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Second Quantization for a System of Fermions
APPENDIX D
Second Quantization for a System of Fermions
The second quantization method is quite useful as a systematic approach to the many-fermion problem. Such a procedure is needed because it is difficult to work directly with a many-electron wave function. A brief survey of second quantization for a system of fermions is given here. The corresponding procedures for phonons and photons were discussed in Section 2.3 and Appendix B, respectively. For a more complete account, consuh Schweber (1961, Chap. 6). We begin with a set of single-particle wave functions, including spin, which will be denoted here by W A T W - These functions wih be assumed to be orthonormal, but they are otherwise arbitrary. Frequently, the space part of Ui^ is taken to be a plane wave. Alternately, Bloch functions may be em ployed. The index k designates the single-particle state, while χ represents both space and spin coordinates. Now consider a set of TV noninteracting particles, such that each particle wih be in one of the states w^. Let nj, be the number of particles in k\ This can be either 1 or 0. The quantity is called the occupation number. Such quantities, rather than the particle coordinates, will be the fundamental variables of the theory. A wave function for the system of noninteracting particles can be constructed as an antisymmetrized product of single-particle functions. Such antisymmetrized functions can be expressed as determinants. In order not to have an ambiguity of algebraic sign, we agree to arrange the indices k of the single-particle states in a definite order, such that k^ occurs before k2y etc. There must be TV states having nonzero occupation numbers: The 935
936
APPENDIX D
wave function for the system is
Ψ ( χ , · · · χ ; , ) = (7ν!) - 1 / 2
.
(D.l)
Note that no two states can have the same quantum numbers (no two ki are equal). Likewise, the coordinates of two particles cannot be equal; otherwise, the wave function vanishes. If the basis functions form a complete set, the set of ah possible independent determinantal functions ( D . l ) is a complete set for the expansion of the many-body function for interacting particles. It can be verified easily that ( D . l ) is normahzed. This is accomphshed by the factor (Ν1)~^^^. Likewise, functions with any differences in the occupied single-particle states are orthogonal to ( D . l ) . It is necessary to determine the effect of operators on determinantal functions. The operators to be considered are of two types: single-particle and two-particle operators. A single-particle operator is one which depends on the coordinates of a single particle, e . g . , / ( x , ) ; however, in considering a system of identical particles, we always encounter sums of such operators, identical except for the coordinate, over all the particles of the system. Such a sum wih be referred to as a single-particle operator, the sum being implied. Thus,
Similarly a two-particle operator G wih refer to a sum of operators g(x/, \j) that depend on the coordinates of a pair of particles. Ah distinct pairs are included in the sum. However, a particle does not interact with hself, so there is no term of the form g(x/, X / ) in G: (D.3) ij>i
Consider first a single-particle operator F, Its matrix elements wih be zero unless: (1) all occupation numbers are unchanged (diagonal element) or (2) the states differ in regard to a single occupation number. In the second case, consider the element (B\F\A), \A} and \B} being represented by wave functions of the form ( D . l ) . These states are specified by the occupation numbers of the single-particle states. Suppose that in Β the occupation number of state / is 1 and that of η is 0, while in A that of / is 0 and Λ is 1. It is necessary to introduce fi„, the matrix element between
Second Quantization for System of Fermions
937
single-particle states: fln =
unx)f(x)u„(x)d'x.
(D.4)
Integrations include summations over spin coordinates. It is easy to see that the N-particle matrix element is proportional to / / „ ; however, the sign is difficuh to specify. Some experimentation, which wih be left to the reader, shows that (/ < n) {B\F\A)
= < . . . 1 , . . . 0 , . . . | F | ...Or-- 1„--·> = / / . ( - l ) ^ ^ ' ^ ' ' " - ^
(D.5)
in which S is the sum of the occupation numbers from (and including) / + 1 through η - 1. In general, S(iJ)=
Σ
(D.6)
a=i
If / > the exponent in (D.5) is S{n + \,l - n): We always count the numbers of occupied states between the one of smaller index in our conven tional order and the one of larger index. If / = y ± 1, S = 0. The quantity S specifies the number of permutations of the required to make the determinants line up with all rows identical except for the one substituted. The diagonal matrix element of F (between identical determinants), (A \F\A), is simply <\F\A)= ΣΛ·«/. (D.7) /•
In order to keep the complicated business of algebraic signs in order, we introduce operators c„, etc., defined so that removes row η from the determinant. This operator connects an Λ^-particle state with one containing TV - 1 particles: c„ is said to be an annihilation operator for state n. We define < . . . 0 „ . . . | c j . . . l , . . . > = (-l)^<^''^-i>. (D.8) Other elements of c„ are zero. The operator c„ is not Hermhian. Its adjoint cl evidently adds an electron in state η if none was present inhially. It is therefore a creation operator: < . . . l „ . . . k I | . . . 0 , . . . > = (-l)^^^''^-^>.
