Page 248 - Principles and Applications of NanoMEMS Physics

238 Appendix B

Clearly, all the A = 1 and B = 0 . These examples together with Figs. B-1
n n
and B-2 should provide an intuitive way of assimilating the concept of
bosonization. What we will do next is to finally present an example of the
bosonization procedure, namely, their Delft and Scholler’s [139] application
of the procedure to a Hamiltonian with a linear dispersion.
They begin by assuming a linear dispersion of the form, () vkE = k = ,
F
which measures all energies in units v = , where the total Hamiltonian is,
F
H ≡ ¦ H , (B.44)
0 0 η
η
with,

∞
H ≡ ¦ k + c + c +
0 η + kη kη +
k = −∞
. (B.45)
L 2/ dx
+
= ³ + + ψ η () ∂ix x ψ η () x + +
−L 2/ 2 π

Then, the fact that the Hamiltonian commutes with the number operator
[H 0η , N ˆ ' η ] 0= for all ,ηη ', is exploited as an argument to justify that any
G
N -particle ground state is an eigenstate of H 0 η , in particular,
G G G
N
H 0η N = E 0η N . The eigenvalue is obtained by adding the energy of
0 0
all levels,

­ N η 1 2 1
( δ
−
° ¦
0
η
η
η
b
G G 2 π ° (n δ 2/ ) = N + N 1 − b ) for N ≥ ,
E = N H N = ® n=1 2 2
N
0
η 0
0 η 0 0 L ° ¦ − (n δ 2/ ) = 1 N + 1 N 1 − ) for N < 0 . (B.46)
−
( δ
2
η
η
η
¯ n ° = N η +1 b 2 2 b
= 2 π 1 N η (N +1 − δ b )
η
L 2
This gives the ground state energy of H . When the system is excited, its
0 η
eigenstate energy E , may increase in units of q. This follows from the
commutation relations,
δ
[H 0η ,b q + ' η ]= qb q + η ' ηη ' , (B.4 )
7
and its consequence,