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- Show that the wavefunction of a particle of mass in an infinite one-dimensional square-well of width
returns to its original form after a quantum revival time
.

- A particle of mass moves freely in one dimension between
impenetrable walls located at
and . Its initial wavefunction is

What is the subsequent time evolution of the wavefunction?
Suppose that the initial wavefunction is

What now is the subsequent time evolution? Calculate the probability
of finding the particle between 0 and as a function of time in
each case.

- A particle of mass is in the ground-state of an infinite one-dimensional square-well of width . Suddenly the well expands to
twice its original size, as the right wall moves from to , leaving
the wavefunction momentarily undisturbed. The energy of the particle
is now measured. What is the most probable result? What is the probability
of obtaining this result? What is the next most probable result, and
what is its probability of occurrence? What is the expectation value
of the energy?

- A stream of particles of mass and energy encounter a
potential step of height :
*i.e.*, for and
for with the particles incident from . Show that the fraction
reflected is

where
and
.

- A stream of particles of mass and energy encounter the
delta-function potential
, where
. Show that the fraction
reflected is

where
, and
.
Does such a potential have a bound state? If so, what is its
energy?

- Two potential wells of width are separated by a distance .
A particle of mass and energy is in one of the wells. Estimate
the time required for the particle to tunnel to the other well.

- Consider the half-infinite potential well

where . Demonstrate that the bound-states of a particle of
mass and energy satisfy

- Find the properly normalized first two excited energy eigenstates of
the harmonic oscillator, as well as the expectation value of the potential energy in the th energy eigenstate. Hint: Consider the raising and lowering operators defined
in Eq. (408).

** Next:** Multi-Particle Systems
** Up:** Simple Harmonic Oscillator
** Previous:** Simple Harmonic Oscillator
Richard Fitzpatrick
2010-07-20