It turns out that when a free electron is placed in a magnetic field, its energy levels go from being continuous (by the virtue of being free particles) to being a discrete set. What follows is the calculation of this phenomenom, which is called the Quantum Hall Effect.
Deduction of the quantization of electrons' energy levels
For a free electrons gas in a magnetic field, the Hamiltonian is:
↑The k vector is the unit vector along the Z axis. The Z axis is chosen here (as is standard practice) just for the sake of giving a name to the direction of the magnetic field. X (the i vector) and Y (the j vector) are thus whatever two axes are perpendicular to the magnetic field here and follow the w:right-hand rule.
↑I.e., states of the system, wavefunctions if you like, for which the time independent Schroedinger equation holds.
↑I.e., a number. Operators can be thought of as matrices that multiply the wavefunctions, which would be vectors. That means that in Schroedinger's equation (H*psi) would be a vector, and (E*psi) too, but you cannot drop the psi because H is a matrix while E is a number. In our case up here, we have found a vector for which the Hamiltonian operator is effectivly just multiplying it by a number. Thus that number is equal to E, and we can from that calculate the value of E. This method of solving matricial differential equations is analogous to Diagonalization. The value obtained for E will be the energy of that particular state and, as is common e.g. in Fourier problems, is restricted to a discrete set of values (the energy is quantizised).