(D.9)
Equations (D.8) and (D.9) suffice to enable us to work out the commutation rules algebraically. In particular (for η > /), O^-lr''0„'''\cJcJ-''0r-K'''>
=
=
(-lf'''-'^^^^^ (-i)^('+i.«-i).
(D.lOa)
Note the omission of the contribution from the state / (which is empty
938
APPENDIX D
initially) to the algebraic sign. If we consider the operators in opposite order c„c/, the state / is present when c„ acts, and an additional negative sign is obtained: <...lr--0„...|c,c/|...0,...l,...>
= -(-l)^('-i'-i).
(D.lOb)
The reader should investigate what happens if / > « . In the case / = the operator clc^ is diagonal with unit matrix element if η is occupied initially, and is zero otherwise. In the case of we obtain unity if is initially empty. These results can be combined as
η
c„cl,
cjc„ + c„cj = δ„,.
(D.U)
We can also obtain qc„
+ c,ci
=
cjcl + clcj = 0.
Equations ( D . l l ) and (D.12) are the fundamental anticommutation relations for a system of fermions. The operator c j q is called the number operator n i ^ c j q . (D.13) The basis states of interest are eigenstates of the number operator. Recah that in the case of a Boson system, such as the electromagnetic field, the anticommutation rules become commutation rules. We can now see by comparison of (D.5) and (D.lOa) that the singleparticle operator F can be expressed as F=
(D.14)
IfinCjCn. In
The resuhs for the two-body operator (D.3) are similar. It turns out that G =i l
(D.15)
giuncjclc^ci,
ikln
in which the matrix element gikln
=
giki„
Uf(\i)uti\2)g(\i,
is given by X2)W/(Xi)W„(X2) d \
d%.
(D. 16)
J
It is important to note that the order of the operators in (D.15), (/, k, n, /), is different from that appearing in the matrix element (/, k, /, n). It is frequently convenient to introduce the electron field operators ^ ( x ) , ψ^{χ) through the definitions ^(x) =
Σ Uk(x)c,; k
ψ\χ)
=
Σ k
(D.17)
(
Second Quantization for System of Fermions
939
We will suppose that the single-particle functions u are complete in the sense that Σ u,(x)ut(x') = δ(χ - X'). (D.18) k It is then easy to obtain the anticommutation rules of ψ, ψ^: ψ(χ)ψ\χ')
+ ψ\χ')ψ{χ)
= δ(χ - χ')
(D.19)
and ψ{χ)ψ(χ') + Ψ(ΧΊΨ(Χ) = ψ\χ)ψ\χ')
+ ^^(χ')^^(χ) = 0.
(D.20)
The operator ^(χ) destroys a particle at the point x, while ^^^(x) creates a particle at x. Let us suppose the Hamihonian for the system of interacting particles has the form H= Ho + Hi, (D.21a) with Ηο=Σ [TM + F(x,)] (D.21b) and Hi=
Σ V(^i,^A
(0-21C)
iJ > i
In Eq. (D.21b), Tixf) and F(x,) are the single-particle kinetic and potential energies, respectively, while K(x,, x^) in (D.21c) is the two-body interaction potential, for example, the Coulomb interaction. It is usually convenient to choose the basis functions to be eigenfunctions of the single-particle Hamiltonian: [T(x) + V(x)]u,ix)
= E,u,(x),
(D.22)
If this is done, the Hamiltonian is expressed in second quantized for as EAck + 1 Σ Vukiclcjqc,, (D.23) k ijkl An equivalent expression can be obtained in terms of the field operators /: Η=Σ
Η =
ψ\χ)[Τ(χ)
+ Κ(χ)]^(χ) d^x
-h ^ [ /(x)^^^(x')K(x, χ')ψ(χ' )ψ(χ) d^xd^x',
(D.24)
REFERENCE Schweber, S. (1961). " A n Introduction to Relativisitic Quantum Field Theory." Harper, New York.
