Page 134 - Principles and Applications of NanoMEMS Physics

122 Chapter 3

Ψ σ σ ( , xx 1 2 ) ψ σ () ( ). 123)
=
x ψ
x
σ
1
2
Being the wave function of a boson, Ψ σ σ ( , xx 1 2 ) must satisfy Pauli’s
exclusion principle, wheraby it must be anti-symmetric. Furthermore, since
spins and spatial coordinates operate in different (tensor) spaces, the wave
function must be a product of a spin-dependent factor, and a coordinate-
dependent factor, i.e.,

∆
=
Ψ σ σ (x , x 2 ) E σ σ f (x , x 2 ) E σ σ ⋅ , (124)
=
1
1
g
where E σ σ is the anti-symmetric spin-dependent factor. With this definition,
one can rewrite (124) as,

∂ ψ ( ) tx, = 2
*
i= σ = − ∇ 2 ψ σ + ∆E σ σ ψ . (125)
σ
∂t 2m
Following the same procedure as in the previous section, the dispersion
relation is obtained from the set of equations,

∂ ψ ( ) tx, = 2
*
i= σ = − ∇ 2 ψ σ − E F ψ σ + ∆E σ σ ψ , (126a)
σ
∂t 2m
and
∂ ψ * ( tx, ) = 2
*
i - = σ = − ∇ 2 ψ * σ − E ψ * σ + ∆ E σ σ ψ σ , (126b)
∂t 2 m F
where the energy is now referred to the Fermi energy. Then, postulating
solutions of the form,
+
− iEt/ = ipx/ =
ψ ~ η e , (127a)
σ
σ
and

+
ψ σ * ~ ζ e − iEt/ = ipx/ = , (127b)
σ
it can be shown, upon substitution on (126), that the set of equations,