Page 129 - Principles and Applications of NanoMEMS Physics

3. NANOMEMS PHYSICS: Quantum Wave Phenomena 117

and

*
χ ~ ηe − iEt+ ipx , (107)

substituting them into (104) and (105), together with (101), and solving the
resulting system of equations for E, one obtains the result,

p 4
2
2
E () p = S p + . (108)
4m 2
This is the dispersion relation of a superfluid. Expressing the fluid velocity
in terms of it, we obtain,

v = E = S 2 + p 2 . (109)
p 4m 2

This equation has a positive minimum, occurring at p → 0, and given by
the constant velocity S. Since this velocity is independent of momentum,
E(p) must contain an energy gap. An energy gap in its spectrum, thus, is
another manifestation of superfluidic behavior.
The zero-vorticity property of a superfluid is derived from first principles
as follows. From (103) it may be seen that the wave function for the Bose
condensate in its lowest energy state is a one-particle complex wave.
Generalizing this expression to,

ψ () =x ψ e iχ () , (110)
x

one can express the mass density as =ρ m ψ , where () x and the current
ψ
2
are related, as usual, by,
G i=
j = − (ψ * ∇ ψ − ψ ∇ ψ * ). (111)
2

It then follows that, inserting (110) into (101) one obtains,
G 2 =
j = = ψ ∇ χ = ρ ∇ χ , (112)
m

which, upon comparison with (94) yields,