Graduate Quantum Mechanics II (Physics 524)
Problem Set 1
Due Fri Jan 22, 2010
1. Clebsch-Gordan coefficients
(a) Two particles, each with spin 12 , combine to make a state |ψi with total spin
quantum number stot = 1 and Sz quantum number mtot = 0. In the state |ψi,
what is the probability that particle 1 has ms = + 21 ?
(b) A nucleus of spin quantum number 3/2 has an electron bound to it in an l = 0
state. The total angular momentum of the system is stot = 2, with z-component
mtot = −1. What is the probability that the z component of the spin of the nucleus
is ms = +3/2?
(c) Two spin-1 particles each have zero z-component of their spin (ms = 0). What is
the probability that measuring their total spin will yield stot = 0? What is the
probability that stot = 1? What is the probability that stot = 2?
(d) Consider two nuclei. Nucleus 1 has s = 2 and ms = +2. Nucleus 2 has s = 3/2
and ms = −3/2. What are the possible values of the total angular momentum
quantum number stot and what is the probability of each?
A table of Clebsch-Gordan coefficients is available on the course website http://www.
physics.wustl.edu/~alford/p524/.
{15 points}
2. Textbook errata.
Go to web site http://www.physics.ucla.edu/~abers/QMErrata3.html, the list of
errata for the textbook, and mark the significant corrections in your copy from page
138 to 200 (or further).
Graduate Quantum Mechanics II (Physics 524)
Problem Set 2
Due Fri Jan 29, 2010
1. Two spinors make a 3-vector.
If you want to use Mathematica to do this problem, see the course web page for a
“getting started” guide and a file SO3_rotations.m that will give you some of the
definitions you need.
Consider the operation g(φ) which is a rotation by angle φ about the x axis.
(a) In an arbitrary representation n, g(φ) can be written as an exponential of an
element of the Lie Algebra,
D(n) (g(φ)) = exp(−iθA D(n) (JA )) ,
where A is summed over the values x, y, z. What are the θA in terms of φ?
(b) Write down the 3 × 3 matrix for g in the 3 irrep, i.e. the matrix D(3) (g(φ)) which
performs g on a 3-vector in the (vx , vy , vz ) basis. (“D(3) ” is my notation for Abers
“D(1) ”).
(c) Write down the 3 × 3 matrix which performs g on a 3-vector in the (v+1 , v0 , v−1 )
basis. I called this “D(3s) (g(φ))”.
(d) Write down the generators in the 2 representation (the one that acts on spinors,
see Sec. 4.3.4), i.e. D(2) (Ji ), and from that calculate the 2 × 2 matrix D(2) (g(φ))
1
which performs g on a spinor. (“D(2) ” is my notation for Abers “D( 2 ) ”).
(e) A 3-vector can be constructed from two spin- 12 spinors. If we write the components
of the vector v in the “3s” basis and the components of the spinors in the basis of
eigenstates of Sz ,
vm = Cmαβ ψα χβ .
where m = −1, 0, +1, and spinor indices α, β = −1, +1. What are the coefficients
Cmαβ ?
(f) Show that if we perform the rotation g(φ) on ψ and χ, the effect on the vector v
is the same as acting directly on v with the appropriate 3 × 3 matrix.
{20 points}
2. Consider 3 possible Hamiltonians
H1 =
1 2
L + V (r),
mr2
H2 =
1 2
L + Bz Lz ,
mr2
H3 =
1 4
L + V (r) ,
mr2
where Bz is a constant over all space and ~ = c = 1 units are used. For each
Hamiltonian, say which quantities are conserved. Suppose a spinless particle starts off
with angular wavefunction Y20 (θ, φ) (using the notation Ylm for the sperical harmonics).
For each Hamiltonian, describe what spherical harmonics could be included in its angular
wavefunction at a later time.
{10 points}
Quantum Mechanics II
Problem Set 3
Due Fri Feb 5, 2010
1. Distinction between basis elements and components
In class (and in the file SO3_rotations.m on the course website) we wrote down
the matrix M that converts the Cartesian components (vx , vy , vz ) of a vector to its
“spherical” components (v+1 , v0 , v−1 ). So the change of basis on the components of a
vector is
vq = Mqi vi .
where index i ranges over x, y, z; index q ranges over +1, 0, −1. Any vector can be
written in terms of orthonormal basis elements
v = vi ûi = vq êq
where ûi are the Cartesian basis (ûx = î etc), and êq are the “spherical” basis.
