June 2004
QUANTUM MECHANICS
PROBLEMS
FROM THE
1. The projection theorem states that for
a vector operator ,
,
where the a’s (with and without
primes) denote nonangular quantum numbers, q labels the vector
component, and is the angular momentum operator of the
isolated system.
(a) Briefly (a sentence or so), what is the
physical meaning of this theorem?
(b) Suppose a system has total angular
momentum and total magnetic moment .
Find the expectation values of the components of in the total angular momentum eigenstate ;
that is, J2 has eigenvalue and Jz has eigenvalue .
2. In the Stark effect for a hydrogen atom
in its ground state, the energy change DE associated with a small
applied electric field is proportional to the square of the field strength ,
,
where a
is the polarizability.
(a) Find an expression for a using perturbation
theory.
(b) Neglecting continuum states, use bounds
on energy differences to establish that
,
where a0 is the Bohr
radius. (The ground state wavefunction for hydrogen is ,
with C a normalization constant,
while the ground state energy is -e2/2a0.)
3. We know that for the one-dimensional
oscillator, .
(a) Derive and expression for the commutator
.
(b) Prove that is an eigenstate of the (non-Hermitian)
annihilation operator a, and give its eigenvalue.
(c) Use the identity to establish that .
(d) Prove that these states are complete,
that is, for l = x + iy,
.(Hint:
Use polar coordinates.)
4. A particle of mass m that can
move in three dimensions interacts with a potential V(r) =
-l d(r - a),
that is, a delta function spread over the surface of a sphere of radius a,
like a spherical shell.
(a) Find the minimum value of l for which this
potential has a bound state.
(b) As the strength of the potential is
increased, would you expect to see more bound states? Give a qualitative
argument only.
(c) Find the phase shift for scattering from
this potential in the low energy limit. (Warning: as k ® 0, the phase shift is
linear in k.)
5. A light wave is incident on a hydrogen atom in its ground
state at the origin. The frequency w is sufficient to
liberate the electron into a plane wave state; the problem here is to calculate
the rate at which this occurs. Treat the incoming light wave as a classical
field.
(a) State Fermi’s Golden Rule for
calculating transition rates.
(b) Find an expression for the matrix
element for the electron to be ejected into a plane wave state with momentum .
(c) Give an order of magnitude argument for
ignoring the k-dependence of the incident light wave in the matrix
element, assuming the light is in the visible range.
(d) Find explicitly the angular dependence
of this matrix element, and therefore of the relative probability of emission
into different solid angles dW.
(e) Do the necessary integral to evaluate
the matrix element. You can take the normalization constants of the hydrogen
ground state and the plane wave state to be N1, N2
if you don’t know them.
6. A particle with electric charge q
is confined to move in the xy-plane, with a uniform perpendicular
magnetic field, magnitude B, and a corresponding vector potential = (0, B x, 0).
(a) Write down the time-independent
Schrödinger equation for the particle.
(b) Find the commutation relations of the
Hamiltonian with the momenta px and py. Use
this information to write the eigenfunction in the form y(x, y) = y(x)j(y) where one of
the functions y, j is a plane wave. Which
one? Find the differential equation the other function satisfies, and
write down its lowest energy solution.
(c) Suppose now that the system has finite
size L in the y-direction, and assume periodic boundary
conditions in that direction. How does that affect possible solutions of the x-direction
equation?
(d) If the system can be taken to have
infinite extent in the x-direction, what is the density of lowest energy
states per unit length in that direction?
(e) Choose one particular ground state
wavefunction and find the probability current distribution.
7. A spin one (not spin one-half!)
particle has its component of spin parallel to the positive z-axis equal
to one (that is, it is in the m = 1 state).
(a) Suppose the component of spin is
measured in the direction defined by the vector (1, 1, 1). What is the probability of finding m =
1 in that direction?
(b) In the same (1,1,1) direction, what is
the probability of finding m = -1?
