Appendix B Second Quantization

The method of second quantization is very useful in the study of systems of weakly interacting particles. In particular, the second quantization is an efficient tool to describe collective elementary excitations such as phonons and magnons.

B.1 First Quantization: Symmetric and Antisymmetric Wave Functions

One considers a system of N identical, indiscernible particles. The Hamiltonian is written as

where Hi is a single-particle term such as the kinetic energy of the particle i and F(r,, r,) the interaction between two particles at r, and r j.

In the first quantization, the postulate on the symmetry of the wave function allows us to distinguish bosons and fermions: For bosons, the permutation of two particles in their states does not change the sign of the corresponding wave function, while

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ISBN 978-981-4877-42-8 (Hardcover), 978-1-003-12110-7 (eBook) www.jennystanford.com for fermions the permutation does change its sign. One of the consequences of the symmetry postulate is that in the case of bosons an individual state can contain any number of particles while in the case of fermions, each individual state can have zero or one particle (see Appendix A). This is known as the Pauli principle. In the method of second quantization, it is the symmetiy of the operators which allows one to distinguish bosons and fermions as seen in the following.

B.2 Second Quantization: Representation of Microstates by Occupation Numbers

Since the particles are identical and indiscernible, one can imagine that they have the same "list” of individual states: Each of them takes one state of the list. A state / is characterized by some physical parameters such as the wave vector and the spin state, k, and 07 (/ = 1, • • • , N). This state of the list is occupied by n, particles. One can define a microstate of the system by the numbers of particles (n,} in the individual states (к/, <т,)> / = 1, • • • , N. All possible different particle distributions {n, } constitute the ensemble of microstates of the system. One says that the system is defined by a "state vector" given by

where nk is the number of particles occupying the individual state k. This state vector replaces the wave function of the Schrodinger equation. As for wave functions, one imposes that the state vectors form a complete set of orthogonal states. One has

where 8„'kik is the Kronecker symbol.

B.2.1 The Case of Bosons

One introduces the operators ak and ak defined by the following relations:

As seen in the kets, operator creates a particle while operator ak destroys a particle in the state к when they operate on |Ф>. For this reason, they are called creation and annihilation operators, respectively. By the above definitions, one sees that

This relation shows that operator akak has the eigenvalue nk which is the number of particles in the state к. One calls, therefore, акак operator "occupation number.”

In addition, using (B.4) and (B.5), one gets

Comparing this relation to (B.6), one obtains

Now, if к ф к', by using (B.3) one has

Combining with (B.8) one can write

One can show in the same manner that for arbitrary к and к1, one has

Relations (B. 11) and (B.12) are called "commutation relations.” Hamiltonian (B.l) in the case of bosons can be written in the second quantization as (see demonstration in Ref. [88])

where

The wave function , (r) describes the individual state i of the particle at r. For example, in the case of a plane wave one has ,(r) = exp(;k, • r)//Q. where k, is the wave vector and Q the system volume.

B.2.2 The Case of Fermions

In the case of fermions, one can demonstrate the general Hamiltonian starting from the Schrodinger equation can be written as Eq. (B.26) below (see demonstration in Ref. [88]), using the creation and annihilation operators b^ and b/ defined by

where [/] is, by convention, the number of particles occupying the states on the left of the state / in the ket. It is noted that in some books the coefficients in front of the ket of (B.16)-(B.17) are given by yjTJ and ^/1 - rtf instead of rif and (1 — n/). However, one can verify that they are equivalent because л/ is 0 or 1. One has

from which one has bfbg = —bgbf, or equivalently In the same manner, one obtains for arbitrary / and g and

Now if / = g, one has

where in the last line, one has used nД2 - ny) = rif because

One calls b+^bf operator "occupation number" because its eigenvalue when operating on the ket is n/. Besides,

from which

Comparing (B.24) to (B.21), one obtains bfb^j = 1 — b^bf, namely [.b f, b'f] = 1. Combining with (B.20), one can write

Relations (B.18), (B. 19) and (B.25) are called "fermion anticommutation relations.”

Hamiltonian (B.l) in the case of fermions is written in the second quantization as (see demonstration in Ref. [88])

where

Due to the anticommutation of the operators, one should respect the order of the operators as well as that of the arguments r and r' of ф functions in (W|V|/7). A permutation of the operators should obey the anticommutation relations (B.18), (B.19) and (B.25).

In practice, one can use Hamiltonians (B.13) and (B.26) are starting points to study systems of bosons and fermions if one knows H (r) and V (r, r'). These forms of Hamiltonian are very useful in the study of systems of weakly interacting particles [186].

 
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