SPAT0071 Exam 2019-2020

Introduction to quantum field theory

Problem 1

1. A particle is moving under the influence of a 3 dimensional potential [math]\displaystyle{ V(x,y,z)=\frac{m \omega^2}{2}(x^2 + y^2 + z^2) }[/math]. Calculate the possible energy levels of the particle and discuss their degeneracies.
2. The potential is then modified to [math]\displaystyle{ V^{\prime}(x,y,z)=\frac{m \omega^2}{2}(x^2 + \alpha x + y^2 + z^2) }[/math], where [math]\displaystyle{ \alpha }[/math] is a small parameter. By solving the Schrödinger equation exactly, compute what effect, if any, this has on the energy levels.
3. Solve the same problem (i.e. find the change in energy levels) by treating the additional contribution to the potential [math]\displaystyle{ \alpha x }[/math] as a perturbation to the original potential [math]\displaystyle{ V(x,y,z) }[/math] up to the 2nd order in [math]\displaystyle{ \alpha }[/math]. Comment on the difference(s), if any, between results obtained here and in 1.2 above. What is the range of validity of perturbation theory?

Problem 2

We solved in class the Schrödinger equation for the spectrum of the hydrogen atom as if it were isolated in the Universe. In reality, H is in an environment, which means that the Coulomb potential does not extend to infinity, but gets screened by other charges. Using first-order perturbation theory, explore the simple case of a screened Coulomb potential cut-off at a constant distance R: [math]\displaystyle{ V(r)={-\kappa\over r}\theta(R-r) }[/math], with [math]\displaystyle{ \kappa }[/math] the square of the electron charge (in Gaussian units) and [math]\displaystyle{ \theta }[/math] the Heaviside step function.

1. How would you characterize the strength of the perturbation and under which conditions does perturbation theory give reliable results?
2. How does the cutoff affect the spectrum? Does it lift degeneracies? Give specific answers for three values of [math]\displaystyle{ R }[/math], one for gases under standard conditions, one for liquids and one for solids (and explain how you estimate those values). Is the approximation reliable in all three cases?
3. Give detailed results for the energies and the wave functions in the [math]\displaystyle{ 1s }[/math] and [math]\displaystyle{ 2s }[/math] cases.

Problem 3

Two spin-½ particles of masses [math]\displaystyle{ m_1 }[/math] and [math]\displaystyle{ m_2 }[/math] interact via a potential that may be expressed as:

[math]\displaystyle{ V = \dfrac{-e^2}{r}\left(1 + \dfrac{b}{r}\sigma^{(1)}\cdot \sigma^{(2)} \right), }[/math]

where [math]\displaystyle{ \sigma^{(i)} }[/math] corresponds to the [math]\displaystyle{ i }[/math]-th particle; [math]\displaystyle{ r = \lvert \vec{r}_{1} - \vec{r}_2 \rvert }[/math] represents the magnitude of relative separation between the two particles; and [math]\displaystyle{ b }[/math] is an arbitrary constant. Determine the bound-state energy spectrum of the combined two-particle system. What condition(s) do you need to impose upon [math]\displaystyle{ b }[/math] to avoid unphysical eigenstates?