and hence looks like it's 2
(see page 6).
Fit the data of page 6 to
expected classical motion and see that the quantum is like
Produce your own quantum moving particle following
the methods of page 7.
Express x' and p' in terms of the
raising and lowering operators of page 8.
Use these results to find the following matrix elements:
Use the above results to find x' × p'
in the nth eigenstate.
On page 10 you saw how a set of
degenerate states can be expressed differently in different
coordinate systems. The five degenerate wavefunctions E'=10
could be expressed either in terms of |nrm> wavefunctions
or |nxny> wavefunctions.
For the three examples given, prove these results algebraically.
For the 2 antisymmetric combinations (e.g., |12>-|1-2>)
proceed by integration and orthogonality. Combine all the results
to express each E'=10 |nrm>
in terms of a sum of |nxny> wavefunctions.
On page 11 I claim that the minimum of the effective
potential in the 2-d oscillator is r'=m½. Show this.
Find the classical period of circular orbits in our dimensionless units and
show the result is
I claim on page 11 that my quantum probability blob
moves uniformly in , i.e.,
Prove this statement is true.
On these pages we made a "2-d" SHM problem by including
a new direction (y) and a new
force in the y direction: Fy=-ky.
Consider instead a "2-d" SHM problem where the y direction is
included, but with no force in that direction. (This is how we
made ballistic motion -- a "2-d" falling problem.) Write down the
energy eigenfunctions and eigenenergies. Describe the classical
motion. Make a QM superposition in which a probability lump moves
much like the classical motion suggests the point-particle moves.
On page 14 you saw WKB approximation fit into
the 3-d case. Follow the same analysis for the 2-d oscillator r'
We need not display similar plots for other times because the equation
for shows that plot for any other time is
identical to the above except rotated by 2t' radians. Thus
the lump will make one complete revolution in a period of
[Oddly enough the easiest starting point to change variables to s=ln(r')
is the second equation, even though the third seems closest to WKB form.]
Follow all the way through to find the WKB energies.
Consider a 1-d spring with an additional applied constant force
(e.g., a mass-on-spring hanging in the Earth's gravitational field).
Solve for the classical motion (hint: the equilibrium position just
shifts, the mass will happily oscillate about the new equilibrium
position just exactly as without the additional force). Following
the same hint, exactly solve the Schrödinger's with the
additional potential -Fx. Now treat the the additional
potential as a perturbation and find the new energy levels, correct
to second order.
The (3-d) nuclear potential is often modeled as being flatter than
r2, say r4. Consider the
on the 3-d oscillator. When =1 we'd have our solution
to the r4 potential (but low-order perturbation theory is probably
not valid for such large ). In any case the aim is to see how
the low-energy states (say E'<=7) move as a function of .
Note that since the perturbation is spherically symmetric, V' will not connect
different lm states. Find the energy shifts correct to second order for the
E'=7 degenerate set, and first order for E'<7.
The actual order of filling in nuclei seems to be: 1s, 1p, 1d,
2s. What value of is consistent with the data?
Note that for <0 (even slightly so) the bound states
are not stable, and will "leak". Explain. What is the meaning of the perturbation
theory in this case?
For the quarkonium states, it is believed that the potential for small
r goes like -1/r and for large r goes like +r.
Consider the perturbation:
Find the first-order shifts for the 1s, 2s, and 3s levels.
Calculate the ratio of the (3s-2s) energy difference over
the (2s-1s) energy difference as a function of .
For the bottomonium system the ratio is (10.3553-10.0233)/(10.0233-9.460)=.59.
Find the value of that works. For a pure oscillator this
ratio is 1, what is it for the pure -1/r potential (another possible
start for perturbation theory)?
Consider the perturbing potential V'=(x'2-y'2)
in the case of the 2-d oscillator. There are 4 degenerate E'=8 states, which we may consider either
in the |nxny> basis or in the |nrm> basis.
Follow this problem through as we did on page 15 where we did the 3-d case.