Page 232 - Principles and Applications of NanoMEMS Physics

222 Appendix A

So, the eigenvalue of N is n. For fermion operators, the pertinent
anticommutation relations and number operator are,
+
+
+
a α a β + a β a α = { ,aa α β } 0= , { ,aa α β } 0= { ,aa α β } 0= , (A.40a)
,
+ + +
a α a α = 0 , a α a α = 0, N = a α a , (A.40b)
α
α
N = a α a α a α a = a α ( − a α a α )a = a α a = N . (A.41)
+
+
+
+
+
2
1
α
α
α
α
α
The second quantization formalism is completed by the necessary
expressions for operators, which upon acting on the wavefunction will
measure certain quantities of interest. In this context, the quantities to be
measured are classified according to the number of fundamental particles
producing it. For instance, in a noninteracting system, these quantities may
depend on individual particles, where each particle contributes its share
independently from the others. An example of such a quantity is the kinetic
energy of the system. On the other hand, quantities such as the Coulomb
interaction energy, in an interacting system, depend on two-particle
potentials, thus two-particle operators must be employed. Next, expressions
for one- and two-particle operators are presented.
A typical one-particle operator is the kinetic energy. For a bosonic
system, this is obtained by counting the number of particles in a given state,
multiplying this number by the energy of each particle, and then adding the
energies of all states. If an arbitrary particle occupies state α, following this
prescription, then the one-particle operator is given by,
H = ¦ α K α N = α K α a α a . (A.42)
ˆ
+
α ¦
α
α α
In this expression, α K α is the state energy, given by,
º
2
E α = α K α = φ + () −r G ª = 2 ∇ φ () rdr G G , (A.43)
«
» α
³ α
¬ 2 m ¼
G
where m is the particle mass, and φ α () r is the configuration space
representation of the wavefunction. Notice, that α K α may be computed
in any convenient basis in which the wavefunction are available. Thus, in
momentum-space basis we would have,
G
2
º
G
G
φ
E α = α K α = φ + () p ª p » α () p p d 3 . (A.44)
«
³ α
2π
¬ 2m ¼ ( )