Possibly useful info (possibly not!):
8. Consider a two-level system with
Hamiltonian H = H0 + V(t), where H0
has energy levels E1, E2, E1
< E2. The matrix
elements of the perturbation V(t) in the eigenbasis of H0
are:
(a) Write down Schrödinger’s equation for
the system, and show that it can be expressed in terms of the amplitudes c1(t),
c2(t) for the two states as the coupled differential
equations:
Given that at t = 0 only the lower level is populated so c1(0) = 1, c2(0)=0, find |c1(t)|2
and |c2(t)|2 exactly by solving the
coupled differential equations.
(b) Do the same problem using time-dependent
perturbation theory to lowest non-vanishing order. Compare the two approaches for small values
of g.
9. You have a 1-dimensional harmonic
oscillator, V(x) = ½ mw2x2, with an angular frequency composed of a constant term
plus a small “wobbling” term:
where . The system is in its ground state just before
t = 0.
(a) Use perturbation theory in the small
parameter dw to find the
(time-dependent) amplitude for the probability that the system is in a particular
excited state for times larger than T. Leave your answer in terms of
unevaluated matrix elements and/or unevaluated time integrals.
(b) Evaluate any matrix elements that you
may have found in part (a). To which excited states can transitions occur to
this order of perturbation theory?
10. Two electrons scatter from one another.
The forces between them are such that f(q) would be the
scattering amplitude if the electrons
were distinguishable. (q is the center-of-mass
scattering angle.) But of course real electrons are not distinguishable, and
the following questions apply to real electrons. [Express your answers, when
appropriate, in terms of f(q).].
(a) What is the differential cross section
at q = 90° if the electrons form a
spin singlet?
(b) What is the differential cross section
at q = 90° if the electrons form a
spin triplet?
(c) What is the differential cross section
at q = 90° if the electrons are
unpolarized?
11. Two electrons (electron mass = m,
spin = ½) are confined to move on a circle of radius R (as for example
in a model for the electronic states of [ring-shaped] benzene). The Hamiltonian
is
where is the angular position of particle i
on the circle.
(a) Find the lowest energy, as well as the
corresponding eigenfunctions(s). Give the degeneracy, if any, explicitly. Do
the same for the first excited state.
(b) Include in the energy of part (a) an
electron interaction term of the form
Find the corrected energy and degeneracy (if any) of the ground state(s) to lowest
order of perturbations in V0. Discuss the first order
correction to the wave function(s) of the ground state(s)¾in particular, how does
the zeroth order wave function(s) of the first excited state(s) contribute?
12. It can be shown that the amplitude for
the decay of an initial atomic state |iñ (wave function ) to a final state |fñ (wave function ) with the emission of a single photon is
proportional to the integral (the reduction of a matrix element) ,
where in this expression the photon wave number is .
If the wavelength of the emitted photon is large compared to the atomic
dimensions, then one can set the harmonic factor .
This is known as the dipole approximation, and the emitted photon is dipole
radiation.
(a) Consider hydrogen-like atoms (arbitrary Z
for the nucleus but a single electron). For which such atoms is the dipole
approximation justified? You may suppose that the relevant measure of atomic
size is provided by the size of the smaller of the two states in question.
(b) Assume that the dipole approximation is
justified. You can then integrate the expression above by parts to find the
integral
.
Show that this integral is proportional to the matrix element of the operator ,
i.e. .
(That is why the term dipole approximation is used.)
(c) By writing in spherical coordinates, show that the dipole
radiation is emitted such that the orbital angular momentum quantum number
changes by one.
Hint: Spherical harmonics have the general property that
13. Two identical
spin-1/2 fermions of mass m are
placed in the same one-dimensional harmonic oscillator potential The only interaction between the two fermions
is a small quadratic one of the form ,
where g is a constant.
(a) Determine the lowest three distinct possible total energies
for the 2-fermion system to first order in g. [That means you may ignore O(g2)
effects.] Consider all possible spin
states.
