Consider a non-relativistic free particle of mass m in the d-dimensional space R^d.In order to avoid mathematical diﬃculties one considers ﬁrst the particle in a d-dimensional
cube CL := [−L/2, L/2]^d
∈ R^d of volume L^d and performs the limit L → ∞ at the end of the calculation. For the boundaries ∂CL of the cube CL having a minimal inﬂuence on the result one usually assumes periodic boundary conditions, i.e., CL is a d-dimensional torus by identifying opposite points on the faces of the cube. Under these conditions consider
the wave function of the particle to be given by the plane wave
ψL,a,b(r, t) =1/√L^d exp(i(a · r − bt)), r ∈ CL (1)
with a vector a ∈ R^d
and a scalar b ∈ R.
(a) Derive the condition on vector a which is implied by the periodic boundary
conditions. What happens in the limit L → ∞?
(b) Verify that the wave function in Eq. (1) is properly normalized. Why is the
limit L → ∞ performed only “at the end of the calculation”?
(c) Describe the distribution of the particle in CL. What happens in the limit
L → ∞?
(d) Read oﬀ the wave vector, the frequency, the momentum, and the energy
from Eq. (1).
(e) From the previous step, infer the dispersion relation, i.e., the frequency ω
as a function of the wave vector k.
(f) Calculate the mean r := hri and the variance h(r − r)^2
i of the particle
position. What happens in the limit L → ∞? Compare with the result of step (c)