| Summary: | We investigate the two-dimensional motion of relativistic cold electrons in the
presence of ‘strictly’ spatially varying magnetic fields satisfying, however, no
magnetic monopole condition. We find that the degeneracy of Landau levels, which arises in the case of the constant magnetic field, lifts out when the
field is variable and the energy levels of spin-up and spin-down electrons align
in an interesting way depending on the nature of change of field. Also the
varying magnetic field splits Landau levels of electrons with zero angular momentum from positive angular momentum, unlike the constant field which only
can split the levels between positive and negative angular momenta. Exploring Landau quantization in non-uniform magnetic fields is a unique venture
on its own and has interdisciplinary implications in the fields ranging from
condensed matter to astrophysics to quantum information. As examples, we
show magnetized white dwarfs, with varying magnetic fields, involved simultaneously with Lorentz force and Landau quantization affecting the underlying
degenerate electron gas, exhibiting a significant violation of the Chandrasekhar
mass-limit; and an increase in quantum speed of electrons in the presence of
a spatially growing magnetic field.
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