Fujita S., Ito K., Godoy S. Quantum Theory of Conducting Matter - Superconductivity
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A.3 Fermion Creation and Annihilation Operators 233
A.3 Fermion Creation and Annihilation Operators
In this section we treat a system of identical fermions in a parallel manner.
The quantum states for fermions, by postulate, are represented by antisymmetric
kets:
|α
a
α
b
···α
g
A
≡ A
|α
(1)
a
|α
(2)
b
···|α
(N)
g
, (A.34)
where
A ≡
1
√
N!
P
δ
P
P, (A.35)
is the antisymmetrizing operator, with δ
P
being +1or−1 according to whether P
is even or odd. Each antisymmetric ket in Equation (A.34) is characterized such that
it changes its sign if an odd permutation of particle indices is applied to it, and the
fermion states a, b,...,g are all different. Similar to a boson system, we can intro-
duce observables n
1
, n
2
, . . . each with eigenvalues 0 or 1, representing the number of
fermions in the states α
1
, α
2
,...respectively. The many-fermion occupation-number
state can be represented as shown in Figure A.2.
We can also introduce a set of linear operators (η, η
†
), one pair (η
a
,η
†
a
) for each
state α
a
, satisfying the Fermi anticommutation rules:
{η
a
,η
†
b
}≡η
a
η
†
b
+η
†
b
η
a
= δ
ab
, {η
a
,η
b
}={η
†
a
,η
†
b
}=0. (A.36)
The number of fermions in the state α
a
is again represented by
n
a
= η
†
a
η
a
= n
†
a
. (A.37)
Fig. A.2 A many-fermion
state is represented by the set
of the numbers {n
j
}
occupying the states
{p
j
≡ 2π L
−1
j}. Each n
j
is
restricted to 0 or 1
p
0
p
2
p
1
p
1
p
2
, ,0,1,10, 1,
21012
nnnnn
234 Appendix A Second Quantization
Using Equations (A.36), we obtain
n
2
a
= η
†
a
η
a
η
†
a
η
a
= η
†
a
(1 −η
†
a
η
a
)η
a
= η
†
a
η
a
= n
a
or n
2
a
−n
a
= 0. (A.38)
If an eigenket of n
a
belonging to the eigenvalue n
a
is denoted by |n
a
, then Equa-
tion (A.38) yields
(n
2
a
−n
a
)|n
a
=(n
a
2
−n
a
)|n
a
=0. (A.39)
Since |n
a
= 0, we obtain n
a
(n
a
− 1) = 0, meaning that the eigenvalues n
a
are
either 0 or 1 as required:
n
a
= 0or1. (A.40)
Similarly to bosons, we can show that
|α
a
α
b
···α
g
A
= η
†
a
η
†
b
···η
†
g
|⌽
0
, (A.41)
which is normalized to unity.
Observables describing the system of fermions can be expressed in terms of op-
erators η and η
†
, and the results have the same forms as for the case of bosons,
Equations (A.31) and (A.32).
In summary both states and observables for a system of identical particles
(fermions, bosons) can be expressed in terms of creation and annihilation opera-
tors. This formalism, called the second quantization formalism, has some notable
advantages over the usual Schr
¨
odinger formalism. First, the permutation-symmetry
property of the quantum particles is represented simply in the form of Bose commu-
tation (or Fermi anticommutation) rules. Second, observables in second quantization
are defined for an arbitrary number of particles so that the formalism may apply to
systems in which the number of particles is not fixed but variable. Third, and most
importantly, all relevant quantities (states and observables) can be defined referring
only to the single-particle states. This property allows us to describe the motion
of the many-body system in the 3D space. We shall explain this in the following
section. This is a natural description. In fact, relativistic quantum field theory can be
developed only in second quantization.
