Physics 312: Problem Set 1
We will start with some review problems to get you up to speed, and then turn to perturbation theory, which is review for only some of you.
1. Using the Schr¨ m moving under a potential V (
~
), show that d dt h ~ i =
1 m h ~ i , where h A i = h ψ ( t ) | A | ψ ( t ) i , where A is an observable. Here observables, respectively.
~ and
~ are the position and momentum vector
2. Find the components of the matrix e iαA where
 0 0 1 
A =

0 0 0
1 0 0

3. If [ A, B ] =0, find e iαA
Be
− iαA
(1)
4. (from Sakurai chapter 4 problem 2; you don’t need to reference the book to do these, but you can if you want.):
Which of the following pairs commute and why?
(a) Translation by ~ and translation by ~
0
, where ~ and ~
0 are different directions.
(b) Rotation by an amount φ about an axis ˆ and rotation by an amount φ
0 and ˆ = ˆ
0
.
(c) Translation by an amount ~ and a parity operation.
(d) Rotation by an amount φ about an axis ˆ and a parity operation.
n
0
, where φ = φ
0
5. A particle in a one dimensional square well has quantized energy levels E n
= n
2
π
2
¯
2
2 mL 2
, where L is the length of the well, m is the mass of the particle, and n is its quantum number with possible values n =1,2,3,... A particle exists in the superposition state
Ψ =
1
5
(3 φ
1
+ 4 φ
2
) , where φ
1
E
2
.
is the state the particle is in if it has energy E
1
, and φ
2 is the state the particle is in if it has energy
(a) What is the probability of finding the particle with an energy E
2
?
(b) What is the probability of finding the particle with an energy E
3
?
(c) What is the energy expectation value for a particle in the wavefunction given?
(d) What is the probability that the particle exists in the state
χ =
1
√
2
( φ
1
+ φ
2
) ?
6. A deuterium molecule, D
2
, is known to be in the state
ψ ( θ, φ, t = 0) =
3 Y
1
1 + 4 Y
√ 7
3
26
+ Y
7
1
, at time t = 0. The Y l m are the spherical harmonics. The molecule has moment of inertia I
D
.
(a) Find two possible complete sets of commuting observables (CSCOs) for this system.
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(b) What values of |
~
| and L z will measurement find and with what probabilities will these values occur?
(c) Find h
ˆ x i and h 2 x i for the system existing in the state ψ ( θ, φ, 0)
(d) Considering only the rotational energy, what is the Hamiltonian for the deuterium molecule? Given the initial state above, what is ψ ( θ, φ, t )?
(e) Again considering only the rotational motion, what is h
ˆ i for t > 0?
(f) Now the deuterium molecule is put in a uniform magnetic field,
~
= B
0
ˆ .
i. What is the Hamiltonian now that we consider not only the kinetic energy of rotation for the molecule, but also its interaction with the B field?
ii. Given | Ψ( t = 0) i as before, what is | Ψ( t ) i ?
iii. What is h Ψ( t ) |
ˆ y
| Ψ( t ) i ?
7. A beam of spin-1/2 particles, moving along the y -axis, is incident on two collinear Stern-Gerlach apparatuses; the first with
~ along the z -axis and the second with
~ along the z
0
-axis, which lies in the x − z plane at an angle θ relative to the z -axis.
(a) Using the Pauli matrices, what is the operator, σ z 0
, corresponding to the second Stern-Gerlach apparatus described above? It might be helpful to make a sketch.
(b) If both apparatuses transmit only the lower beam, what fraction leaving the first will pass the second?
8. The bond strength of the
35
Cl
2 molecule is k = 328.6 N/m. What is the spacing between vibrational energy levels for this molecule? What is the energy associated with the n=3 energy eigenstate? Sketch the wavefunction, labelling the axes and the equilibrium location. (1 amu= 1.6
× 10
− 27 kg.)
9. A single mass m moves under the influence of a two-dimensional harmonic oscillator potential with distinct spring constants, k x and k y
.
(a) Write down the Hamiltonian, eigenenergies, and eigenstates for this system.
(b) If k y
= 4 k x
, what is the degeneracy of E n x
,n y
= E
2 , 3
? List the corresponding eigenstates. What parity do these states have? Is there a symmetry degeneracy among these states?
(c) If | Ψ( t = 0) i = N {| 0 1 i + 2 | 1 2 i + | 1 0 i} , what is N ? If an energy measurement is made, what results are possible and with what probabilities? What is | Ψ( t ) i ?
(d) Use ladder operators to find the selection rules between n 0 x matrix elements h n
0 x n
0 y
| L z
| n x n y i are nonzero. ( ˆ z
P y
− and n x
, and between n 0 y
P x
) and n y such that the
10. Consider an electron in a diatomic molecule. When the electron is on nucleus A , it is in the state | ψ
A i . When the electron is on nucleus B , it is in the state | ψ
B i . If we neglect the possibility of a transition from one state to the other, | ψ
A i and | ψ
B i are orthogonal and each has energy E
0
. If coupling is allowed, an additional term in the hamiltonian behaves so that
ˆ
| ψ
A i = − a | ψ
B i and
ˆ
| ψ
B i = − a | ψ
A i .
The total hamiltonian is then ˆ H
0
+ ˆ H
0 is the unperturbed hamiltonian.
(a) Find the exact eigenvalues and eigenvectors of the hamiltonian ˆ .
The electron is localized on nucleus B at time t = 0.
(b) What is the time-evolved state?
(c) What is the probability, as a function of time, that the electron is to be found on nucleus A ? Sketch this function.
(d) ˆ is a quantum mechanical observable which corresponds to an electric dipole moment. It acts so that
ˆ
| ψ
A i = d | ψ
A i and
ˆ
| ψ
B i = − d | ψ
B i .
If ˆ is measured on the time-evolved state, what results are found and with what probabilities? What information does this give about the spectroscopic properties of this diatomic molecule?
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11. Anharmonic oscillator. In class we will consider a system governed by the hamiltonian H
(0) is the one dimensional SHO and H (1) = αx 3
+ H
(1
), where H
(0)
= λ
ˆ
, with λ small. Verify (to order λ 2 ) our results for the estimated ground state energy of the perturbed system.
12. Perturbed Square Well
Consider a one-dimensional system consisting of a particle of mass m and charge e in the potential well V ( x ) shown (see figure).
∞
6
V ( x )
∞
6
a
2
a
4 a
2
6 b =
π
2 h
2
8 ma 2
x
0 a
4
The potential energy function is
V ( x ) =
 ∞ , x < − a
2
, x >