(b) What are the corresponding normalized
2-particle wave functions to zeroth-order in g? You must show the spin as
well as spatial parts of the wave functions.
[You may express your wave function in terms of normalized 1-particle
eigenfunctions of the 1-particle harmonic oscillator problem
and needn't give explicit formulas for .]
14. A particle of mass m starts (t = ) in the ground state of a one-dimensional
harmonic oscillator At time t
= 0, the harmonic oscillator potential suddenly disappears, and then it
reappears later at a time t = t. That is, the potential is
A long time later (t ® ¥), the energy of the
particle is measured. Solve exactly for
the probability that this measurement yields
The ground state wave function of a harmonic oscillator is An integral you may or may not find useful
is
15. Two electrons feel the same potential V(r)
(for example, the Coulomb potential of the nucleus). Ignore the Coulomb repulsion between the
electrons as well as all spin-dependent forces.
Let and be two (one-particle) energy eigenstates of
the potential problem. Suppose one
electron is in state and one is in state . An example of a two-electron wave function
with this property is
(a) Write the analogous two-electron wave
function if the two electrons have total spin s = 0.
(Total spin refers here to )
Correctly normalize your wave
function.
(b) Write the analogous two-electron wave
functions for total spin s = 1 for
the three cases ms = 1, ms = 0, ms = .
(c) Consider the expectation of the square of the separation between the two electrons. For the s
= 0 case, show that this expectation can be written in the form
where (with i,
j = a or b) denotes the
one-particle spatial matrix elements
As part of your derivation, find the numerical values of the coefficients
(d) Do the same as part (c) but for the s = 1 states.
(e) Show that electrons with s = 1 are generically further apart on
average (and never closer on average) than electrons with s = 0, as measured by This is an example of what fundamental
physical effect?
16. A long-wavelength, polarized,
electromagnetic wave is incident on a hydrogen atom. Long wavelength means Ignoring spin, the electron's interaction
with the external radiation field can be treated as a perturbation,
(in the Coulomb gauge, ),
to the atomic Hamiltonian. Neglect the A2
term in parts (a)-(c). Treat the
electromagnetic wave classically and
the electron quantum mechanically.
(a) Explain why, under these circumstances,
the probability for the electron to make a transition from an initial
(unperturbed) atomic eigenstate to a final one is proportional to the square of some linear
combination of the matrix elements What determines which linear combination of and is relevant?
(b) Derive an expression for Wfi in terms of and the (unperturbed) energy eigenvalues of
the states, Ei and Ef. As always, show your work.
(c) Use parity, and the fact that r is a vector operator, to derive the
selection rules, in the above approximations, for radiative transitions between
states of angular momentum and . Explain your reasoning: a simple statement of the selection rules is not an adequate response.
(d) When the intensity of the
electromagnetic wave is small enough to treat Hint as a perturbation to the atomic problem, then one
often ignores the A2 term
in Hint compared to the A×p term, as you have
above. Justify this. Explain any caveats
to your justification.
17. The standard l = 1 spherical harmonics can be written in the form
where the unit vector has been used to specify direction, and refer to its components.
(a) What are the l = 1 eigenfunctions of Lx
in terms of and ? Clearly specify the eigenvalue of Lx associated with each
eigenfunction you write down.
[Hint: Simply use symmetry and/or
simply rotate coordinates.]
(b) Consider the electron of an idealized
hydrogenic atom. [“Idealized” means the
simple non-relativistic Coulomb problem:
ignore spin-orbit interactions, center of mass corrections, hyperfine
splitting, the Lamb shift, etc.] An
experimenter has a large collection of these atoms. For each atom, he measures L2 and Lz , and he throws the atom away if the result does not
correspond to l = 1 and mz = 1. For each atom he keeps, he then does the
following:
Simultaneously measure L2
and Lx (an instant after
the earlier measurement).
What are the possible values of L2
and Lx , and what fraction
of his large number of measurements has each value?
