Page 139 - Principles and Applications of NanoMEMS Physics

3. NANOMEMS PHYSICS: Quantum Wave Phenomena 127

G G G G G G G G G G G G
with R = (r + r ) 2, r = r − r , P = p + p , and p = p ( − p 2 / ) .
1 2 1 2 1 2 1 2
Then, expressing the solution as,
G G G G
ψ = e i K ⋅R ¦ h e k i ⋅r , (146)
G k
k

and taking into consideration the symmetry properties of the problem, in
G
G
G
G
G
G
particular, upon interchange of r and r , R → R , r → r − , and h G = G h ,
−
1
2
k
G k
and in the frame of reference in which the system is at rest K = ,
0
substitution of (146) into the Schrödinger equation, ψ = E ψ , yields,
H
= 2 k 2 G G G G ' G G G
¦ k G h e k i ⋅r + ¦ V ()er k i ⋅r G h k ' = E ¦ G h k e k i ⋅r
G
G
G
k 2m k ' k . (147)
§ = 2 k 2 · 1 G G G G G ' G
Ÿ ¨ E − ¸h G = ¦ ³ d r e − ki ⋅r V ()er k i ⋅r G h = ¦ V GG ' h G
¨ 2m ¸ k G Ω k ' G k k k '
© ¹ k ' () k '
Ω
Since the electron-electron interaction is mediated by phonons, and the
=
phonon energies lie between 0 and ω , where ω is the Debye energy, the
D D
electrons will be under the influence of the binding potential as long as the
their excitation energy of the pair is lower than the Debye energy, i.e.,
ε G − ε G ' < = ω , ε = = 2 k 2 2 m . In this context, we have,
k k D k
V GG ' = − V (148)
k k
and we can write (147) as,
§ = 2 k 2 · '
¨ −E ¸ G h = −V ¦ ' h , (149)
G
G
¨ ¸ k k k '
© 2m ¹
which, may be expressed as,
'
¦ ' G h G = ¦ ' h G ' ' G V , (150)
G
¦ '
k
k
k
k
k
§ = 2 k 2 ·
¨ − ¸
E
¨ ¸
© 2m ¹
which may be factored as,