Simple Nature - Light and Matter

12 This problem builds on the results of problems 13-21 (p. 944)
and 13-38 (p. 947).
Suppose we have a three-dimensional box of dimensionsL × L ×
L/2. Let the box be oriented so that the shorter dimension is along
thezdirection. For convenience, define the quantity=h^2 / 8 mL^2 ,
which has units of energy.
(a) What are the five lowest energies allowed in this box, expressed
in terms of? Give the quantum numbers for each energy, and find
the degeneracy (p. 920) of each.
(b) Suppose we put five electrons in this box such that they have the
lowest possible total energy. (Keep in mind that there is a limit to
how many electrons can have the same spatial wavefunction.) What
is the total energy of this state?

√

(c) What are the two lowest-energy photons that can excite one

of the five electrons (from the situation described in part b) to an

excited state?

√

[Problem by B. Shotwell.]

13 In a helium nucleus, each particle feels a potential due to the

attractive forces from the other three. This potential can be well

approximated as

U=

1

2

kr^2 ,

wherekis a constant andris the distance from the center of mass
in three dimensions. You will need the result of problem 23, p. 944.
Refer also to sec. 14.2.4, p. 962.
(a) Show that the Schr ̈odinger equation is separable in terms ofx,
y, andzin this example.
(b) Find the ground-state wavefunction, expressed in terms ofrand
the constantbdefined in problem 23, p. 944.

√

(c) Find the energy of the ground state, expressed in terms of the

classical frequencyω.

√

14 An entangled state of three particles is prepared, described

by the wavefunction,

Ψ =k[− 2 |↑↑↑〉+ 2i|↑↓↑〉+|↓↓↓〉]

where the arrows are thez-components of the spins of particles A,
B, and C, respectively.
(a) The wavefunction is not properly normalized. What value of|k|
will normalize the wavefunction?

√

(b) What is the probability that measuring thez-component of par-

ticle A’s spin will give spin up?

√

(c) Suppose that we first measure the spin of particle C and find

that it is down, and then we measure the spin of particle A. What

is the probability that we will find A’s spin to be up?

√

[Problem by B. Shotwell.]

Problems 1011