(a) Using the fact that the vector v itself is independent of basis, calculate the matrix
M̃ which gives the transformation of the basis elements,
êq = M̃qi ûi .
Calculate M M̃ T and explain the result you get.
(b) How should one convert the angular momentum generators Ji to the spherical
versions Jq —should one use M or M̃ ? Are the Ji a basis or are they components?
(c) In eqn (5.16), the book describes how to take three operators Vi that form a vector,
and express them in the spherical basis as Vq . Are the Vi and Vq components of a
vector, or basis elements for a vector?
{10 points}
2. (a) Consider an operator A obeying [Jz , A] = nA. Show that when A acts on an
eigenstate of Jz with eigenvalue m, it yields an eigenstate with eigenvalue m + n.
(b) From eqn (5.15) derive eqn (5.17a).
(c) Explain the relevance of the previous two parts to eqn (5.25).
{10 points}
3. In this question we use the notation |n, l, mi for Hydrogen atom energy eigenstates of a
spinless electron. V is an arbitrary vector operator.
(a) Explain why hn0 , l, m|x|n, l, mi = 0 for any n0 , n, l, m.
(b) Explain why hn0 , l, m|z|n, l, mi = 0 for any n0 , n, l, m.
(c) Calculate h3, 2, 0|V0 |3, 1, 0i/h3, 2, +2|V+1 |3, 1, +1i.
(d) Show that hn, l, m + 1|V+1 |n, l, mi/hn, l, m + 1|J+1 |n, l, mi is independent of m, and
hence calculate how h2, 1, m + 1|V+ |2, 1, mi depends on m.1
{10 points}
1
Note that in the book the Jq for q = ±1 are normalized differently from the “J± ” (see eqn (5.20)).
Quantum Mechanics II
Problem Set 4
Due Fri Feb 12, 2010
1. (a) Calculate how the operator A = (x + iy)/r changes the eigenvalue of Lz when it
acts on an eigenstate of Lz . It is best to work in spherical co-ordinates and use the
result of question (2a) on the previous homework.
(b) Using the notation |n, l, mi for Hydrogen atom energy eigenstates of a spinless
electron, show that any matrix element hn0 , l0 , m0 |A|n, l, mi can be factorized in to
a radial integral and an angular integral.
(c) Calculate h2, 0, 0|x+iy|2, 1, mi for all allowed values of m, and explain your results.
(If you decide to use Mathematica to evaluate some of the integrals, there are some
relevant examples in the file H_atom_states.m on the course website.)
(d) How does the operator B = (x2 − y 2 + i(xy + yx))/r2 affect the Lz eigenvalue of a
state it acts on?
(e) Calculate h3, 0, 0|A|3, 2, mi for all allowed values of m, and explain your results.
{15 points}
2. The state of a spin- 21 particle is described in the basis of eigenstates of Sz by a spinor
ψ = (ψ↑ , ψ↓ ). The action of the time reversal operation is T ψ = ησy ψ ∗ , where σi are
the Pauli matrices, and η is an arbitrary complex phase.
(a) Show that T 2 ψ = −ψ.
(b) Show that T σi T −1 ψ = −σi ψ for any ψ.
(c) Show that the orbital angular momentum operator transforms in a similar way, as
T Li T −1 = −Li (where T is now in the appropriate representation to act on spatial
wavefunctions).
{7 points}
3. Write down the most general possible 2 × 2 unitary matrix with determinant 1. By
performing an arbitrary isospin rotation simultaneously on all the constituent quarks,
show that the π + can be isospin-rotated into a state that contains a component of π 0
and into a state that contains a component of π − , but it will never contain a component
of η 0 .
{8 points}
Quantum Mechanics II
Problem Set 5
Due Fri Feb 19, 2010
1. Express the following in “natural” (~ = c = 4πε0 = kB = 1) units, using MeV as your
energy unit.
{4}
(a) The volume of a proton (taking its radius to be 1 fm).
(b) The time for light to cross a Hydrogen atom (ie travel twice the Bohr radius,
aBohr = 0.53 × 10−10 m).
(c) The surface tension of Mercury (0.49 Nm−1 ).
