1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 569

∗b. Derive eqns. (14.6.8) through (14.6.10).

14.3. Derive eqn. (14.6.16).

14.4. A quantum harmonic oscillator of massmand frequencyωis subject to a
time-dependent perturbationHˆ 1 (t) =−αˆxexp(−t^2 /τ^2 ),t∈(−∞,∞). At
t 0 =−∞, the oscillator is in its ground state.
a. To the lowest nonvanishing order in perturbation theory, calculate the
probability of a transition from the ground to the first excited state as
t→∞.

b. To the lowest nonvanishing order in perturbation theory, calculate the

probability of a transition from the ground to the second excited state as

t→∞.

14.5. The time-dependent Schr ̈odinger equation for a single particle of massmand
charge−emoving in a potentialU(r) subject to an electromagnetic field is
{
−

1

2 m

[

−i ̄h∇−

e

c

A(r,t)

] 2

−eφ(r,t) +V(r)

}

ψ(r,t) =i ̄h

∂

∂t

ψ(r,t).

Show that the Schr ̈odinger equation is invariant under a gauge transformation

A′(r,t) =A(r,t)−∇χ(r,t)

φ′(r,t) =φ(r,t) +

1

c

∂

∂t

χ(r,t)

ψ′(r,t) =e−ieχ(r,t)/ ̄hcψ(r,t).

14.6. Consider the free rotational motion of a rigid heteronuclear diatomic molecule
of (fixed) bond lengthRand moment of inertiaI =μR^2 , whereμis the
reduced mass, about an axis through its center of mass perpendicular to the
internuclear bond axis. The molecule is constrained to rotate in thexyplane
only. One of the atoms carries a chargeqand the other a charge−q.
a. Ignoring center-of-mass motion, write down the HamiltonianHˆ 0 for the
molecule.

b. Find the eigenvalues and eigenvectors ofHˆ 0.

c. The molecule is exposed to spatially homogeneous, monochromaticradi-

ation with an electric fieldE(t) given by

E(t) =E(ω)eiωtxˆ,

wherexˆis the unit vector in thex-direction. Write down the perturbation

HamiltonianHˆ 1.