1549380323-Statistical Mechanics Theory and Molecular Simulation

392 Quantum mechanics

9.7. Consider an unbound free particle in one dimension such thatx∈(−∞,∞).

An initial wave function Ψ(x,0), where

Ψ(x,0) =

[

1

2 πσ^2

] 1 / 4

e−x

(^2) / 4 σ 2
is prepared.
a. Determine the time evolution of the initial wave function and the corre-
sponding time-dependent probability density.
b. Calculate the uncertainties in ˆxand ˆpat timet. What is the product
∆x∆p?
∗9.8. A charged particle with chargeqand massmmoves in an external magnetic
fieldB= (0, 0 ,B). Letˆrandˆpbe the position and momentum operators for
the particle, respectively. The Hamiltonian for the system is
Hˆ=^1
2 m

(

pˆ−

q

c

A(ˆr)

) 2

,

wherecis the speed of light andA(r) is called thevector potential.Ais

related to the magnetic fieldBby

B=∇×A(r).

One possible choice forAis

A(r) = (−By, 0 ,0).

The particles occupy a cubic box of sideLthat extends from 0 toLin each

spatial direction subject to periodic boundary conditions. Find theenergy

eigenvalues and eigenfunctions for this problem. Are any of the energy levels

degenerate?

Hint: Try a solution of the form

ψ(x,y,z) =Cei(pxx+pzz)/ ̄hφ(y)

and show thatφ(y) satisfies a harmonic oscillator equation with frequency

ω=qB/mcand equilibrium positiony 0 =−(cpx/qB). You may assumeLis

much larger than the range ofy−y 0.

∗9.9. Consider a system ofNidentical particles moving in one spatial dimension.

Suppose the Hamiltonian for the system isseparable, meaning that it can be

expressed as a sum

Hˆ=

∑N

i=1

ˆh(ˆxi,pˆi),

where ˆxiand ˆpiare the coordinate and momentum operators for particlei.

These operators satisfy the commutation relations

[ˆxi,xˆj] = 0, [ˆpi,pˆj] = 0, [ˆxi,pˆj] =i ̄hδij.