Page 240 - Principles and Applications of NanoMEMS Physics

230 Appendix B

∞ ∞
+
+
+
N ˆ η ≡ ¦ + kη c kη + + = ¦ [c kη c kη − 0 c + kη c 0 ], (B.10)
c
kη
k = −∞ k = −∞
G G
and operates on states of the form N . The ground state, denoted N ,
0
represents the state in which there are no particle-hole excitations, and it is
constructed as follows.
G
N ≡ () ( ) ...CC 1 N 1 2 N 2 (C M ) N M 0 , (B.11)
0
where,

­ + + +
0
° c N η η c ( η −1 ) η ...c 1η for N η > ,
N
( ) N η ≡ ° 1 for N η = , , (B.12)
C
0
®
η
°
+
+
° c ( η +1 η c + N +2 ) η ...c 0η for N η < .
0
N
) ( η
¯
G
5) Given the fixed number of particles in every N -particle Hilbert space,
their excitations are construed as particle-hole excitations of the ground state
G
N , and captured by bosonic creation and annihilation operators defined
0
by,
+ i ∞ + − i ∞ +
b qη = ¦ c k +qη c , b qη = ¦ c k −qη c kη , (B.13)
kη
n = −∞ n = −∞
q k q k
2 n π
+
where = q > 0 , and n q ∈ Z is a positive integer. Then, operating on
q
L
G
+
any state N with b or b causes an aggregate of particle-hole
q η q η
excitations, where each excitation’s momentum differs, from that in the
+
ground state, by q units. This permits interpreting b and b as
η
η
q
q
momentum raising and lowering operators, which obey the following
commutation relations:
[b ,b ] [b= + ,b + ] 0 N= [ , ,b ] [N= ,b + ] 0= , for all q q , ' , η ' ,η (B.14)
q η q ' 'η q η q ' 'η q η q ' 'η q η q ' 'η