Energy Levels of Polarons in a Magnetic Field

A variational method closely related to the intermediate coupling method of Lee, Low, and Pines is used to calculate the ground-state energy and low-lying excited states of the Fr\"ohlich Hamiltonian with a uniform time-independent magnetic field. The energy is calculated in a power series in ωcω to order (ωcω)², where ωc is the cyclotron resonance frequency of the electron in the absence of electron-phonon interaction and ω is the frequency of the longitudinal optical phonons. It is shown that in the presence of electron-phonon interaction the energy of the nth magnetic level is no longer proportional to n and that the effective mass for motion along the direction of the magnetic field is a function of n. The calculated variational energies approach the weak field result expected from the calculation of Lee, Low, and Pines (LLP) when ωcω→0, and in the weak coupling limit the ground-state energy becomes exact to order (ωcω)².