(d) A magnetic field of 1 Tesla.
2. Express the following in SI units.
{4}
(a) The mass of the pion, mπ = 140 MeV (convert to kg).
(b) A flux of particles J = 2.55 × 10−25 MeV3 (convert to m−2 s−1 ).
(c) The approximate energy density of a nucleus, 5 × 108 MeV4 (convert to Jm−3 ).
(d) A temperature of 0.001 MeV (convert to Kelvin).
3. Consider a two-dimensional harmonic oscillator,
1 2 1
1 2 1
px + 2 mω 2 x2
Hy =
py + 2 mω 2 y 2 .
2m
2m
(a) Write Hx in terms of creation and annihilation operators ax and a†x .
H = Hx + Hy
Hx =
(b) Write the eigenstates and eigenvalues of H in terms of the occupation numbers nx
and ny and their eigenstates |ψ(nx , ny )i. Draw a diagram showing the first three
energy levels, and the states |ψ(nx , ny )i in each level.
√
(c) Transform to the basis a± = (ax ∓ iay )/ 2 and repeat parts (a) and (b) in the
new basis, using eigenstates |φ(n+ , n− )i.
(d) Show mathematically that Lz commutes with H and calculate Lz in terms of a± .
(e) Which eigenvalues of Lz occur in the first three energy levels?
{10}
4. Consider a spinless electron in a magnetic field B oriented in the z direction. (See Abers
problem 6.4 for more information).
(a) Show that the classical motion of the electron is a helix along the z direction.
Calculate the energy of the particle in terms of the position (x0 , y0 ) of the center
axis of the helix, the radius R of the helix, the frequency ωc with which the particle
circles the center axis, and its velocity vz in the z direction.
(b) Show that for a quantum mechanical particle, x0 and y0 are conserved.
(c) Calculate hn+ , n− |x0 |n+ , n− i and hn+ , n− |y0 |n+ , n− i.
(d) Calculate the variance (“dispersion”) of x0 and y0 in a general energy eigenstate
|n+ , n− i. In which state is the product of the two variances smallest?
{12}
Quantum Mechanics II
Problem Set 6
Due Fri Feb 26th, 2010
1. (a) In the basis of eigenstates of Sz , write the eigenstates of Sx as two-component
spinors.
(b) Write down the density matrix for the spin state of electron that has probability p
of being in a state with spin up in the x direction and probability 1 − p of being in
a state with spin down in the x direction. You should use the basis of eigenstates
of Sz to write the matrix. What is the polarization vector P?
(c) Suppose the electron is in the state described above at time t = 0, and is in a
magnetic field B in the z direction. At a later time t what is the polarization
vector and density matrix? (Ignore the spatial state of the electron)
(d) For the electron in the previous part, at a given time t what is the probability that
a measurement of Sx will yield + 21 , and that a measurement of Sz will yield + 21 ?
{10}
2. A one-dimensional system has Hamiltonian H =
p2
mω 2 x2
+
+ λmx4 .
2m
2
(a) What are the energy levels of the system when λ = 0?
(b) Calculate the shift in those levels to first order in λ.
(c) Explain why, even if λ is very small, first-order perturbation theory can only be
used for the lower energy levels. Above roughly what energy level will perturbation
theory become unreliable?
(d) How could a classical physicist have arrived at a reasonably accurate answer to the
previous part, using only classical intuition about a particle moving in a potential?
{10}
3. Obtain Abers (7.16) from (7.15) and (7.5).
{5}
Quantum Mechanics II
Problem Set 7
Due Fri March 5th, 2010
1. Consider an isotropic harmonic oscillator in two dimensions, with Hamiltonian
p2y
mω 2 2
p2x
+
+
x + y2 .
H0 =
2m 2m
2
√
0
(a) √
Show that for
a
one-dimensional
harmonic
oscillator,
hn
|x|ni
=
(
n + 1 δn0 ,n+1 +
√
n δn0 ,n−1 )/ 2mω where |ni is an eigenstate of H0 with n quanta of energy.
(b) For the two-dimensional oscillator, what are the energies of the three lowest lying
states? Is there any degeneracy?
(c) We now apply a perturbation λV where V = mω 2 xy, and 0 < λ 1. Find
the energies of the three lowest states through first order in λ, as well as the
corresponding “zero-th order” eigenkets which define the appropriate basis for this
perturbation.
