Overall
The first part of the exam is closed book, close everything. It consists of multiple choice and
true-false questions. It will try to test the concepts and definitions in the first part. The second
part of the test is work problem or short answer. There will be some problems that could be just
“set it up” and some that will require a complete solution. You will be allowed to use your book
or a page of eqns. for this part. The most important concepts are those covered in this review and
in the homework that you have done up to this point.
You should be familiar with some of the history of the development of Quantum Mechanics.
Specifically, blackbody radiator, photoelectric effect (work problems), two slit electron
experiment (Davidson and Germer). How does deBroglies hypothesis fit in? What are these,
and how did they help define the theory of small particles/waves? Find for small particles that it
matters if we measure the momentum or position first. How does this correlate with the idea of
commutation of operators. Matricies? What does it mean if two operators are said to
“commute” What is an operator? Be able to determine the value of the commutator. Why use
operators to represent the physical observables? What are the Hamiltonian and the Laplacian?
What are an eigenvalue eqn., an eigenvalue, an eigenstate and a linear combination. What is the
relationship between the Heisenberg uncertainty principle and operators? What are the
Correspondence Principle and the Complimentary Principle?
Quantum Theory can be written down in terms of the postulates. You should be familiar with
the postulates and understand all that they define. Postulate one tells about the wavefunction and
the idea that all the information about the system is in the wavefunction. Postulate 2 defines the
operators in the position representation. Postulate 3 links observables to Hermitian operators in
terms of the average or expectation value of an observable D, specified as <D>. What is a
Hermitian operator? Why is it special? Postulate 4 describe what observation is expected if
the system is in an eigenstate of the operator. Postulate 5 describes what happens if it is in some
arbitrary state which can be written as a linear combination (a superposition) of the eigenstates
and gives the probability that the result is a particular eigenvalue based upon the coefficients in
the linear combination. Postulate 6, the Born interpretation, tells us about *  d
What
are the restrictions on the wavefunctions that are solutions to an eigenvalue eqn of a physical
observable. What is d? Is it the same for all coordinate systems? What is a linear operator?
Postulate 7 tell us about the time dependence of a wavefunction; gives time dependent
Schrodinger eqn.
ANOTHER postulate based on the commutator relations says that if two
observables are to have simultaneous precisely defined values, then their corresponding
operators must commute. Two observables that cannot be determined simultaneously are said to
be complementary (Heisenberg Uncertainty Principle again). Be familiar with the Dirac “bra”
“ket” notation for the Schrodinger eqn. and expectation values. Know the definitions of
normalization and orthogonality as it pertains to wavefunctions.
What is the curvature of the wavefunction? Be able to predict its sign based on the values of ,
E and V. Can quantization occur for a free 1d particle, a 1d particle with a potential barrier only
on one side, a 1d particle surrounded by barriers? Looking at the form of the wavefunction for
translational motion, should be able to indicate the momentum direction.
Complex
wavefunctions correspond to definite states of linear momentum. What is a wavepacket? You
should understand how to set up the Schrodinger eqn for a free particle, a finite barrier problem,
an infinite barrier problem, and a particle in a box. What are the continuity equations (amplitude
and slope), and how do they help get the probability of reflection or probability of transmitting
through the barrier? Can you sketch the transmission probability as a function of E/V for cases
where E<V and E>V? You should be familiar with how the energy eigenvalues were determined
for the particle in the box and how they are spaced. What are the allowed values for the quantum
number. How about a two dimensional box. How do we get degeneracy (define) for a two or
higher dimensional box? This models translational motion; translation of electrons in dye
molecules for instance.
What is the ideal model for vibrational motion. Be able to set up the Schrodinger equation.
What mathematical “tricks” are use to solve the Schrodinger eqn. How do the solutions in this
case get to be quantized? What polynomial makes up part of the eigenfunctions. What is the
spacing between the energy eigenvalues. What are the allowed values of the quantum #. Be able
to use the recursion relationships and determine the normalized eigenfunctions. Be able to show
if two eigenfunctions are orthogonal. Remember an alternate way to solve the Harmonic
Oscillator Schrodinger Eqn (ladder operators). Be able to use the operators introduced in this
other derivation. What is the virial eqn?
Rotational Motion of a diatomic molecule (Rigid Rotor) is described by the Schrodinger Eqn.
