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- Demonstrate directly from the fundamental commutation relations for angular momentum, (300), that
,
, and
.

- Demonstrate from Equations (363)-(368) that

where
,
are conventional spherical polar angles.

- A system is in the state
. Evaluate
,
,
, and
.

- Derive Equations (385) and (386) from Equation (384).

- Find the eigenvalues and eigenfunctions (in terms of the angles
and
) of
.

- Consider a beam of particles with
. A measurement of
yields the result
. What
values will be obtained by a subsequent measurement of
, and with what probabilities? Repeat
the calculation for the cases in which the measurement of
yields the results 0
and
.

- The Hamiltonian for an axially symmetric rotator is given by

What are the eigenvalues of
?

- The expectation value of
in any stationary state is a constant.
Calculate

for a Hamiltonian of the form

Hence, show that

in a stationary state. This is another form of the *Virial theorem*. (See Exercise 8.)

- Use the Virial theorem of the previous exercise to prove that

for an energy eigenstate of the hydrogen atom.

- Demonstrate that the first few properly normalized radial wavefunctions of the hydrogen atom take the form:

** Next:** Spin Angular Momentum
** Up:** Orbital Angular Momentum
** Previous:** Energy Levels of Hydrogen
Richard Fitzpatrick
2013-04-08