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Physics 4610, Quantum Mechanics
Prof. Schroeder
Spring 2015
Name
Problem Set 9
(due Monday, March 30, 4:00 pm)
1.
Consider a particle of mass m confined by an attractive central potential in three
dimensions that is directly proportional to the distance from the origin: V = αr, for
some constant α. Use the matrix diagonalization method to find the low-lying energies
and wavefunctions for l = 0, l = 1, and l = 2. Please use units in which ~2 /m = 1 (not
~2 /2m = 1) and α = 1. Plot an energy-level diagram showing a few of the lowest levels
for each of these l values. Also plot both the reduced radial wavefunctions u(r) and
the actual radial wavefunctions R(r) for a few of the lowest levels for each l. (Make
the plots small, and plot several at once using the Mathematica Table function or
something similar, so you don’t use a huge amount of paper.)
2.
It is sometimes accurate to model the force between quarks as constant, so the potential
energy is linear. The light quarks that make up protons and neutrons are relativistic,
but the c and b quarks are heavy enough for nonrelativistic quantum mechanics to
apply to their bound states with reasonable accuracy. (Even though these quarks
are unstable, they live long enough to form well-defined bound states.) A recently
published (Godfrey and Olsen, arXiv:0801.3867 [hep-ph], 2008) energy level diagram
for “charmonium,” the system of a c quark and c antiquark, is shown below.
In the diagram the letters S, P, and D correspond to l = 0, 1, and 2, respectively. The
subscripts and superscripts refer to different spin states, which I’d like you to ignore
for the purpose of this problem, so I’ve added broad gray lines to the diagram to guide
your eye to “average” over the various spin states. The vertical axis in the diagram is
the total energy of the state, including the rest energies of the two particles.
(a) Compare the qualitative features of the charmonium energy level diagram to the
energy levels that you found for a linear potential in the previous problem. Is the
linear potential a reasonable model for this system? Explain in some detail.
(b) Use the two lowest l = 0 levels to make a quantitative estimate of the c quark
rest energy and the strength α of the (approximately) linear potential. Express your
answers in GeV and in GeV/fm, respectively. (Hints: First compare the spacing between these two levels to what you found for the two lowest l = 0 levels in the previous
problem, and use this comparison to determine where zero on the Charmonium energy
scale “should” be; assume that any remaining energy comes from the rest energies of
the two particles. Then determine what one unit of energy, in the previous problem,
is in terms of ~, m, and α. Note that for a system of two moving particles, the mass
m that enters the Schr¨
odinger equation is actually the reduced mass, which in this
case is half the actual mass of either of the particles (see Problem 5.1 in Griffiths if
you want details). Because m is now known, you can use the values in the diagram to
determine α.)
3.
Use the shooting method to verify the energies, and to plot the reduced radial wavefunctions, for the following states of a hydrogen atom: (a) the three lowest-energy
states for l = 0; (b) the two lowest-energy states for l = 1; and (c) the lowest-energy
state for l = 2. Use atomic units.
4.
Look up the exact formula for the n = 3, l = 0 hydrogen radial wavefunction, and (a)
plot it using atomic units. (b) Also plot the corresponding reduced radial wavefunction
and check that this plot agrees (aside from normalization) with the one you made in
the previous problem. (c) Show (either by hand or using Mathematica) that this exact
function is properly normalized.
5.
Look up the formula for the n = 3, l = 1 hydrogen radial wavefunction, and then write
down the corresponding reduced function u(r). You may omit the overall normalization constant. Check explicitly (by hand) that this u(r) satisfies the reduced radial
Schr¨
odinger equation.
6.
Use Mathematica to produce density plots showing a slice, in the xz plane (with x
horizontal and z vertical), through the probability density function for each of the
n = 2 hydrogen wavefunctions (there are four of them). Note that you’ll need to
express θ in terms of x and z (and set y = 0 for your slice). Then use your colored
pencils to sketch three-dimensional “cloud” diagrams of each of these wavefunctions,
using colors to represent phases as usual. It’s difficult to draw the wavefunctions
accurately in this way, but do the best you can. Take the z axis to point upward,
with the viewer’s location somewhat above the xy plane for perspective. I recommend
that for practice you also plot and sketch all the n = 3 wavefunctions, but that’s not
required for this assignment.