Be able to use the Schrodinger eqn in either Cartesian or Spherical Polar Coordinates. What is
d in each of these? What are the moment of inertia, center of mass, and reduced mass? What
is the angular momentum. What is the magnitude of the angular momentum, and the magnitude
of the zth component of the angular momentum? What is the Schrodinger eqn for the particle on
a ring and a particle on a sphere. What happened to the potential energy part? What is the form
of the eigenvalues for the energy of a rigid rotator? What mathematical “trick” was used to find
the solution to the Schrodinger eqn. for this system. What are solutions to the  and  parts.
What are the limits on r,  and ? The overall wavefunction is the Spherical Harmonic which
is governed by 2 quantum #s. What are the allowed values for those two quantum #s. What do
the boundary surfaces look like corresponding to  and  for the rigid rotor wavefunctions?
Be able to set up the Schrodinger eqn for a general case of electrons interacting with a nucleus.
The solution to the Schrodinger eqn for the hydrogenic atom follows more easily if one works in
center of mass coordinates, where it is separated into two pieces. Again the trick of Separation
of Variables is used, and then the solution to the angular part is found to be just the spherical
harmonics. What about the radial part. What mathematical tricks are used again here to solve
the differential eqn. How does the radial part become quantized. What polynomial corresponds
to part of the radial solution? What quantum #s need to be specified for radial, theta, and phi
parts of the eigenfunctions for the Schrodinger eqn. for the hydrogen atom? What are the
allowed values of these quantum #s n, l, ml ? Be able to give the wavefunction in r , and 
corresponding to, for instance 321 etc. What do the various quantum # tell us about the
electronic states in the hydrogenic atoms. How many nodes are there for the wavefunctions. Be
able to sketch the radial part of the wavefunction, the probability of finding an electron at a
specified location, and the probability of finding the particle at a given radius (radial distribution
function). How about the expressions for the mean radius, mean square radius, or the most
probable radius? You should be able to specify the atomic orbitals which correspond to some
values of the quantum #s. You should also be familiar with the different ways that the atomic
orbitals are represented and their general shapes. What are the values of ml for a px, a py, and a pz
orbital? What does the virial eqn. look like for this system. What is the degeneracy of a given
atomic shell?
You should be able to manipulate commutation relations to determine values like [l x, ly]. Do l2
and lz operators commute. These are important to define since once they are defined we can
subject newly defined operators to the same commutations and if the new operator gives the
same results it is an angular momentum operator. We do this with the spin operators and the
overall momentum operator for a composite system. What does it mean if they do not commute?
Does the l2 operator commute with all its component operators? You should be able to derive
the basic commutation relationships that define angular momentum. What are the shift
operators? What do they do. How can they be used to reduce a second order differential
equation into a first order one? Do the shift operators commute with the l2, lx, ly, or lz operators.
What are the values of l2/l,ml> and lz/l,ml>? What is the effect of the l+ and l- operator on the
eigenfunction, what are the c+ and c- coefficients? What happens when l+ is applied to an
eigenstate where ml is already its maximum value? This was part of the trick used to come up
with the 1st order differential equation that resulted in the spherical harmonics as the eigenvalues
angular part of the hydrogenic Schrodinger eqn. You should know the definitions of  and  in
terms of their s and ms values. You should be able to apply the shift operators to them as well.
Be able to show that the nonzero matrix elements of the shift operators are </s+/ > and </s/>.
For a composite system there are two ways of specifying the combined states. What is the
difference between the coupled and uncoupled pictures? What are the good quantum #s in each.
Remember that the boundary conditions determine if the l and ml , s and ms values are integral or
½ integral. The allowed values of J for the composite system are determined from the Clebsch
Gordan series. You should be able to use the Clebsch Gordan series for coupling of two or more
angular momenta. Also you should be familiar enough with the figures which show the different
coupling to sketch one. The differences are due primarily to the differences in the “good”
quantum #s. It is possible to express the coupled state in terms of a linear combination of the
uncoupled states. The values of the coefficients for this linear combination, the Wigner
coefficients or Clebsh Gordan coefficients or simply vector coupling coefficients are tabulated,
but they can be calculated in some simple cases from the shift operators. You should understand
this procedure well. You should be able to use the table to give the Wigner coefficients.