Second Quantization for a System of Fermions
The second quantization method is quite useful as a systematic approach to the many-fermion problem. Such a procedure is needed because it is difficult to work directly with a many-electron wave function. A brief survey of second quantization for a system of fermions is given here. The corresponding procedures for phonons and photons were discussed in Section 2.3 and Appendix B, respectively. For a more complete account, consuh Schweber (1961, Chap. 6). We begin with a set of single-particle wave functions, including spin, which will be denoted here by W A T W - These functions wih be assumed to be orthonormal, but they are otherwise arbitrary. Frequently, the space part of Ui^ is taken to be a plane wave. Alternately, Bloch functions may be em ployed. The index k designates the single-particle state, while χ represents both space and spin coordinates. Now consider a set of TV noninteracting particles, such that each particle wih be in one of the states w^. Let nj, be the number of particles in k\ This can be either 1 or 0. The quantity is called the occupation number. Such quantities, rather than the particle coordinates, will be the fundamental variables of the theory. A wave function for the system of noninteracting particles can be constructed as an antisymmetrized product of single-particle functions. Such antisymmetrized functions can be expressed as determinants. In order not to have an ambiguity of algebraic sign, we agree to arrange the indices k of the single-particle states in a definite order, such that k^ occurs before k2y etc. There must be TV states having nonzero occupation numbers: The 935
936
APPENDIX D
wave function for the system is
Ψ ( χ , · · · χ ; , ) = (7ν!) - 1 / 2
.
(D.l)
Note that no two states can have the same quantum numbers (no two ki are equal). Likewise, the coordinates of two particles cannot be equal; otherwise, the wave function vanishes. If the basis functions form a complete set, the set of ah possible independent determinantal functions ( D . l ) is a complete set for the expansion of the many-body function for interacting particles. It can be verified easily that ( D . l ) is normahzed. This is accomphshed by the factor (Ν1)~^^^. Likewise, functions with any differences in the occupied single-particle states are orthogonal to ( D . l ) . It is necessary to determine the effect of operators on determinantal functions. The operators to be considered are of two types: single-particle and two-particle operators. A single-particle operator is one which depends on the coordinates of a single particle, e . g . , / ( x , ) ; however, in considering a system of identical particles, we always encounter sums of such operators, identical except for the coordinate, over all the particles of the system. Such a sum wih be referred to as a single-particle operator, the sum being implied. Thus,
Similarly a two-particle operator G wih refer to a sum of operators g(x/, \j) that depend on the coordinates of a pair of particles. Ah distinct pairs are included in the sum. However, a particle does not interact with hself, so there is no term of the form g(x/, X / ) in G: (D.3) ij>i
Consider first a single-particle operator F, Its matrix elements wih be zero unless: (1) all occupation numbers are unchanged (diagonal element) or (2) the states differ in regard to a single occupation number. In the second case, consider the element (B\F\A), \A} and \B} being represented by wave functions of the form ( D . l ) . These states are specified by the occupation numbers of the single-particle states. Suppose that in Β the occupation number of state / is 1 and that of η is 0, while in A that of / is 0 and Λ is 1. It is necessary to introduce fi„, the matrix element between
Second Quantization for System of Fermions
937
single-particle states: fln =
unx)f(x)u„(x)d'x.
(D.4)
Integrations include summations over spin coordinates. It is easy to see that the N-particle matrix element is proportional to / / „ ; however, the sign is difficuh to specify. Some experimentation, which wih be left to the reader, shows that (/ < n) {B\F\A)
= < . . . 1 , . . . 0 , . . . | F | ...Or-- 1„--·> = / / . ( - l ) ^ ^ ' ^ ' ' " - ^
(D.5)
in which S is the sum of the occupation numbers from (and including) / + 1 through η - 1. In general, S(iJ)=
Σ
(D.6)
a=i
If / > the exponent in (D.5) is S{n + \,l - n): We always count the numbers of occupied states between the one of smaller index in our conven tional order and the one of larger index. If / = y ± 1, S = 0. The quantity S specifies the number of permutations of the required to make the determinants line up with all rows identical except for the one substituted. The diagonal matrix element of F (between identical determinants), (A \F\A), is simply <\F\A)= ΣΛ·«/. (D.7) /•
In order to keep the complicated business of algebraic signs in order, we introduce operators c„, etc., defined so that removes row η from the determinant. This operator connects an Λ^-particle state with one containing TV - 1 particles: c„ is said to be an annihilation operator for state n. We define < . . . 0 „ . . . | c j . . . l , . . . > = (-l)^<^''^-i>. (D.8) Other elements of c„ are zero. The operator c„ is not Hermhian. Its adjoint cl evidently adds an electron in state η if none was present inhially. It is therefore a creation operator: < . . . l „ . . . k I | . . . 0 , . . . > = (-l)^^^''^-^>.
