Page 127 - Principles and Applications of NanoMEMS Physics

3. NANOMEMS PHYSICS: Quantum Wave Phenomena 115

∇ × v = 0 . (95)
s

Eq.(95) signifies that a superfluid is irrotational, i.e., it exhibits no vorticity.
The quintessential example of a superfluid is embodied by a Bose liquid,
which consists of atoms of integral-value spins, in particular, liquid helium
4
(He ), which does not solidify at absolute zero and flows through capillaries
without dissipation.
Landau’s arguments, presented above, while successfully explaining
liquid helium behavior, were of an intuitive and phenomenological nature.
Elements for a first-principles theory to explain superfluid behavior began
taking shape with observations by Fritz London [155], to the effect that the
constitution of He atoms, which are composed of an even number of
elementary particles (2 protons, 2 neutrons, and 2 electrons) suggested that
they should be described by a symmetric wavefunction and, consequently,
should obey Bose statistics, together with the earlier observation by Einstein
that, at appropriately low temperatures and mass and density conditions, a
gas of non-interacting Bose particles condenses with the remarkable property
that a nonzero fraction of the condensed atoms occupies a single one-particle
state. Such a state, in particular, is a coherent state and has come to be
known as a Bose-Einstein condensate (BEC) [155]. A fundamental theory
capturing this behavior is the Gross-Pitaevskii (GP) model. The GP equation
models the general Bose gas by the equation [78],

∂ ψ = 2
i= = − ∇ 2 ψ + U mf ψ , (96)
∂t 2 m

where m is particle mass,
ψ () dx 2 ′ x
U = e 2 ³ , (97)
mf
x − ′ x

is the mean field for Coulomb interaction between atoms, and may be
expressed as,

′
) ( ) dx′
′
′
x′
)() dx′
′
=
2
U mf ³ V (x − x ψ 2 x = ³ gδ (x − x ψ 2 x = g ψ () . (98)
Substituting (98) into (96) one obtains a nonlinear Schrödinger equation,
∂ ψ = 2 2 = 2
2
i= = − ∇ 2 ψ + g ψ () ψx () x = − ∇ 2 ψ + g ψ * ψ . (99)
t ∂ 2m 2m