Bose-Einstein Condensates in Rotating Traps and Optical ... - BEC

Chapter 4
Single particle in a periodic potential
This chapter reviews some standard results (see for example [19, 96]) concerning a single
particle in one dimension subject to an external potential V (x) which is periodic in space with
period d
V (x) =V (x + d) . (4.1)
The aim of this chapter is to provide some basic concepts such as quasi-momentum, band
structure, Brillouin zone, Bloch functions and Wannier functions. Starting from the single
particle case we can then conveniently extend and generalize these notions to the case of a
Bose-Einstein condensate in the following chapters.
4.1 Solution of the Schrödinger equation
Bloch states and Bloch bands
The one-dimensional motion of a particle in the periodic potential (4.1) is described by the
Schrödinger equation
i¯h ∂ϕ(x)
∂t =
− ¯h2 ∂
2m
2
+ V (x) ϕ(x) . (4.2)
∂x2 Due to the periodicity of the potential this equation is invariant under any tranformation
x → x+ld where l is any integer. Thus, if ϕ(x) is the wavefunction of a stationary state, then
ϕ(x + ld) is also a solution of the Schrödinger equation. This means that the two functions
must be the same apart from a constant factor: ϕ(x + ld) = constant × ϕ(x). It is evident
that the constant must have unit modulus; otherwise, the wave function would tend to infinity
when the displacement through ld was repeated infinitely. The general form of a function
having this property is
ϕjk(x) =e ikx ˜ϕjk(x) , (4.3)
where ¯hk is the quasi-momentum and ˜ϕjk(x) is a periodic function
˜ϕjk(x) = ˜ϕjk(x + d) . (4.4)
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