(c) Another experimenter has a similar
collection of atoms. For each atom, she
also measures L2 and Lz and throws the atom away
unless the result corresponds to l =
1 and mz = 1. For each atom she keeps, she then does the
following additional sequence of measurements:
[1.] Simultaneously measure L2 and Lx ( an instant after the earlier measurement). An instant later, make measurement 2 below.
[2.] Simultaneously measure L2 and Lz.
What are the possible values of L2 and Lz from the final measurement (measurement 2) above, and
what fraction of her large number of measurements has each value?
18. Consider a system of three spin-½ particles.
Let be the total spin.
(a) What are the eigenvalues of S2?
(b) How many independent states (of the
spins) are there with each of those eigenvalues?
(c) Explicitly write a complete set of
orthonormal S2
eigenstates, writing each as a superposition of the eigenstates ,
,
,
etc. Normalize your states.
A formula which may or may not be useful to you:
19. Consider the bound-state problem for the
attractive one-dimensional delta-function potential . You can convert the usual stationary
Schröedinger equation for this into a momentum space equation by multiplying
the Schröedinger equation by and integrating over x.
(a) Using the fact that the momentum-space
wave function j(p) is the integral over
x of fy(x), find the equation obeyed by j(p).
(b) Solve this equation.
(c) Once you have solved for j(p), find y(x) through the inverse transform.
(d) Show that evaluating y(0) provides a relation
for the energy. How many bound states
are there? Solve the energy relation (if
you can) to find the allowed bound state energy (or energies).
20. Use the Rayleigh-Ritz variational method
to estimate the lowest energy level of the one-dimensional anharmonic
oscillator defined by the Hamiltonian
(Hint: try a Gaussian trial wave
function with one variational parameter.)
21. A particle of mass M is restricted to move only along the circumference of a fixed
circle of radius R lying in the xy plane. The particle moves freely along the
circumference.
(a) What are the eigen-energies of this
system? What are the corresponding wave
functions as functions of the position variable ?
(b) Write the Hamiltonian in terms of the
momentum ps conjugate to s.
(c) Now suppose that the particle is an
electron and that there is an external uniform magnetic field in the +z
direction. Write down the
Hamiltonian. Include spin, but ignore
spin-orbit coupling. [Note: .]
(d) What are the eigen-energies of this
system? What is the ground state energy E0 as a function of B?
Sketch E0 vs. B.
22. (a) Derive the commutation relations of with Jz,
where is an angular momentum operator. (You may assume the standard results for or, if necessary, derive everything you need
from scratch.)
(b) Derive
the normalization of (That is, what is if and Then phrase your final result as a formula
for in terms of other ’s (using any consistent phase
convention). In your derivation, you may
assume the standard result for the eigenvalues of .
(c) A two-particle system consists of a spin
1 particle and a spin particle.
The total spin state can, of course, be represented either by a basis
with good S1z and S2z quantum numbers or alternatively
by a basis with good and
Sz = S1z + S2z quantum
numbers. Write the expansion of each
possible state in terms of states.
[Hint: Start with the state with
highest m1 and m2.]
23. We define the Heisenberg representation
as the one in which time dependence is put into operators and not into states.
The time-dependent operator A(t) is defined through the expectation
value, as
áy,t|A|y,tñ = áy,0|A(t)|y,0ñ,
where A is the (time-independent)
operator that appears in the Schrödinger representation.
(a) Give an explicit expression for A(t)
in terms of A and any other operators
or algebraic quantities you might like to use.
(b) Find the equation of motion obeyed by
the operator A(t).
(c) Consider a harmonic oscillator, for
which the hamiltonian is
H = h?(a†a + 1/2),
where a† and a are the creation and destruction
operators. Show that in the Heisenberg
picture the commutation relations between the time-dependent operators are the
same as those at t = 0, namely
[a†(t), a(t)] = -1.
(d) Solve the equations of motion for a†(t) and a(t) in terms of their values at t = 0.
(e) Calculate the commutator [a†(t), a(0)].
24. Consider a 1-dimensional
potential consisting of two identical 1-dimensional square wells side by side.