(d) Find the exact energy eigenstates of the full Hamiltonian H = H0 + λV , and
show that your results for the energies agree to the proper order with those of the
previous part.√ To get the exact
√solution, you will need to change to new variables
0
0
x = (x − y)/ 2, y = (x + y)/ 2, and appropriately defined px0 and py0 , such that
[x0 , px0 ] = [y 0 , py0 ] = i, and [x0 , y 0 ] = [px0 , py0 ] = [y 0 , px0 ] = [x0 , py0 ] = 0.
(e) The exact result for the ground state energy can be used to solve a very simple
model of the van der Waals attraction between two hydrogen atoms, a distance
r apart, in their ground states. Treat the electron in each atom as if it were
in a one-dimensional harmonic oscillator. The x and y coordinates now become
the displacements from equilibrium of the electrons in the two atoms (see figure):
they are displacements in the same direction, but of different electrons, so the
relationship with the previous parts of this problem is mathematical, not physical.
x
y
r
The dipole moments of the two atoms are Px = ex and Py = ey (note that
they are parallel in real space). The electric field at the right-hand atom,
due to the left-hand atom, is E = −Px /r3 , so the interaction Hamiltonian is
λV = −E · Py = Px Py /r3 = e2 xy/r3 . Show, from your exact result for the ground
state energy in the previous part, that the energy of interaction between the two
atoms in their ground states goes like 1/r6 . (The force then goes like 1/r7 .) Check
that the force is attractive.
{15}
This model shows the essential features of the van der Waals force: it is due to a second order
perturbative effect in which the quantum fluctuations of one atom result in a dipole electric field
that polarizes the other atom, resulting in a net attraction.
2. (a) Write down an expression for the Stark effect shift in the energy of a Hydrogen atom
in its ground state, to quadratic order in the electric field E. Your expression can
include an explicit sum over principal quantum number n, but you should evaluate
the sum over l and m. You do not have to evaluate the matrix element that depends
on n.
(b) If one evaluates the sum, the quadratic energy shift is ∆100 = −(9/4)a3 |E|2 . Make
a table showing what fraction of the full result is obtained if you evaluate your
expression including values of n up to nmax for nmax = 2, 3, and 4. How quickly
is this series converging? (The necessary integrals can easily be performed using
Mathematica, with the wavefunctions specified in the file H_atom_states.m on the
course website.)
{10}
3. Textbook errata.
Make sure you have marked in your copy of the textbook all the significant errata (from
http://www.physics.ucla.edu/~abers/QMErrata3.html) up to p287 (or further).
Quantum Mechanics II
Problem Set 8
Due Fri March 19th, 2010
1. Consider a Hydrogen atom in a weak magnetic field B.
(a) Roughly how strong would the magnetic field have to be, in Gauss or Tesla, for
the Zeeman splitting to become comparable to the fine structure splitting of the
n = 2 level? How does this value compare to the earth’s magnetic field, and to the
maximum sustained fields that can be reached in laboratories?
(b) Show that, for the n = 2 levels of Hydrogen, it is not necessary to use degenerate
perturbation theory when calculating the effect of the weak magnetic field using
first-order perturbation theory.
{5}
2. The electron in a hydrogen atom is subject to a small, inhomogeneous, external electric
field E = β(−2xx̂ + 2y ŷ) where β is a constant. (Ignore electron spin in this problem.)
(a) What is the perturbation H 0 to the Hamiltonian?
(b) Consider the five 3d states (n = 3, ` = 2). Using the Wigner-Eckart theorem
and Clebsch-Gordon coefficients (available on the class website), calculate all nonzero matrix elements of H 0 between these five states. (Hint: start by expressing
x2 − y 2 in terms of the Ylm .) The radial wavefunctions are specified in the file
H_atom_states.m on the course website.
(c) Diagonalize the Hamiltonian within the n = 3, ` = 2 degenerate subspace, and find
the exact eigenvalues. Is the degeneracy completely lifted by the perturbation?
