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Quantum Mechanics 3:

the quantum mechanics of many-particle systems

Common part of the course (3rd quarter)

W.J.P. Beenakker

Academic year 2018 – 2019

Contents of the common part of the course:

1) Occupation-number representation

2) Quantum statistics (up to § 2.5)

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

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

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.

• Due to the permutation degeneracy, it seems to be impossible to fix the quantum state of an identical-particle system by means of a complete measurement. Nature

has bypassed this quantum mechanical obstruction through the

symmetrization postulate: identical-particle systems can be described by means

of either totally symmetric state functions if the particles are bosons or totally

antisymmetric state functions if the particles are fermions. In the non-relativistic

QM it is an empirical fact that no mixed symmetry occurs in nature.

1

In this chapter the symmetrization postulate will be reformulated in an alternative

way, giving rise to the conclusion that only totally symmetric/antisymmetric state

functions will fit the bill.

Totally symmetric state functions can be represented in the so-called q-representation by

ψS(q1, · · · , qN , t), where q1 , · · · , qN are the “coordinates” of the N separate identical par- ticles. These coordinates are the eigenvalues belonging to a complete set of commuting

1-particle observables q̂ . Obviously there are many ways to choose these coordinates. A

popular choice is for instance qj = (spatial coordinate ~rj , magnetic spin quantum number

msj ≡ σj , · · · ), where the dots represent other possible internal (intrinsic) degrees of free- dom of particle j . For a symmetric state function we have

∀̂ P

P̂ ψS(q1, · · · , qN , t) = ψS(qP (1), · · · , qP (N) , t) = ψS(q1, · · · , qN , t) , (1)

with P̂ a permutation operator that permutes the sets of coordinates of the identical

particles according to

q1 → qP (1) , q2 → qP (2) , · · · , qN → qP (N) . (2)

Particles whose quantum states are described by totally symmetric state functions are

called bosons. They have the following properties:

- bosons have integer spin (see Ch. 4 and 5);

- bosons obey so-called Bose–Einstein statistics (see Ch. 2);

- bosons prefer to be in the same quantum state.

Totally antisymmetric state functions can be represented in the q-representation by

ψA(q1, · · · , qN , t), with

∀ P̂

P̂ ψA(q1, · · · , qN , t) = ψA(qP (1), · · · , qP (N) , t) =

+ ψA(q1, · · · , qN , t) even P̂

− ψA(q1, · · · , qN , t) odd P̂ . (3)

A permutation P̂ is called even/odd if it consists of an even/odd number of two-particle

interchanges. Particles whose quantum states are described by totally antisymmetric state

functions are called fermions. They have the following properties:

- fermions have half integer spin (see Ch. 5);

- fermions obey so-called Fermi–Dirac statistics (see Ch. 2);

- fermions are not allowed to be in the same quantum state.

2

If the above (anti)symmetrization procedure results in spatial symmetrization, the particles

have an increased probability to be in each other’s vicinity. Note that this can apply to

identical fermions if they happen to be in an antisymmtric spin state. On the other hand,

spatial antisymmetrization gives rise to a decreased probability for the particles to be in

each other’s vicinity. Note that this can apply to identical spin-1 bosons if they happen to

be in an antisymmtric spin state.

Isolated non-interacting many-particle systems: many-particle systems with negi-

gible interactions among the particles are called non-interacting many-particle systems.

The properties of such sys