The depth of each well is V0 and the width of each well is a, with the product V0a2 large enough so that
several bound states would be possible for the single well. The separation between the wells is b, as in the figure.
(a) Sketch
the ground state wave function y0 and the first excited
state wave function y1 in the three cases b = 0, b >> a,
and b = O(a).
(b) How do the corresponding energies E0 and E1 vary as b
varies? Refer to the three cases for b
in part (a) and sketch E0
and E1 as functions of b.
(c) If you view this model as a model of the
potential energy of the electron in H2+; the nuclei an
move to minimize the energy. Does the electron serve to bring the nuclei closer
or to push them apart?
25. F
is the scattering amplitude of a very low energy plane wave, wave number k, with kL<<1 (where L is the range of the potential), traveling
along the positive z-axis, scattering
from a single scattering center.
(a) What is the q
- dependence of F, where q is the scattering angle,
measured from the positive z-direction?
(b) If there are two such scattering centers
located respectively at (x, y, z)
= (-a, 0, 0) and (+a, 0, 0) what is the scattering
amplitude at the same k as in part
(a)?
(c) What is your answer to (b) if the two
scattering centers are at (0, 0, -a)
and (0, 0, +a)?
26. Atoms are made up of nuclei that consist
of protons and neutrons with masses M @ 940 MeV/c2 and electrons with masses
described by me @ 0.5 MeV/c2. Suppose that we lived in a world in which the
electrons had mass 5.0 MeV/c2
but with the nucleons retaining the masses they have now. You can continue to regard the nucleons as
essentially infinitely heavy, and you should also pretend that there are no
weak interactions in this hypothetical world (if there were, there would be no
world!).
(a) What is the density of liquid water?
(b) At what temperature would paper
burn? (normal paper burns @ 451° Fahrenheit)
(c) Given that the rate for the 2p ® 1s transition in real hydrogen is 0.6 ´ 109 s-1,
what would be the rate for this transition in our hypothetical world?
(d) Would the Planck radiation law be
changed in the hypothetical world, and if so, how?
27. A three-dimensional harmonic oscillator
is perturbed by a constant force. Calculate the energies En to second
order and compare with the exact results for energies and states. Explain.
28. The separation R between the nuclei of two hydrogen atoms in their ground state is
much larger than the Bohr radius.
Starting with the classical Coulomb interaction terms show that there is
the small quantum-mechanical interaction between the atoms is attractive, and
that it varies as e4/R6. [Hints: It is useful to place one nucleus at the origin, to
define the position of the second as lying on the z-axis, and to measure the position of each electron with resect to
its parent nucleus.]
29. (a) Consider
a rotator consisting of two identical particles
(for the more sophisticated among
you, you can consider them to be bosons) of mass m connected by a massless rigid rod of length 2R. The center of the rod is
attached to a point, so that the system is constrained to move in the xy-plane. When we ask about angular momentum,
therefore, we are really asking about the z-component
of angular momentum. What is the angular
wave function, and what are the possible values of the angular momentum?
(b) We can use an extension of the above
result to make a "perfectly smooth" cylinder by considering a large
chain of atoms aligned in a circle. (We consider only rotations about the axis
perpendicular to the plane of the ring.) Therefore suppose you have an array of
N identical equally spaced particles,
each of mass m, forming a ring of
radius R. The total mass of the ring
is M = Nm. Assume throughout that M and R are fixed. What are the possible values of the angular
momentum of this system? (Hint: Think
about invariance of the wave function under rotations of the system).
(c) Find the energy difference between the
ground state (that of zero angular momentum) and the first excited rotational
state - in particular, what is the difference in the "smooth" limit,
the case N ®
¥
?
(d) Repeat part (c) for a marked cylinder,
in which, say, one of the atoms is missing or is different from the
others. Explain the difference, if any,
between your answers to parts (d) and (c).
30. Consider the bound states of two massive
particles whose interaction V is central, attractive, and everywhere finite.
