PHYS 5322 – Quantum Mechanics
S. Abachi
csula – Spring 2021
Midterm - 2
“Due on Monday March 22 – 6 pm (post online)”
Note: Equation numbers given in prentices are good for 2nd edition of textbook.
1. Look at chapter 13 of the Liboff’s book. In table 13.1, you see Examples of several perturbation
Hamiltonians.
a) Choose four of them and describe briefly what they entail.
b) Justify their given Hamiltonian, in moderate detail.
⃗ 𝑜 = 𝐵𝑜 𝑧̂ is subjected to an additional field given by 𝐵
2. A Sodium nucleus in magnetic field 𝐵 ⃗ 1 = 𝐵1 𝑥̂ as a
perturbing effect (B1 << Bo).
a) Look up the spin of the nucleus of sodium.
b) Write down Ho matrix.
c) Write down V matrix.
d) Is there a 1st order correction to the energy? If so, find it.
e) Find the system energy correct to the 2nd order.
f) Find the state vectors correct to the 1st order.
3. In the textbook, there is a treatment of relativistic corrections to the Hydrogen atom given by an
expression in equation 5.101 (5.3.7).
a) Derive this equation in detail starting from Eq. 5.96 (5.3.2).
b) Use the recommended approach in textbook for exact calculations of the second and third terms
in this expression. For this part, as well as for the following parts, you may consult Townsend’s
book, chapter 11; especially problems 11.15 – 11.17.
c) Prove Eq. 5.102 (5.3.8).
d) Prove Eq. 5.104 (5.3.10).
4. For a particle inside a sphere of radius R;
a) Write down the Schrodinger equation for the wave function. (No need to solve it.)
b) Instead of the real solutions, use a trial function (r) = 1 – (r/R)2, to find an approximate ground
state energy.
5. Consider an Intermediate Zeeman effect correction, where the effect is comparable to the Fine Structure
correction, and thus we have to treat the sum of both effects as the perturbation to the 0th Bohr energy. We
(𝑜)
then have to diagonalize this perturbation Hamiltonian in each degenerate 𝐸𝑛 energy subspace.
a) Which basis is more appropriate and why: L2, S2, Lz, Sz simultaneous eigenkets ( |n, l, s, ml, ms > ),
or L2, S2, J2, Jz simultaneous eigenkets ( |n, l, s, j, mj > )? In fact, argue that neither one is perfect
and will not fully diagonalize, and thus there is no obvious preferred basis.
b) Why in the Zeeman weak field case this issue did not arise?
1
6. In the textbook, equation 5.190 (5.5.18) describes a two-level system Hamiltonian and time dependent
perturbation.
a) Mathematically justify how H can be written the way it is given.
b) Justify the same for V.
c) If you combine Ho and V as H in one formula, how would that be written (with justification).
d) Derive Rabi’s formula 5.193 (5.5.21), with any method you like.
7. A time dependent potential is turned on with a gaussian time dependence in the form of
2/2𝜏2
𝑉(𝑡) = 𝑉𝑜 𝑒 −(𝑡−𝑡𝑜 )
Where is the perturbation characteristic time, and the perturbation is peaked at t = to, while it is
insignificant a few time constants to the right or left of to. Assuming that long before to the stateket was
| i >, which is one of the energy eigenkets of Ho, then:
c) What is the transition probability that after the perturbation the system is found in the final state
| f >?
d) To have appreciable probability for this transition, what constraint we should have on
e) Argue that in the Adiabatic limit, where the perturbation is varying too slowly, the transition does
not occur. How do you write this condition that leads to this adiabatic limit?
Hint; You can use transition probability derived from Dyson series for this problem.
