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- Monochromatic light with a wavelength of passes
through a fast shutter that opens for sec. What is the
subsequent spread in wavelengths of the no longer monochromatic light?

- Calculate
,
, and , as
well as
,
, and ,
for the normalized wavefunction

Use these to find
. Note that
.

- Classically, if a particle is not observed then the probability of finding it
in a one-dimensional box of length , which extends from to , is a constant per unit length.
Show that the classical expectation value of is , the expectation value of
is , and the standard deviation of is .

- Demonstrate that if a particle in a one-dimensional stationary state is bound then the
expectation value of its momentum must be zero.

- Suppose that is complex. Obtain an expression for
and
from Schrödinger's equation. What does this tell us about a complex ?

- and are normalized eigenfunctions corresponding to
the same eigenvalue. If

where is real, find normalized linear combinations of and
which are orthogonal to (a) , (b) .

- Demonstrate that
is an Hermitian
operator. Find the Hermitian conjugate of
.

- An operator , corresponding to a physical quantity , has
two normalized eigenfunctions and , with eigenvalues
and . An operator , corresponding to another physical
quantity , has normalized eigenfunctions and ,
with eigenvalues and . The eigenfunctions are
related via

is measured and the value is obtained. If is then measured and then again, show that the probability of obtaining
a second time is .

- Demonstrate that an operator which commutes with the
Hamiltonian, and contains no explicit time dependence, has an expectation
value which is constant in time.

- For a certain system, the operator corresponding to the physical
quantity does not commute with the Hamiltonian. It has
eigenvalues and , corresponding to properly normalized eigenfunctions

where and are properly normalized eigenfunctions of the
Hamiltonian with eigenvalues and . If the system is in the
state at time , show that the expectation value of
at time is

** Next:** One-Dimensional Potentials
** Up:** Stationary States
** Previous:** Stationary States
Richard Fitzpatrick
2010-07-20