Lunds Universitet Spring 2016
Tineke van den Berg, Johannes Bjerlin
Exercises to Quantum Mechanics FYSN17/FMFN01, Week 5
Homework to be handed in on February 23
Exercise 1: Energy shifts of atoms in magnetic fields
Aim: Understanding the Hamilton operator in a magnetic field
Let B be a constant magnetic field. Consider the Hamiltonian
1 2
Ĥ = p̂ − qA(r̂) + qφ(r̂) .
2m
It can be rewritten:
~2 q q2 2 2
Ĥ = − ∆ + qφ(r̂) − B L̂z + B (x̂ + ŷ 2 )
2m 2m 8m
where it is assumed that B = Bez . Estimate the size of the last two terms for an electron in an atom if the
field has the strength of 10 T.
Exercise 2 (Homework): Electron in inversion layer with magnetic field
Aim: Quantifying the energy splitting due to B; addressing a common task in solid state physics
In solid state physics one can establish two-dimensional systems at the interface of a semiconductor. Consider
a single electron confined to such a layer, so that it can only move freely in x and y-direction. A magnetic
field points in z-direction, perpendicular to the interface.
a) Construct the Hamiltonian for this particle.
b) Find the energy spectrum for a magnetic field of 5 T. Use the effective mass mef f = 0.067me which is
appropriate for a conduction band electron in GaAs.
Exercise 3: First-order correction of energy
Aim: Learning how to apply the formalism of stationary perturbation theory
2
Consider a one-dimensional harmonic oscillator with frequency ω which is perturbed by Ĥp = λe−αx̂ . Cal-
culate the first order correction to the ground state energy and to the energy of the first excited state.
Exercise 4: Perturbation of two dimensional oscillator
Aim: Getting familiar with degeneracies in perturbation theory
A two dimensional harmonic oscillator is perturbed by Ĥp = λmω 2 x̂ŷ.
a) Use first order perturbation theory to calculate the energy shift for the ground state and the first excited
state.
b) Calculate the ground state energy to second order.
c) Solve the full problem exactly and compare the result with the approximation you obtained in a) and b).
See other side
Exercise 5 (Homework): Molecule vibration
Aim: Applying stationary perturbation theory; understanding basic features of molecular vi-
brations
A diatomic molecule can rotate and vibrate. To describe vibrations one can use a potential as in the figure
where r denotes the distance between the nuclei. As a first approximation we may consider this as a one-
dimensional problem (in x direction) with the parabolic potential
1
V0 (x) = mω 2 x2 − De
2
where m is the reduced mass for the system of both atoms and x
is the deviation from the equilibrium position re .
For higher excitation energies the potential deviates from the
parabolic approximation and one can add a third order term:
2mω 23 3
Ĥp = α~ω x̂ .
~
We shall in this exercise study the effect of Ĥp as a perturbation.
a) Express Ĥp in terms of a and a† of the harmonic oscillator
corresponding to V0 (x).
b) Determine all matrix elements hm|Ĥp |ni between unperturbed Graphical depiction of the Morse potential with a har-
harmonic oscillator states. monic potential for comparison, CC-BY-SA Mark So-
moza (2006)
c) Consider the second excited state. Calculate the change in energy in second order perturbation theory
and its state in the first order.
d) Calculate hxi with the new state and discuss its relation to thermal expansion of a crystal
Exercise 6: Rabi’s resonance experiment
Aim: Getting familiar with an important feature of general two-level systems
We will in this example discuss Rabi’s famous spin resonance experiment. The
particles (say neutrons) enter the apparatus from the left with a selected spin.
A magnet gives a homogeneous magnetic field in the z−direction, B0 = B0 ez .
In the area where this magnetic field is, there is also a so called RF-loop which
creates an oscillating field BRF = B1 [cos(ωt)ex +sin(ωt)ey ]. When the particles
enter the field B = B0 + BRF from the left side they are in the spin state | ↑i.
Only those particles that changed their spin directions (flips) will be detected
on the right hand side by an appropriate detector.
a) The magnetic moment of the particles is given by µ̂ = γ Ŝ. Write down the
time-dependent Schrödinger equation valid in the magnetic field B in spinor
representation, i.e. for the component a(t), b(t) of the time-dependent state
|Ψi(t) = a(t)| ↑i + b(t)| ↓i.
b) With the initial condition |Ψi(0) = | ↑i one finds (after tedious but straightforward algebra – not required)
γB0 + ω γB1
a(t) = cos(Ωt) + i sin(Ωt) e−iωt/2 and b(t) = i sin(Ωt)eiωt/2
2Ω 2Ω
q
with the Rabi frequency Ω = γ 2 B12 + (γB0 + ω)2 /2. For which field B0 can the probability to observe a
spin flip be equal to one? How does the probability evolve in time?
c) In an experiment with neutrons one observes the maximal signal at B0 = 0.54T for a frequency of
ωresonance = 9.92 × 107 rad/s. Determine γ and deduce the Landé factor gn of the neutron.