A.4 Heisenberg Equation of Motion
In the Schr
¨
odinger Picture (SP), the energy eigenvalue equation is
H |E=E |E, (A.42)
where H is the Hamiltonian and E the eigenvalue. In the position representation
this equation is written as
A.4 Heisenberg Equation of Motion 235
H(x
1
, −i⭸/⭸x
1
, x
2
, −i⭸/⭸x
2
, ···)⌿(x
1
, x
2
, ···) = E⌿, (A.43)
where ⌿ is the wave function for the system. We consider a one-dimensional motion
for conceptional and notational simplicity. [For a three-dimensional motion, (x, p)
should be replaced by (x, y, z, p
x
, p
y
, p
z
) = (r, p).] If the number of electrons N
is large, the wave function ⌿ contains many electron variables (x
1
, x
2
, ···). This
complexity in dealing with many electron variables can be avoided if we use the
second quantization formulation and the Heisenberg Picture (HP), which will be
shown in this section.
If the Hamiltonian H is the sum of single-particle Hamiltonians:
H =
j
h
( j)
, (A.44)
this Hamiltonian H can be represented by
H =
a
b
α
a
|h |α
b
η
†
a
η
b
≡
a
b
h
ab
η
†
a
η
b
, (A.45)
where η
a
(η
†
a
) are annihilation (creation) operators associated with particle-state a
and satisfying the Fermi commutation rules.
In the HP a variable ξ (t) changes in time, following the Heisenberg equation of
motion:
−i
dξ(t)
dt
= [H,ξ] ≡ Hξ −ξ H. (A.46)
Setting ξ = η
†
a
, we obtain
−i
dη
†
a
dt
= [H,η
†
a
], (A.47)
whose Hermitian conjugate is given by
i
dη
a
dt
= ([H,η
†
a
])
†
=−[H,η
a
]. (A.48)
By postulate the physical observable ξ is Hermitian: ξ
†
= ξ . Variables η
a
and η
†
a
are not Hermitian, but both obey the same Heisenberg equation of motion.
We introduce Equation (A.45) into Equation (A.47), and calculate the commu-
tator [H,η
†
a
]. In such a commutator calculation, identities Equations (A.9) and the
following identities:
236 Appendix A Second Quantization
[A, BC] ={A, B}C − B{A, C},
[AB, C] = A{B, C}−{A, C}B ,
(A.49)
are very useful. Note: The negative signs on the right-hand terms in Equations
(A.49) occur when the cyclic order is destroyed. We obtain from Equations (A.47)
and (A.48)
−i
dη
†
a
dt
=
c
b
h
cb
[η
†
c
η
b
,η
†
a
] =
c
b
h
cb
η
†
c
{η
b
,η
†
a
}=
c
h
ca
η
†
c
(A.50)
i
dη
a
dt
=
c
h
ac
η
c
. (A.51)
Equation (A.50) means that the change of the one-body operator η
†
a
is determined
by the one-body Hamiltonian h. This is the main advantage of working in the HP.
Equations (A.50) and (A.51) are valid for any single-particle state {a}.
In the field operator language Equation (A.51) reads
i
⭸ψ(r, t)
⭸t
= h(r, −i⭸/⭸r)ψ(r, t), (A.52)
which is formally identical to the Schr
¨
odinger equation of motion for a particle. The
ψ represents a field operator, however.
If the system Hamiltonian H contains an interparticle interaction
V =
1
2
d
3
r
d
3
r
v(r −r
)ψ
†
(r, t)ψ
†
(r
, t)ψ(r
, t)ψ(r, t), (A.53)
the evolution equation for ψ(r, t) is nonlinear (Problem A.4.2):
i
⭸ψ(r, t)
⭸t
= h(r, −i⭸/⭸r)ψ(r, t)
+
d
3
r
v(r −r
)ψ
†
(r
, t)ψ(r
, t)ψ(r, t). (A.54)
In quantum field theory the basic dynamical variables are particle-field operators.
The quantum statistics of the particles are given by the Bose commutation or the
A.4 Heisenberg Equation of Motion 237
Fermi commutation rules satisfied by the field operators. The evolution equations
of the field operators are intrinsically nonlinear when the interparticle interaction is
present.
Problem A.4.1. Verify that the equation of motion Equation (A.50) holds for
bosons.
