Charge in a magnetic field: quasi-classical states

Abstract

We propose an elementary approach for introducing the quasi-classical quantum states of a spinless charged particle in a uniform magnetic field. By exploiting the similarity with the case of the two-dimensional harmonic oscillator as well as a property of the solutions of the pertinent Schrödinger equation, we derive two basic solutions. One of them is a stationary, minimum-uncertainty wavepacket centred at an arbitrary point. This corresponds to a classical particle at rest. The other solution is a minimum-uncertainty wavepacket that rotates at the classical angular speed. The expected value of the energy agrees with the classical prediction within the zero-point energy. These results are obtained without any knowledge of the energy eigenstates. An appendix suggests a brief and self-contained procedure for writing the Hamiltonian of a charged particle under the Lorentz force.