Introduction to quantum field theory/Final Exam

Introduction to quantum field theory

Problem 1

A particle is moving under the influence of a 2 dimensional harmonic oscillator with the potential [math]\displaystyle{ V=\tfrac{m \omega^2}{2}(X^2 + Y^2) }[/math].

1. Write down the Hamiltonian in terms of the raising and lowering operators [math]\displaystyle{ \cal{H}=\cal{H} \left( \hat{a}_x,\hat{a}_x^{\dagger},\hat{a}_y,\hat{a}_y^{\dagger} \right) }[/math]. What is the explicit form of all these operators in terms of the position and momentum operators?
2. Using [math]\displaystyle{ \hat{a}_x \hat{a}_y\vert \Psi_{0,0}\rangle = 0 }[/math] find the normalized wave function [math]\displaystyle{ \vert\Psi_{0,0} (p_x, p_y)\rangle }[/math]. Please pay attention to the momentum dependence of the wave function.
3. Using the previous answers, calculate [math]\displaystyle{ \vert \Psi_{0,1}(p_x, p_y)\rangle }[/math], [math]\displaystyle{ \vert \Psi_{1,0}(p_x, p_y)\rangle }[/math], [math]\displaystyle{ \vert \Psi_{1,1}(p_x, p_y) \rangle }[/math], and [math]\displaystyle{ \vert \Psi_{3,1}(p_x, p_y)\rangle }[/math], where [math]\displaystyle{ \vert \Psi_{n_x,n_y}(p_x, p_y)\rangle }[/math] is the wave function in momentum space while [math]\displaystyle{ n_{x} }[/math] and [math]\displaystyle{ n_{y} }[/math] label the energy levels.

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?