Problem A.4.2. Use Equation (A.46) to verify Equation (A.54)
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Index
A
Abrikosov, 103, 104, 105, 108, 132, 138, 209
Abrikosov structure, 105, 108
Ac Josephson effect, 143, 150–153, 154
Acoustic phonon, 42, 103, 111, 161, 178, 194,
195, 200, 221
Affinity between electrons and phonons, 19
Anderson, 143, 144, 214, 215
Anderson and Rowell, 143, 144, 214
Angle-Resolved PhotoEmission Spectroscopy
(ARPES), 88, 196
Angular frequency, 17, 18, 116, 125, 150
Anisotropic optical-phonon-exchange, 193
Antiferromagnetic, 173, 174, 182, 205, 215
Antiferromagnetic insulator, 173, 182,
205, 215
Antisymmetric state, 65, 69
B
Bardeen, Cooper and Schrieffer (BCS), 10, 11,
12, 18, 29–43, 45, 46, 50, 53, 55, 58, 59,
60, 70, 79, 86, 88, 95, 105, 111, 114, 115,
132, 135, 136, 137, 155, 162, 163, 164,
167, 168, 173, 194, 210–213, 215, 220, 222
BCS coherence length, 105, 167
BCS energy gap equation, 38, 137
BCS formula, 38, 46, 58, 59, 87
BCS Hamiltonian, 12, 18, 29, 30, 32, 39, 53,
59, 111, 114, 155, 162, 163, 210, 211,
212, 222
BCS-like Hamiltonian, 162
BCS picture, 115, 220
Bednorz, 9, 157, 197, 215
Bednorz and M
¨
uller, 9, 157, 215
Bessel function, 152
Binding energy, 37, 39–40, 75, 92, 96, 140,
169, 205, 211, 221
Bloch system, 31, 34, 37, 211
Bloch theorem, 159
Blurred Fermi surface, 43
Bohr magneton, 184
Boltzmann distribution, 188, 201
Boltzmann factor, 97
Bose commutation rule, 21, 61, 65, 79, 227
Bose-condensed state, 55
Bose distribution function, 80, 188
Bose–Einstein Condensation (BEC), 73,
79–83, 84–86, 88, 157, 178, 187,
213–214, 215
Boson enhancement effect, 147
Bound electron-pair, 27
Brillouin zone, 18, 43, 161, 175, 184, 195
Bulk limit, 17, 37, 38, 54, 57, 80, 84, 165, 211
C
Cassimir, 207
Cause of superconductivity, 10, 18, 19
Center-of-mass (CM), 52, 61, 62, 63, 64, 65,
68, 69, 70, 73, 94, 113, 178
Charge state, 97, 159
Chu and Wu, 215
Circumference, 114, 118, 127, 134, 220
Classical (Boltzmann) distribution func-
tion, 188
CM motion, 62, 65, 69, 70, 178
Coherence length, 105, 108, 132, 136, 138,
157, 161, 167, 168, 204, 209, 213, 223
Coherent, 116
Complex long-range order, 6
Complex order parameter, 131, 208
Concentration, 105, 168, 173, 175, 181, 183,
189, 197, 215
Condensate density, 95, 136
Condensed pairon, 45, 56, 59, 60, 86, 91, 95,
96, 103, 106, 120, 126, 127, 131, 132, 133,
134, 135, 136, 137, 140, 152, 153, 154,
207, 208, 209, 220, 221, 224
Constriction, 126, 144, 145
243
244 Index
Continuous with no energy gap, 79
Cooper, 4, 10–12, 15, 27, 30, 39, 41–42,
51–54, 59, 60, 61, 70, 75–77, 79, 108, 135,
139, 140, 157, 162, 166, 173, 175, 181,
187, 193–196, 210, 213, 219, 221, 223
Cooper Hamiltnian, 53
Cooper pair, 4, 10, 12, 15, 27, 30, 41–42, 51,
59, 61, 75–77, 108, 135, 140, 157, 162,
173, 175, 181, 187, 193–196, 210, 219,
221, 223
Cooper’s equation, 54, 76, 77, 210
Copper plane, 157, 159, 160, 161, 163, 166,
173, 174, 175, 176, 177, 178, 182, 183,
189, 190, 195, 202, 205
