Joel Broida
UCSD Fall 2009
Phys 130B
QM II
Homework Set 6
THIS IS A COMBINATION LAST HOMEWORK SET AND FINAL EXAM. YOU
ARE ON YOUR HONOR NOT TO DISCUSS ANY OF THESE PROBLEMS
WITH ANYONE BUT ME. YOU ARE FREE TO USE YOUR NOTES AND
ANY OF THE STANDARD TEXTBOOKS, BUT THAT’S ALL. IN PARTICULAR, NO INTERNET SEARCHES.
PLEASE DO NOT TURN IN SCRATCH PAPER. FIGURE OUT HOW TO DO
THE PROBLEM, WORK IT OUT, AND THEN WRITE IT UP VERY NEATLY.
CAREFULLY EXPLAIN YOUR WORK; YOU DON’T HAVE TO SHOW EVERY
MINOR ALGEBRA STEP IF IT’S OBVIOUS TO ME THAT YOU KNOW HOW
TO GO FROM ONE LINE TO THE NEXT.
READ THE PROBLEMS CAREFULLY. DON’T LOSE POINTS BECAUSE OF
CARELESSNESS.
WHETHER YOUR FINAL ANSWER IS RIGHT OR WRONG, IF I CAN’T EASILY UNDERSTAND WHAT YOU HAVE DONE, YOU AREN’T GOING TO GET
ANY CREDIT.
YOU MUST DO ALL CALCULATIONS BY HAND, EXCEPT THAT YOU MAY
USE THE INTEGRALS LISTED IN THE TABLE POSTED ON THE WEBSITE
BY THE HW5 LINK. NONE OF THESE PROBLEMS REQUIRES AN EXTENSIVE AMOUNT OF ALGEBRA.
THIS IS TO BE TURNED IN TO ME AT MY OFFICE AT 11:30 AM ON TUESDAY, DECEMBER 8. THERE WILL BE NO EXTENSIONS. IF YOU WANT TO
TURN IT IN BEFORE THEN, PLEASE MAKE ARRANGEMENTS WITH ME
TO DO SO.
1. Consider a system comprised of two different spin-1/2 particles with the
Hamiltonian H0 = AS1 · S2 . Let the particles also have magnetic moments
µ1 = αS1 and µ2 = βS2 , and suppose that an external magnetic field B = Bẑ
is applied.
(a) In order to solve this problem exactly, what would you choose as your
basis states and why?
(b) Find the exact energy eigenvalues of the system. Hint: Order your basis
states so that the matrix of H becomes block-diagonal.
1
2. Suppose two spin-1/2 particles are in the state
1
|ψi = √ (| + +i + | − −i)
2
where the states |±i are eigenstates of Sz . Let â be the unit vector with
usual spherical coordinates θa and φa . If you make a measurement of the first
particle’s spin along â, what is the probability of obtaining +~/2? (You must
show your work, not make a guess.)
3. Suppose you have a particle of mass m moving under the potential
V (r) = −V0
e−r/a
.
r
Consider the trial function ψ(r) = N e−λr where N is a normalization constant
and λ is a variational parameter.
(a)
(b)
(c)
(d)
Find the value of N .
What is the corresponding variational integral W (λ)?
Find an equation for λ that minimizes the variational integral.
For what values of a will this trial function yield a bound state? Express
this value of a in terms of ~, m and V0 .
4. Two distinguishable non-interacting spin-one particles are in the ground state
wave function ψ0 (r) of some spherically symmetric potential. A small perturbation
H ′ = λS1 · S2
is turned on, where Si is the spin operator for particle i.
(a) Find all possible spin states, and determine their first-order energy shifts
due to the perturbation.
(b) Which states are allowed if the two particles are identical and why?
5. Suppose you have a spin 3/2 particle with the Hamiltonian
H = αSz2 + β(Sx2 − Sy2 ) .
(a) Find the energy levels of the system. Hint: Order your basis to make the
matrix representation of H block-diagonal.
(b) Describe how you would go about finding the corresponding eigenstates,
and what their general form will be in this case. You need not actually
work them out.
(c) Now assume that α = β, and let a perturbation H ′ = γ~Sx (with γ ≪ 1)
be turned on at t = 0. If the system is in the ground state at t = 0, find
the probability that it is still in the ground state at t = T .
2
6. Consider a hydrogen atom in its ground state
1 −3/2
ψ0 = √ a0 e−r/a0 .
π
Suppose that at t = 0 it is placed in a harmonically time-varying electric field
E(t) of magnitude
E (t) = E0 e−iωt .
(a) What is the minimum frequency (in Hz) required of the electric field in
order that ionization occur?
(b) Find the probability per unit time that the electron is ejected into the
solid angle dΩ. Hints: (i) Use the golden rule
2π
2
Γ=
ρ(Ef ) |Vf i | .
~
Ef =Ei +~ω
(ii) Take the final ejected electron state to be a plane wave with box
normalization. Thus the final state of the electron is
1
ψp = √ e(i/~)p·r .
V
(iii) Let p̂ = ẑ and let θ′ be the angle between p and r. (iv) Let θ and φ be
the polar angles of p with respect to E(t). (v) Let θ′′ be the angle between
E(t) and r. See the figure below. Note that φ lies in a plane perpendicular
to E(t), while φ′ lies in a plane perpendicular to p. This convention is in
fact somewhat simpler than letting E(t) lie in the ẑ direction.
z
p
E(t)
r
θ′
θ
θ′′
y
χ
φ′
x
(vi) You can now derive an expression for cos θ′′ in terms of cos θ, sin θ,
cos θ′ , sin θ′ , and cos(φ′ − χ). To do so, let the projection of E(t) in the
xy-plane be in the û direction, and let the projection of r in the xy-plane
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be in the v̂ direction. Now look at two different ways of writing E · r.
(vii) When you evaluate the matrix element, do the φ′ integral first. The
remaining integral is actually not at all hard to do. You get an integral
of the form
Z 1
x dx
(α
+
iβx)4
−1
which is easily done with the change of variables y = α + iβx. Note
that the limits of integration are now complex conjugates of each other,
and recall that z − z ∗ = 2i Im z. (viii) For the density of states, use
equation (3.24) from the notes where d3 n = ρ(E) dE with E = p2 /2m,
and d3 p = p2 dp d cos θ dφ.
(c) What is the total rate of ionization?
7. We have seen that the wave function for a system of two electrons must be
antisymmetric. But do we really have to worry about this when one electron
is on earth and the other is on the moon? If they are both in the ground
state, do we have to make sure that their spins are opposite? In this exercise
you will show that we don’t have to worry about such things if the electrons
are widely separated. In other words, you will examine the difference between
the uncorrelated wave function for two electrons
Ψ(u) (x1 , x2 ) = ψa (x1 )ψb (x2 )
and the antisymmetrized wave function
Ψ(a) (x1 , x2 ) =
1
[ψa (x1 )ψb (x2 ) − ψa (x2 )ψb (x1 )] .
N
Here the wave functions are normalized so that
Z
Z
2
2
|ψa (x)| dx = |ψb (x)| dx = 1
but the wave functions ψa and ψb are not necessarily orthogonal.
(a) Find an expression for the normalization constant N in terms of the overlap integral Sab = hψa |ψb i.
(u)
(b) For Ψ(u) , what is the probability Pa (R) that the electron with the a
label is found in some region R?
(a)
(c) For Ψ(a) , what is the probability Pa (R) that the electron with the a
label is found in some region R?
(d) What do you conclude from these calculations and why?
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