(a) Show by variation that the ground state
of the system is an S state.
(b) Use the WKB approximation to show that,
if with >
2 for ,
then the system has only a finite number of bound states.
(Hint: Use the variational reasoning of
a) as a justification for considering only S-states and recall .)
31. Consider an electron moving in the
periodic potential
where V0 > 0. There is a solution to Schrödinger’s
equation having the form
Y(x)
= eikxUk(x),
where
(a) Show that Y(x) in the
interval 0 < x < 1 has the form Y(x) = Aeiqx
+ Be-iqx and give q in
terms of the electron energy E.
(b) From the continuity of the wavefunction
at the origin (and the periodicity of U) find an equation relating A,
B, q and k.
(c) Find another relationship among V0,
A, B, q and k by considering the derivatives of the
wavefunction on approaching the origin from either side.
(d) Deduce that the E, k
relation is determined by
.
(e) Do real k solutions exist for
energies E such that q is close to but slightly larger than p ? Explain.
32. In the ammonia molecule NH3,
the three H’s form an equilateral triangle and the nitrogen atom lies on a line
through the centroid of the triangle and perpendicular to its plane. Points of minimum energy are at distances ± a
from the plane. Refer to the state with
the and at -a as . Suppose the Hamiltonian has the form H = Ho
+ V where
(a) Find the energy of the ground state and
first excited state of the system. What
is the form of the corresponding wavefunctions?
(b) Give a very rough estimate for the
electric dipole moment of NH3 in configuration
(c) Derive an expression for the energies of
the two eigenstates of NH3 in an electric field, and sketch your
result as a function of field strength.
33. Derive an expression for the differential
scattering cross section for the elastic scattering of electrons by hydrogen
atoms. Use the Born approximation. For hydrogen, here ao is the Bohr radius. (Ignore
exchange.)
34. A charged particle in a potential well (for instance, an atom with one electron outside of closed shells) has the spectrum shown in the figure: If the particle is in the second excited state (energy E2), and emits a photon, which transition is most likely? Calculate the transition rate for this photon emission. The radial parts of the wave functions are:
Carry out
the integrations if time permits.
35. (a) Considering
only Coulomb interactions, what is the binding energy of the hydrogen atom in
the ground state?
(b) The effective Hamiltonian for the ground
state of hydrogen may be written
where Ho is the Coulomb Hamiltonian, A is a constant, B is a
magnetic field in the Z direction, and are respectively the electron’s and proton’s
spin vectors. What physical quantities
are represented by and ? What is the order of magnitude of the ratio ?
(c) If B
= 0, what is the splitting of the ground state in terms of A? What is the splitting called?
(d) What is the splitting of the ground
state in the limit ?
(e) Sketch qualitatively how the energy
levels of the system change as B is increased from 0 to .
36. A plane wave is incident from the left on the potential shown below
(a) Calculate the reflection coefficient for
all incident energies.
(b) What happens to a wavepacket incident on
this potential? Give a qualitative
discussion of what happens as a function of the average energy of the
wavepacket, including the possibility of resonance with quasibound states in
the well.
37. (a) Find
the rotational energy levels of the HI molecule knowing that the interatomic
spacing is .
(b) Estimate (to within 1%) the energy of
excitation from the ground state to the first level. In what part of the optical spectrum does
this energy fall?
(c) What is the degeneracy of each level?
(d) Explain why there is no zero point
energy for rotational motion.
38. An electron is contained inside a sphere
of radius R. What is the pressure P exerted on the surface of the sphere, if
the electron is in the lowest S state?
39. An electron with charge e and mass m is
confined to move on a circle of radius r. It is perturbed by a uniform electric
field F parallel to one of the diameters of the circle. Find the perturbation of the energy levels up
to terms of the order of F2.
Notice in particular the anomalous behavior of the first excited state.
40. An electron of charge e and mass m is
constrained to move on a ring of radius R.
(a) What are the energies and eigenfunctions
for the system?