{12}
3. The hyperfine splitting in hydrogen arises from the energy of the electron’s magnetic
moment in the magnetic field due to the proton’s motion (contact and dipole terms)
and the proton’s magnetic moment in the magnetic field due to the electron’s motion
(orbital term). The perturbing Hamiltonian is
0
0
0
0
HHFS
= Hcontact
+ Hdipole
+ Horbital
ge gp α 8π
0
se · sp δ 3 (r)
Hcontact
=
4mM 3
ge gp α 1 X
0
Hdipole
=
sei spj (3ri rj − δij r2 )
5
4mM r ij
gp α 1
0
sp · L .
Horbital
=
2mM r3
(1)
Here se and sp are, respectively, the spins of the electron and proton, m and M are their
masses, ge and gp are their gyromagnetic ratios (ge ≈ 2, gp ≈ 5.6).
(a) In first order perturbation theory, show that only the contact term is relevant to
the energy of an s state (l = 0).
(b) Find the hyperfine energy (in first order perturbation theory) for an arbitrary s
state (n arbitrary). It is simplest to do this using the eigenstates of F 2 and Fz ,
where F is the total angular momentum of the system, F = J + sp , with J the
total angular momentum of the electron only, J = se + L. For s states, the F 2
quantum number F is 0 or 1 (why?). You will need to use the fact that the radial
wave function at the origin, Rn0 (0) is proportional to n−3/2 .
(c) For states other than s states, the contact term does not contribute. Why not?
{13}
Quantum Mechanics II
Problem Set 9
Due Fri April 2nd, 2010
1. Calculate the energies of the hydrogen 2P levels in a magnetic field B = B ẑ, without
assuming that the magnetic splittings are either much smaller or much bigger than the
fine-structure splitting. Assume that: (i) only n = 2 states are relevant (the magnetic
splittings are much less than the splittings between states of different n); (ii) hyperfine
splitting is negligibly small; (iii) the electron’s spin g factor is exactly 2 and the orbital
g factor is 1. Work in the basis |n`jmj i so that the fine structure (Abers (7.81)) is
diagonal.
(a) Write down the perturbation H 0 to the Hamiltonian from the magnetic field, to
linear order in B. We want to diagonalize it in the n = 2 subspace, and obtain the
energies of the ` = 1 (2P ) states. Can we ignore the 2S states and just work in
the 2P subspace? Explain your answer.
(b) Which states in the 2P subspace are still eigenstates when H 0 is added to the
original Hamiltonian? What are their energies in a magnetic field?
(c) Diagonalize the full Hamiltonian in the subspace of the remaining 2P states. If
you formulate the problem correctly, you should not need to diagonalize anything
larger than a 2 × 2 matrix. Obtain the energies of the remaining states.
(d) Show that the energies you have obtained agree with Abers’s results in the limit
of small B (where the magnetic splittings are small compared to the fine structure
splittings, see (7.3.5)) and of large B (where the fine structure can be neglected
entirely, see (6.1.5)).
{15}
2. Consider a particle of mass m in a one-dimensional quartic well, V (x) = λx4 .
(a) Show that the energy of any state can be written E = (λ/(4m2 ))1/3 Ē where Ē is
a dimensionless number, independent of λ and m.
(b) Estimate the ground state energy using the variational method, with a oneparameter family of wavefunctions ψκ (x) = N exp(−κ2 x2 /2). What is the
numerical value of Ē in this case?
(c) Estimate the ground state energy using the variational method, with a oneparameter family of wavefunctions ψκ (x) = N/(1 + κ2 x2 )2 . What is the numerical
value of Ē in this case?
(d) Which family gives a better estimate, and why?
{10}
Quantum Mechanics II
Problem Set 10
Due Fri April 9th, 2010
{12}
1. Abers question 7.17, parts (a) and (b).
2. Consider a particle of unit mass in one dimension with a wavefunction given by
Z x
k(x)dx ,
ψ(x) = f (x) exp i
0
where k(x) and f (x) are both real functions.
dψ (a) Calculate the probability flux j(x) = Im ψ ∗
.
dx
(b) Show that if ψ(x) is an eigenstate of the Hamiltonian, j(x) must be equal to a
constant, independent of x.
(c) Calculate what f (x) must be in terms of k(x) in order for this condition to be
obeyed.
{8}
3. Using the WKB method, calculate the energies of the lowest three energy states of a
particle of mass m in a one-dimensional quartic well, V (x) = λx4 .
