.6 Superfluidity for weakly repulsive spin-0 bosons (part 2) . . . . . . 40

    1.6.7 The wonderful world of superfluid 4He: the two-fluid model . . . . 42

    1.7 Examples and applications: fermionic systems . . . . . . . . . . . . . . . . 44

    1.7.1 Fermi sea and hole theory . . . . . . . . . . . . . . . . . . . . . . . 44

    1.7.2 The Bogolyubov transformation for fermions . . . . . . . . . . . . . 46

    2 Quantum statistics 49

    2.1 The density operator (J. von Neumann, 1927) . . . . . . . . . . . . . . . . 52

    2.2 Example: polarization of a spin-1/2 ensemble . . . . . . . . . . . . . . . . 55

    2.3 The equation of motion for the density operator . . . . . . . . . . . . . . . 58

    2.4 Quantum mechanical ensembles in thermal equilibrium . . . . . . . . . . . 59

    2.4.1 Thermal equilibrium (thermodynamic postulate) . . . . . . . . . . . 61

    2.4.2 Canonical ensembles (J.W. Gibbs, 1902) . . . . . . . . . . . . . . . 61

    2.4.3 Microcanonical ensembles . . . . . . . . . . . . . . . . . . . . . . . 65

    2.4.4 Grand canonical ensembles (J.W. Gibbs, 1902) . . . . . . . . . . . . 66

    2.4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

    2.5 Fermi gases at T=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

    2.5.1 Ground state of a Fermi gas . . . . . . . . . . . . . . . . . . . . . . 74

    2.5.2 Fermi gas with periodic boundary conditions . . . . . . . . . . . . . 76

    2.5.3 Fermi-gas model for conduction electrons in a metal . . . . . . . . . 77

    2.5.4 Fermi-gas model for heavy nuclei . . . . . . . . . . . . . . . . . . . 80

    2.5.5 Stars in the final stage of stellar evolution . . . . . . . . . . . . . . 84

  • 2.6 Systems consisting of non-interacting particles . . . . . . . . . . . . . . . . 87

    2.7 Ideal gases in three dimensions . . . . . . . . . . . . . . . . . . . . . . . . 90

    2.7.1 Classical gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

    2.7.2 Bose gases and Bose–Einstein condensation (1925) . . . . . . . . . 93

    2.7.3 Fermi gases at T 6= 0 (no exam material) . . . . . . . . . . . . . . . 98 2.7.4 Experimental realization of Bose–Einstein condensates . . . . . . . 101

    3 Relativistic 1-particle quantum mechanics 103

    3.1 First attempt: the Klein–Gordon equation (1926) . . . . . . . . . . . . . . 104

    3.2 Second attempt: the Dirac equation (1928) . . . . . . . . . . . . . . . . . . 109

    3.2.1 The probability interpretation of the Dirac equation . . . . . . . . . 111

    3.2.2 Covariant formulation of the Dirac equation . . . . . . . . . . . . . 112

    3.2.3 Dirac spinors and Poincaré transformations . . . . . . . . . . . . . 113

    3.2.4 Solutions to the Dirac equation . . . . . . . . . . . . . . . . . . . . 116

    3.2.5 The success story of the Dirac theory . . . . . . . . . . . . . . . . . 118

    3.2.6 Problems with the 1-particle Dirac theory . . . . . . . . . . . . . . 122

    4 Quantization of the electromagnetic field 123

    4.1 Classical electromagnetic fields: covariant formulation . . . . . . . . . . . . 123

    4.1.1 Solutions to the free electromagnetic wave equation . . . . . . . . . 125

    4.1.2 Electromagnetic waves in terms of harmonic oscillations . . . . . . . 127

    4.2 Quantization and photons . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

    4.2.1 Particle interpretation belonging to the quantized electromagnetic field130

    4.2.2 Photon states and density of states for photons . . . . . . . . . . . 131

    4.2.3 Zero-point energy and the Casimir effect . . . . . . . . . . . . . . . 132

    4.2.4 Continuum limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

    4.3 A new perspective on particle –wave duality . . . . . . . . . . . . . . . . . 134

    4.4 Interactions with non-relativistic quantum systems . . . . . . . . . . . . . 135

    4.4.1 Classical limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

    4.4.2 Spontaneous photon emission (no exam material) . . . . . . . . . . 139

    4.4.3 The photon gas: quantum statistics for photons . . . . . . . . . . . 141

    5 Many-particle interpretation of the relativistic QM 144

    5.1 Quantization of the Klein–Gordon field . . . . . . . . . . . . . . . . . . . . 145

    5.2 Quantization of the Dirac field . . . . . . . . . . . . . . . . . . . . . . . . . 149

    5.2.1 Extra: the 1-(quasi)particle approximation for electrons . . . . . . . 153

    A Fourier series and Fourier integrals i

    A.1 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

    A.2 Fourier integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

  • A.2.1 Definition of the δ function . . . . . . . . . . . . . . . . . . . . . . . iii

    B Properties of the Pauli spin matrices iv

    C Lagrange-multiplier method v

    D Special relativity: conventions and definitions v

    E Electromagnetic orthonormality relations ix

  • 1 Occupation-number representation

    In this chapter the quantum mechanics of identical-particle systems will be

    worked out in detail. The corresponding space of quantum states will be con-

    structed in the occupation-number representation by employing creation and

    annihilation operators. This will involve the introduction of the notion of quasi

    particles and the concept of second quantization.

    Similar material can be found in Schwabl (Ch. 1,2 and 3) and Merzbacher

    (Ch. 21,22 and the oscillator part of Ch. 14).

    1.1 Summary on identical particles in quantum mechanics

    Particles are called identical if they cannot be distinguished by means of specific intrinsic

    properties (such as spin, charge, mass, · · · ).

    This indistinguishability has important quantum mechanical implications in situations

    where the wave functions of the identical particles overlap, causing the particles to be ob-

    servable simultaneously in the same spatial region. Examples are the interaction region of

    a scattering experiment or a gas container. If the particles are effectively localized, such as

    metal ions in a solid piece of metal, then the identity of the particles will not play a role.

    In those situations the particles are effectively distinguishable by means of their spatial

    coordinates and their wave functions have a negligible overlap.

    For systems consisting of identical particles two additional constraints have to be imposed

    while setting up quantum mechanics (QM).

    • Exchanging the particles of a system of identical particles should have no observ- able effect, otherwise the particles would actually be distinguishable. This gives

    rise to the concept of permutation degeneracy, i.e. for such a system the expecta-

    tion value for an arbitrary many-particle observable should not change upon inter-

    changing the identical particles in the state function. As a consequence, quantum

    mechanical observables for identical-particle systems should be symmetric functions

    of the separate 1-particle observables.

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