Franz Schwabl Quantum Mechanics 4th Edit · Problem 11.2

Relativistic Corrections

Physics — Quantum Mechanics · coupled-rotators-perturbation-theory

Problem

Two coupled, identical rotators are described by the Hamiltonian

H = A\left(p_{\Theta_1}^2 + p_{\Theta_2}^2\right) - B\cos(\Theta_1 - \Theta_2) \quad ,

where

A

and

B > 0

p_{\Theta_i} = (\hbar/\mathrm{i})\partial/\partial\Theta_i

, and

\Theta_i + 2\pi

is equivalent to

\Theta_i

. For the case

B \ll A\hbar^2

, calculate the energy eigenvalues using perturbation theory. In the opposite case

B \gg A\hbar^2

, assume that only small oscillations occur, and construct the solution in analogy to the harmonic oscillator. Hint: Introduce new variables

x = \Theta_1 + \Theta_2

and

y = \Theta_1 - \Theta_2

. In terms of these variables, the Hamiltonian becomes

H = -2A\hbar^2\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right) - B\cos y \quad .