Page 128 - Principles and Applications of NanoMEMS Physics

116 Chapter 3

Condensation is captured when in (99) one imposes the conditions for
obtaining the lowest possible state, ψ , namely, that the wave function be
0
homogeneous, i.e.,

∇ 2 ψ → 0. (100)
0

This leads to the relation,

ψ 2 = n = N , (101)
0
V
where N is the number of atoms and V is the volume. In turn, substitution of
(100) and (101) into ( 99 ) leads to a simplified equation of motion, namely,

∂ ψ
i= 0 = gn ψ , (102)
t ∂ 0

with a solution of the form,

− gnt
ψ = Ce = . (103)
0

The dispersion relation for low-level excitations are obtained by
linearizing (99), in particular, writing ψ = ψ + χ , where χ << ψ , and
0 0
substituting into (99), one obtains,

∂ χ = 2
2 ψ
*
i= = − ∇ 2 χ + g 0 2 χ + g ψ 0 2 χ . (104)
∂t 2m
*
Since this equation contains the two unknowns χ and χ , we generate a
second equation by taking its complex conjugate,

∂ χ * = 2
i - = = − ∇ 2 χ * + 2g ψ 0 2 χ * + g ψ 0 2 χ . (105)
∂t 2m

Then, postulating solutions of the form,

− iEt+ ipx
χ ~ ξe , (106)