{10}
Quantum Mechanics II
Problem Set 11
Due Fri April 23rd, 2010
1. (a) The electrostatic potential of a point charge q at the origin is φ(r) = q/r. Use this
fact, and the relevant Maxwell equation, to show that ∇2 (1/r) = −4πδ (3) (r).
(b) Show by direct substitution that the Green’s function G(r, r 0 ) given in Abers (8.29)
is a solution to Abers (8.26).
{5}
2. Consider a “soft-sphere” potential of radius a, V (r) = V0 Θ(a − r) where Θ is the
Heaviside step function.
(a) Find the scattering amplitude and differential cross-section for scattering off a softsphere potential using the Born approximation for spherically symmetric potentials
(Abers (8.40)).
(b) The Born approximation for the scattering amplitude gives the first term f (1) in a
series. That term is linear in V0 . In the limit of very low momentum scattering,
calculate the second term f (2) , which is quadratic in V0 .
(c) Like any series approximation, the Born series is an expansion in powers of some
dimensionless quantity . By examining the low-momentum limit of the first two
terms, find an expression for in terms of the physical parameters of the problem.
(Don’t worry about arbitrary numerical factors: an expansion in is equivalent to
an expansion in 2 etc.)
(d) Suppose one sent V0 to infinity, making a “hard sphere” potential. In this limit,
how would you intuitively expect the (low-momentum) total cross-section of the
potential to behave? (I.e. how should it depend on m and a?) How would you
expect the scattering amplitude to behave?
(e) Explain why the result of part (d) does not agree with the V0 → ∞ limit of the
result of parts (a) and (b). As V0 gets bigger, at roughly what value of V0 does the
Born approximation for low-momentum scattering become inaccurate?
{20}
3. Consider a Yukawa potential (Abers (8.41)).
(a) The scattering amplitude is given, in the Born approximation, by Abers (8.42).
Calculate the total cross section σtot .
(b) Show that, if we allow V0 to depend on a in the appropriate way, the a → ∞ limit
of the Yukawa potential is the Coulomb potential for an electron scattering off a
nucleus of charge Ze (see Abers (8.43)). Do the scattering amplitudes then agree
also? (You can use Abers (8.46).)
(c) Show that, for a beam of any momentum q, the total cross section for Yukawa
scattering is finite, but for Coulomb scattering it is infinite. Explain the physical
origin of this divergence: what property of the Coulomb potential produces the
divergence?
{10}
Quantum Mechanics II
Problem Set 12
Due Fri April 30th, 2010
1. (a) Write down the first Born approximation to fk (θ = 0) and σtot for the Yukawa
potential and show that they do not satisfy the optical theorem. (These quantities
were calculated in class and/or previous homeworks).
(b) Show that, for any real V (r), the optical theorem is obeyed if one calculates σtot in
the first Born approximation and fk (θ = 0) to second order. To do this, write both
sides of the optical theorem as double spatial integrals and manipulate them until
they are
same. You may find it useful to remember that if f (r0 , r00 ) = −f (r00 , r0 )
R the
then f (r0 , r00 )d3 r0 d3 r00 = 0.
{13}
2. (a) Show that Abers (8.87) is equivalent to Abers (A.118).
(b) Write an alternative version of Abers (A.118) expressed in terms of a sine instead
of a cosine.
(c) From Abers (8.84) (with cl = exp(iδl ) as stated between (8.85) and (8.86)) and
Abers (A.118) show that
lim Rl (kr) − jl (kr) =
r→∞
exp(ikr)
exp(iδl ) sin(δl ) exp(−iπl/2)
kr
(d) Show that the result obtained above is equivalent to Abers (8.88).
{6}
3. (a) Calculate the exact value of the s-wave phase shift δ0 for an attractive soft-sphere
potential V (r) = −V1 Θ(a − r), where V1 > 0. This will require matching the
interior and exterior wavefunctions at r = a. Your result should be a function of
k, the exterior wave number.
(b) From the result of the previous part calculate the low-momentum scattering
amplitude for the attractive soft-sphere potential.
(c) In the previous homework, we calculated the second-order Born approximation to
the low-momentum scattering amplitude for the (repulsive) soft sphere potential.
By expanding in powers of V1 (and remembering that V0 = −V1 ) show that your
answer to part (b) agrees with that result.
{16}