Page 137 - Principles and Applications of NanoMEMS Physics

3. NANOMEMS PHYSICS: Quantum Wave Phenomena 125

contains no off-diagonal terms of O(D). In particular, if we take
H = H + H , where H gives rise to the solutions n when H = , so
'
'
0
0
0
~
that H n = E n , then H may be expanded as, [131],
0 n
§
~
H = e − S (H + H ' )e = § 1 ¨ − S + S 2 + ...¸ · (H + H ' ) 1 ¨ + S + S 2 + ...¸ ·
S
0 ¨ ¸ 0 ¨ ¸
© 2 ¹ © 2 ¹
= H + [ S,H ]+ 1 [ [ ,H ] ] ...S, S + . (136)
2
= H + H + [H S , ] [H ++ 1 ' [H ] ] [ ,HS, S , + 1 ' S ] ...+
'
0 0 0
2 2

If we select H + [H 0 S , ] 0= , then the second and third terms in (136)
'
vanish and we have a prescription for S, namely,
n ' H ' n
n ' H ' n + (E − E ) n ' S n = 0 Ÿ n ' S n = , (137)
n ' n E − E
n ' n

which yields the desired Hamiltonian as,

~
H = H + 1 [ ,H ' S + O S 2
] (). (138)
0
2
Now, in this diagonal formulation, effective electron-electron interaction is
'
elucidated by considering the case in which the perturbation H causes the
following transitions: Either the electron in state k emits a phonon –q and
this is absorbed by the electron in state k’, or the electron in state k’ emits a
phonon q and this is absorbed by the electron in state k. These transitions
may be mathematically represented as occurring from an initial state i to a
final state f via a virtual state m , in terms of which the expectation
value of the commutator in (138) may be expressed as,

f [ H,S ' ]i = ¦ ( Sf m m H ' i − f H ' m m S ) i . (139)
m
Following [154], consideration of the phonon system at absolute zero, so that
one of the phonon states refers to the vacuum, the matrix element calculation
(134) over the phonon operators yields, without loss of generality,