(b) Consider a perturbing potential ,
where z is along a diameter of the ring, and find the energies of the two
lowest levels to second order in perturbation theory.
41. Neutrons can be scattered in the Coulomb
field of a nucleus because of their magnetic moment. Write down the Hamiltonian of the interaction
and calculate the spin-averaged differential cross section in the Born approximation.
(Treat nonrelativistically.)
42. Consider
an electron in the potential . There is also a magnetic field present, Neglecting spin, calculate the energy and wave
function for the ground state and the three lowest excited states when B0
≠ 0. (By lowest excited states, we mean
those three states corresponding to the lowest excitation when B0
= 0). Include terms of order in the Hamiltonian. Note:
Recall that for the linear harmonic oscillator in 1-dimension, and where N0 and N1
are normalization constants.
Note:
43. Consider a hydrogen atom in its ground
state in a static electric field There is also incident radiation of intensity
I(?) per unit ?, i.e. has dimensions ergs/(cm2 sec). Let the radiation of I(?) have polarization .
(a) At what frequencies does absorption take
place? (Assume Eo is
sufficiently small that its effect on the energy can be neglected beyond first
order in Eo. Do not
include first-order and higher-order perturbations on the wave function.)
(b) Calculate the transition probability/sec
at each frequency:
44. Radiation I(?) traveling along and polarized along is incident on a hydrogen atom in the ground
state. Calculate the transition rate for
a transition.
45. Consider a particle of mass m in a
one-dimensional potential
(a) What is the form of the wave function in
each region of space - < x < ?
(b) What are the transcendental equations to
determine the bound states?
(c) How does the number of solutions depend
on ?
46. (a) State
the variational theorem.
(b) Consider i.e. the Coulomb potential is screened. Use the variational theorem to estimate the
lowest (in energy) state with l = 1. Use the trial
function
You will get a complicated algebraic equation for .
Do not carry beyond that point.
(c) What should the result for be when What is the corresponding energy? (Hint: you
may know this without calculating.) You
may use these results to check your equation for .
Note:
47. Consider a nonrelativistic quark of mass M
and spin ½ in a 3-dimensional potential V(r) = ½ Mw2 r2.
(a) What is the energy of the
ground state?
(b) What is the energy of the first excited
state?
(c) Suppose the levels are perturbed by a spin-orbit
interaction
where g is a constant. In using first-order perturbation theory to
calculate energy shifts in the presence of Vso, what are the
appropriate quantum numbers to use with Vso?
(d) Calculate the first-order shift due to Vso
for the ground state. Calculate the radial integral if necessary.
(e) Calculate the first-order shift due to Vso
for the first excited state. Calculate the radial integral if necessary.
48. Consider a particle of mass M constrained
to move on a vertical hoop of radius R = 1.
(a) Suppose What is at a later time t? (Neglect gravity in this
part.)
(b) Suppose the effect of gravity is
included by considering the perturbation What are the energies of the lowest three
levels correct through second order in the perturbation ?
49. Two nucleons each of mass m (938 ) scatter at an energy of 1 in the center of mass system.
Assume that the nucleon-nucleon potential is
cm
What is the total scattering cross section?
FIRST BORN APPROXIMATION:
50. Consider a hydrogen atom in the 3d level
with .
What is the lifetime of this state?
51. Consider an electron in a rectangular box
of sides with lengths a, b, c.
a = 10b and b = c.
The electron is in the second excited state of this box.
(a) What are the energies of the photons
emitted in transitions from the second excited state to lower states?
(b) If there are many identical boxes each
with one electron, what are the relative numbers of photons corresponding to
each transition from the second excited state?
Hint: Consider the transition rates from the second excited state.
52. Consider an electron in the lowest state
of a 1-dimensional well of depth Vo
and width a.
. 
