*October 5, 2007*

1. Suppose an electron is attracted towards the origin by a
force _{}, where *C* is a
positive constant, *r* is distance from
the origin.

(a) What is the radius of the smallest circular Bohr orbit?

(b) What is the energy in the *n*^{th} circular Bohr orbit?

(c) Now suppose the electron oscillated on a line through
the origin, so _{}. Use the Uncertainty
Principle to *estimate* the linear
spread _{}of the wave function.

2. (a) Suppose the inner product for real functions defined
for _{} is

_{}

Begin with the set 1, *x*,
*x*^{2}, … and construct the
first *three* elements of an
orthonormal basis.

(b) Find the eigenvalues and the eigenvectors of:

_{}

3. Consider the
one-dimensional Schrodinger equation for an electron of mass *m* in a potential

_{}

For what values of *g*
is there a bound state with extremely weak binding energy *ε*?

(*Hint*: there are
two values, one of which is extremely small, the other isn’t.)