3.1. Quantum mechanical description of particle motion
143
from regarding the states in k−space.
Separately, it is interesting to see what happens when we apply
the operator for the canonical momentum to the energy eigenfunctions. This gives
−ih ¯ vu k (x) = ¯
hk u k (x).
(3.74)
Evidently, the energy eigenfunctions u k (x) are also the eigenfunctions of the canonical mometum operator, with eigenvalues h ¯k.
Having assumed that the magnetic vector potential A is zero, we
therefore identify ¯
hk with the kinetic momentum. This momentum
is proportional to the gradient of u k (x). It follows that the vector
k is perpendicular to the surfaces u = const, and therefore points
in the direction of wave propagation, as expected. The wavelength
is given by
2π
λ =
.
(3.75)
k
This is the de Broglie wavelength, given by λ = h/p, where p is
the momentum.
Including the time dependence (3.18), we have
1
−iH k t/¯ h
i(k·x−ω k t)
ψ k (x, t) = u k (x) e
= √ e
,
(3.76)
V
where H k = hω k .
¯
This is a traveling plane wave, propagating in
the direction k. From (3.57) the wave vector k and the angular
frequency ω k are related by the energy-momentum equation as
¯
hk
2
ω k =
,
(3.77)
2m
where k
2 = |k|
2 = k · k. This is called the dispersion relation. The
wave propagates with phase velocity v p given by
ω k
¯
hk
v p =
=
.
(3.78)
k
2m
Evidently, states with higher k (shorter wavelength) propagate
faster than states with lower k.