Radiation of energy hw with electric field vector polarized along the x-direction (the
single dimension of the problem) is incident upon the electron. Calculate the cross section for
photoionization. (photo-effect)
Make the following approximations:
i) Assume Vo is deep enough that you can describe the lowest level by the lowest level of an infinite square well of width a.
ii) Assume the dipole approximation.
iii) Assume the continuum states are plane waves.
53. Consider an electron in a magnetic field At t=0 the electron spin is determined to
point along What is the direction of the electron's spin
at a later time T? (Take .)
54. Let be the intrinsic angular momentum of a
particle of spin 1. (Let h = 1.)
x = i 2 = 1(1+1) = 2.
(Note: Since 2 = 2 you are limited to the space of functions f with S 2 f = 2f.)
(a) Show that ,
and give arguments to show where is a unit vector in an arbitrary direction.
(b) Show that the rotation operator for
rotation by an angle f about is given by
.
(You may assume the usual exponential form for R.)
(c) Let be a vector of norm 1 such that and let and be the vectors obtained from it by rotations
of + about and about
respectively.
Prove
Also show that form an orthonormal basis.
55. Electrons of energy E are incident from
the left on a potential given by
What is the probability that a given electron is transmitted to the right,
i.e., into the region x > a?
56. Consider the Hamiltonian H = H(o) + H' for a three-level system. The Hamiltonian H is represented by the following matrix
where the matrix elements are ,
with and being eigenstates of H(o). Assume .
(a) Calculate the energies of all three
levels through second-order in .
(b) Calculate the third-order correction for
the lowest level.
57. Consider
an electron in the infinite square well
Suppose the electron is known to be in the first excited state for t
< 0. At t = 0, the walls are
suddenly removed.
(a) What is Ψ(x, t)
for t > 0? (You may leave your
answer in the form of an integral but be as explicit as possible.)
(b) If a measurement is made
of the electron’s momentum at t > 0, what is the probability that the
measurement yields ? (You may leave your answer as a dimensionless
integral. However, estimate it if you
can).
58. Consider two electrons in the potential The
system is in the excited state with one electron in the ground state and the second electron in the second excited
state with and . What is the transition rate directly to the
ground state due to 1-photon emission?
Note:
59. Consider an electron with spin
"up" along .
(a) What is the spinor describing this
situation?
(b) What is the probability of measuring
spin “up” along
60. Calculate the cross section for
two-photon ionization of a hydrogen atom in the 1s level. The photon energy is hwo and
The photon flux is N = 1014
photons/cm2 -sec.
(Hint: You may find it useful to note .)
Do not evaluate integrals unless you have sufficient time.
61. Calculate in lowest order correction of
the ground state energy for a hydrogenic atom, due to a finite nuclear
size. Assume spherically symmetric
nuclear charge uniformly distributed within the nuclear radius R. If the
nuclear radius is what is the relative correction for Z=50?
Calculate also the second order energy correction approximating the denominator
Eo-En in the
expression for the energy correction by Eo
for all n. Use closure to evaluate the sum.
Does the result make sense?
Explain!
The ground state wave function and energy are
62. Consider a hydrogen atom in a homogeneous
magnetic field along z axis. Note that
the vector potential can be written as .
What is the appropriate Hamiltonian? (Include the spin term.) How does the
presence of the magnetic field affect…
(a) The energy of the ground state?
(Calculate results through 2nd order in B.)
(b) The energy of 2p states? Neglect spin-orbit interaction.
(c) The frequency observed for 2p®1s transition, when B¹0.
63. A p-state electron is in one of the
eigenstates of the component of angular momentum along the direction lying in
the yz plane making an angle q with the z-axis. Find, for each possible such eigenstate, the
probability that a measurement of Lz. will give the value h.
64. The Hamiltonian of the positronium atom
in the 1S state in a magnetic field B along the z axis is to a good
approximation
if all higher energy states are neglected.
The electron is labeled as particle 1 and the positron as particle 2.
Using the coupled representation in which and are diagonal, obtain the energy eigenvalues
and eigenvectors and classify them according to the quantum numbers associated
with constants of the motion of the states without the presence of magnetic
field.