Question

1. Consider a spin-0 particle of mass m and charge q moving in a symmetric three-dimensional harmonic oscillator potential with natural frequency W.Att-0 an external magnetic field is turned on which is uniform in space but oscillates with temporal frequency W as follows. E(t)-Bo sin(at) At time t>0, the perturbation is turned off. Assuming that the system starts off at t-0 in the ground state, apply time-dependent perturbation theory to estimate the probability that the system ends up in an excited state after time t a) Using the standard expression for the interaction of a charged particle with the magnetic field given below, what is the probability that the system ends up in the n'th excited level, where En-hwo(n+3/2) b) So far in this course, we assumed that the above generic form V for the potential of a charged particle in a magnetic field is all there is. In fact, this is the correct classical expression, for one particular choice of gauge. However, gauge invariance requires that when taking the electromagnetic interaction from classical E+M over to quantum mechanics, a second term is needed which is quadratic in the strength of the electromagnetic field. For this problem, that means the in addition to V, there is a second interaction term V2 that must be included in the Hamiltonian. What excited states are allowed for this perturbation? You may label excited states by the usual quantum numbers nn, n, for the 3D harmonic oscillator. c) What is the transition probability to the two allowed excited states found in part (b) What is the resonance frequency for this transition? Hint: the resonance frequency is the value of w that maximizes the transition probability for fixed field amplitude B d) Extra credit: What conditions are required for the validity of perturbation theory for this problem?

Answer 1

2. Linear crystal optics for monochromatic plane waves

D =

1

2

De

−i(ωt−k·r) + D∗

e

i(ωt−k·r)

. (2.3)

The actual fields are represented by the real quantities P, E, and D, but

for many calculations it is more convenient to use the complex envelope

functions D, E, and P . These complex vectors are in general functions of

(x, y, z, t). However, for monochromatic plane waves, they are independent

of space and time so we omit these arguments in this chapter, and treat

the envelope functions as simple complex vectors.

We substitute the expansions of Eqs. (2.1)-(2.3) in the wave equation,

Eq. (1.13)

∇ × ∇ × E = −µ◦

∂

2

∂t2 D = −µ◦

∂

2

∂t2

ǫ◦E + P

. (2.4)

The operator ∇ becomes (±ik) when it operates on the exponent (±ik ·

r). Similarly the operator (∂/∂t) becomes (±iω) when it operates on the

exponent (±iωt). Making these substitutions in Eq. (2.4) and equating the

positive (or negative) frequency components on each side of the equation

yields

k × k × E = −µ◦ω

2D. (2.5)

In deriving this equation we assumed that E and D do not change on prop-

agation, so this is a wave equation for eigenpolarized light. This expression

implies that D must be normal to k, but D is not necessarily parallel to

E. However, k, D, and E must lie in a single plane.

We use a similar procedure to rewrite the Poynting vector equation to find

the energy flow for monochromatic plane waves. We start with Eq. (1.16),

the general Poynting vector equation,

S =

1

µ◦

E × B = E × H. (2.6)

For monochromatic plane waves the third Maxwell equation, Eq. (1.3),

relates H to E for eigenpolarized light in a nonmagnetic material by

H =

k × E

µ◦ω

. (2.7)

Substituting the expansions for E and H in Eq. (2.6), equating equal fre-

quency components, and using Eq. (2.7) plus

ǫ◦µ◦c

2 = 1, (2.8)

we arrive at

S =

nǫ◦c

2

|E|

2 eˆ× kˆ × eˆ,