(D.9)
Equations (D.8) and (D.9) suffice to enable us to work out the commutation rules algebraically. In particular (for η > /), O^-lr''0„'''\cJcJ-''0r-K'''>
=
=
(-lf'''-'^^^^^ (-i)^('+i.«-i).
(D.lOa)
Note the omission of the contribution from the state / (which is empty
938
APPENDIX D
initially) to the algebraic sign. If we consider the operators in opposite order c„c/, the state / is present when c„ acts, and an additional negative sign is obtained: <...lr--0„...|c,c/|...0,...l,...>
= -(-l)^('-i'-i).
(D.lOb)
The reader should investigate what happens if / > « . In the case / = the operator clc^ is diagonal with unit matrix element if η is occupied initially, and is zero otherwise. In the case of we obtain unity if is initially empty. These results can be combined as
η
c„cl,
cjc„ + c„cj = δ„,.
(D.U)
We can also obtain qc„
+ c,ci
=
cjcl + clcj = 0.
Equations ( D . l l ) and (D.12) are the fundamental anticommutation relations for a system of fermions. The operator c j q is called the number operator n i ^ c j q . (D.13) The basis states of interest are eigenstates of the number operator. Recah that in the case of a Boson system, such as the electromagnetic field, the anticommutation rules become commutation rules. We can now see by comparison of (D.5) and (D.lOa) that the singleparticle operator F can be expressed as F=
(D.14)
IfinCjCn. In
The resuhs for the two-body operator (D.3) are similar. It turns out that G =i l
(D.15)
giuncjclc^ci,
ikln
in which the matrix element gikln
=
giki„
Uf(\i)uti\2)g(\i,
is given by X2)W/(Xi)W„(X2) d \
d%.
(D. 16)
J
It is important to note that the order of the operators in (D.15), (/, k, n, /), is different from that appearing in the matrix element (/, k, /, n). It is frequently convenient to introduce the electron field operators ^ ( x ) , ψ^{χ) through the definitions ^(x) =
Σ Uk(x)c,; k
ψ\χ)
=
Σ k
(D.17)
(
Second Quantization for System of Fermions
939
We will suppose that the single-particle functions u are complete in the sense that Σ u,(x)ut(x') = δ(χ - X'). (D.18) k It is then easy to obtain the anticommutation rules of ψ, ψ^: ψ(χ)ψ\χ')
+ ψ\χ')ψ{χ)
= δ(χ - χ')
(D.19)
and ψ{χ)ψ(χ') + Ψ(ΧΊΨ(Χ) = ψ\χ)ψ\χ')
+ ^^(χ')^^(χ) = 0.
(D.20)
The operator ^(χ) destroys a particle at the point x, while ^^^(x) creates a particle at x. Let us suppose the Hamihonian for the system of interacting particles has the form H= Ho + Hi, (D.21a) with Ηο=Σ [TM + F(x,)] (D.21b) and Hi=
Σ V(^i,^A
(0-21C)
iJ > i
In Eq. (D.21b), Tixf) and F(x,) are the single-particle kinetic and potential energies, respectively, while K(x,, x^) in (D.21c) is the two-body interaction potential, for example, the Coulomb interaction. It is usually convenient to choose the basis functions to be eigenfunctions of the single-particle Hamiltonian: [T(x) + V(x)]u,ix)
= E,u,(x),
(D.22)
If this is done, the Hamiltonian is expressed in second quantized for as EAck + 1 Σ Vukiclcjqc,, (D.23) k ijkl An equivalent expression can be obtained in terms of the field operators /: Η=Σ
Η =
ψ\χ)[Τ(χ)
+ Κ(χ)]^(χ) d^x
-h ^ [ /(x)^^^(x')K(x, χ')ψ(χ' )ψ(χ) d^xd^x',
(D.24)
REFERENCE Schweber, S. (1961). " A n Introduction to Relativisitic Quantum Field Theory." Harper, New York.