0 , − a
2
< x < − a
4 a
2
,

 b, − a
4
< x < a
4
.
a
4
< x < a
2
Suppose that b =
π
2 h
2
8 ma 2 and is sufficiently small that we can treat this problem by means of perturbation theory.
That is, think of the problem as being an infinite square well between − a/ 2 and a/ 2, with the bump in the middle treated as a perturbation.
(a) Write expressions for the unperturbed ( b =0) eigenfunctions and energy eigenvalues.
(b) Obtain an expression for the first-order perturbed energy of the ground state of the system, expressing your result in terms of b and fundamental constants.
(c) Write an expression for the first-order perturbed wave function of the ground state in terms of the complete set of unperturbed functions of part (a). (You need not evaluate matrix elements in your expression.)
13. Finite Spatial Extent of the Nucleus (See C-TDL Complement D
XI
)
In all we have done so far, the nucleus has been treated as a positively charged point particle. In fact, the nucleus does possess a finite size with a radius given approximately by the empirical formula
R ' r
0
A
1 / 3
, where r
0
= 1 .
2 × 10
− 13 cm (i.e., 1.2 Fermi) and A is the atomic weight (essentially, the total number of protons and neutrons in the nucleus). A reasonable assumption is to take the total nuclear charge + Ze as being uniformly distributed over the nuclear volume
(a) Electrostatics then gives the following expression for the potential energy of an electron in the field of the
“finite” nucleus of charge + Ze :
V ( r ) =
( − Ze
2
Ze
2 r
R
, r
2
2 R 2
− 3
2 r > R
, r < R .
Draw a graph comparing this potential energy and the point nucleus potential energy.
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(b) Since you know the solution of the point nucleus problem, choose this as H (0)
H (1) such that the total Hamiltonian contains the V ( r and construct a perturbation
) derived above; write an expression for H (1) .
(c) Calculate the first-order perturbed energy for a 1 s state, obtaining an expression in terms of Z and fundamental constants. Do not assume a particular value for the electron mass.
(d) Use your result from part (c) to compare the effect of this finite nuclear-size correction on the ground-state energies of a hydrogen atom and a one-electron carbon ion ( Z = 6). Compare the percentage correction to the energies due to the perturbation. Explain your results.
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