The Landau energy levels E of an electron in a constant magnetic field B (pointed in z direction) are as follows. E = +- sqrt( m^2*c^4 + pz^2*c^2 + e*B*c^2*hbar*(2*n+1-s) ) Here n=0,1,2,3,... is an integer; +- is for positron and electron states; pz is an arbitrary real momentum in z-direction; s is +1 or -1 representing electron spin times 2, e is the charge of the positron, m is mass of electron, c = lightspeed, hbar = planck const/2pi, B=magnetic field. Source: see EQ 21 of http://xxx.tau.ac.il/pdf/0705.4275v2.pdf or exercise 7.6 in W.Greiner+J.Reinhardt: Quantum electrodynamics 2nd ed. Springer 1994, or EQ13 of http://merlin.phys.uni.lodz.pl/concepts/2007_1/2007_1_141.pdf . The lattermost source has the advantage it derives it using a cylindrically symmetric wavefunction, which is just what I want to think about re also imposing a "cylindrical box." But you pay the price of a more complicated derivation. Also, the lattermost source claims the Landau levels are countably-infinitely degenerate, not merely 2-fold like the other two sources claim (More precisely, these states are 2-fold degenerate except when n=0 there is only 1). All these sources derived the energy levels by exactly solving Dirac equation in various ways. If the cylindrical box has a radius that is just right (exactly equal to something arising from a zero of a confluent hypergeometric) then their unboxed wavefunction still should be valid even if the box is there. More generally by taking a linear combination of the 2 degenerate landau states you should be able to choose the coefficients to make it be a solution in any-radius box (at least if that radius is large enough) i.,e. meet the boundary condition that the wavefn by 0 at the box wall. Consequently, I conclude that the landau levels in an aligned-oriented cylindrical box are exactly the same as the unboxed levels, except that: (a) the pz must be chosen to fit an integer number of wavelengths in 2*(box height). (b) for a fixed-radius box, only certain landau level n-parameters may be allowed? (c) the n=0 ground landau level, is not permitted to exist with box confinement. (d) the 2-fold degeneracy for unboxed landau levels reduces to 1 with boxing.