Cosine law, 159
Coulomb interaction, 24, 27, 30, 31, 193, 212
Coulomb repulsion, 10, 11, 29, 30, 163,
168, 222
Creation operator, 47, 67, 70, 73, 134, 230, 235
Critical current, 116, 127, 221, 223–224, 225
Critical field, 8, 9, 39–40, 103, 104, 107, 115,
127, 136, 138, 167, 224
Critical temperature, 1, 7, 8, 9, 11, 12, 29, 39,
45, 49, 57, 58, 59, 73, 81, 84, 86, 87, 103,
104, 107, 131, 136, 157, 166–168, 173,
176, 177, 183, 204, 209, 211, 213, 215,
219, 221, 222, 223
Cu-Fermi sheet, 162, 175, 190, 195
Cu-Fermi surface, 175, 177, 178, 205
Cuprate superconductor, 9, 100, 157, 166, 169,
170, 215, 222
Current density, 94, 99, 114, 122, 126, 132,
199, 209, 220
Curvature inversion, 177, 178, 181, 184, 189,
190, 195, 196, 204, 205
Curvature sign change, 176, 184
Cusp, 101
Cyclotron frequency, 187, 188, 189
Cyclotron resonance, 159
D
dc Josephson effect, 4
Deaver, 4, 115
Debye’s continuum model, 18
Deformation potential approximation, 20
Deformation potential model, 194
de Haas–van Alphen (dHvA) oscillation, 159
Delta function, 184, 187, 189
Delta-function singularity, 189
Density operator, 25, 46, 47, 48, 50, 68,
113, 123, 124, 125, 131, 134, 135, 140,
153–154, 209
Density of states, 11, 17, 75, 77, 79, 87, 94,
136, 178, 181, 184, 185, 186, 188, 189,
190, 204, 211
Density-wave modes, 19
Discontinuous drop, 85
Dispersion relation, 73, 75, 80, 87, 88, 110,
153, 155, 157, 161, 166, 178, 187, 189,
194, 195, 200, 213, 214, 219, 220, 221
Displacement, 15, 19, 20, 21, 108, 110, 121
Distinguishable particle, 65
Dome-shaped curve, 175
Doping, 87, 157, 173–179, 183, 184, 189, 190,
195, 197, 198, 199, 202, 204, 205, 215
Down-spin, 51, 125, 185, 186
d-wave, 157, 162, 173, 175, 178, 193–196
d-wave Cooper pair, 157, 162, 175, 193–196
d-wave type, 193, 195
E
Effective Hamiltonian, 24, 25, 121
Effective interaction, 27, 193
Effective mass, 29, 30, 32, 78, 114, 159, 160,
163, 184, 198, 204
Ehrenfest–Oppenheimer–Bethe (EOB) rule,
61–62, 68, 70
Eigenvalue equation, 46, 48, 49, 60, 76, 77, 234
Electric current density, 114, 126
Electron, 10, 11, 12, 15–27, 29, 30, 31, 32, 36,
37, 40, 42, 43, 45, 46, 47, 50, 51, 56, 59,
60, 62, 64, 65, 69, 70, 73, 75, 78, 79, 89,
90, 91, 97, 99, 101, 103, 111, 114, 115,
124, 137, 147, 159, 161, 162, 163, 168,
173, 175, 176, 177, 178, 184, 185, 190,
193, 194, 195, 197, 198, 204, 205, 207,
208, 209–210, 211, 212, 213, 215, 216,
219, 220, 221, 222, 223, 235
Electron band, 12, 13, 30, 42, 89, 95, 173, 222
Electron gas system, 23, 26
Electronic heat capacity, 6, 7, 167, 168
Electron-pair operator, 79
Electron–phonon system, 23, 25
“Electron” wave packet, 162, 195
Elementary fermion, 61, 68, 70, 71
Energy-eigenvalue equation, 48, 60, 76
Energy gap, 6, 7, 11, 12, 29, 36, 37, 38, 39,
40–41, 45–60, 79, 88, 89, 90, 91, 92, 95,
96, 97, 99, 100, 101, 131, 133, 136, 137,
138, 140, 157, 165, 168–170, 209, 211,
212, 216, 219, 221, 222
Energy gap equation, 36, 38, 40–41, 45–60,
137, 165, 211
Energy momentum relation, 75, 79, 111, 120,
137, 139