On the scaling properties of the energy spectrum of hydrogen in a uniform magnetic field

The magnetic-field scaling relation between the coordinates, momenta and the energy in the Hamiltonian for the Kepler motion in a uniform magnetic field, i.e. H( gamma -1 3/p, gamma 2 3/r; gamma 1 3/m, gamma =1)= gamma -2 3/H(p,r; m, gamma ), where gamma =B/B 0 is the normalised field strength and m the constant value of the angular moment,um B-parallel component is shown to ensure a representation of the semiclassical energy spectrum: E=(n+ 1 / 2 ) -2 f( gamma 1 3/(n+ 1 / 2 ), gamma 1 3/m), m=0, +or-1, ...; n=0, 1, .... The authors discuss how the function f, in the two-dimensional approximation (z=p z =0, z//B), can be constructed.