Degeneracy of Cyclotron Motion

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Abstract
The classical problem of planar cyclotron motion of a charged particle in a uniform magnetic field possesses symmetries which account for the ``accidental'' degeneracies of the analogous nonrelativistic Schrödinger equation, as found by Johnson and Lippman. The essentially quadratic nature of the Hamiltonian is not changed by considering the particle moving in a harmonic oscillator potential, a ``Zeeman effect'' for the harmonic oscillator. The transitions to the limiting cases of a weak magnetic field (pure harmonic oscillator) or a strong field (pure cyclotron motion) involve the contraction of the corresponding symmetry groups, yielding Larmor precession of the oscillator orbits in the first case, and the drift of the cyclotron orbit in the second. The constants of the motion generate the unitary unimodular group SU2 in all cases except for pure cyclotron motion, in which case one obtains the commutation rules of creation and annihilation operators. Only for certain ratios of magnetic field strength to the oscillator frequency does one obtain bounded closed orbits, and presumably only in these cases do degeneracies exist quantum‐mechanically. A transition to a rotating coordinate system reduces the problem to that of a plane harmonic oscillator; however, the time dependencies of the transformation must be allowed for interpreting the constants thereby arising. Moreover, the velocity‐dependent forces introduce gauge transformations which also affect the interpretation of the symmetries. There are two kinds of symmetries—inner symmetries involving the canonical coordinates and governing the shape of the orbits, and outer symmetries involving the mechanical coordinates and governing the location of the orbits.

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