Landau quantization
In quantum mechanics, Landau quantization refers to the quantization of the cyclotron orbits of charged particles in a uniform magnetic field. As a result, the charged particles can only occupy orbits with discrete, equidistant energy values, called Landau levels. These levels are degenerate, with the number of electrons per level directly proportional to the strength of the applied magnetic field. It is named after the Soviet physicist Lev Landau. Landau quantization is directly responsible for the electronic susceptibility of metals, known as Landau diamagnetism. Under strong magnetic fields, Landau quantization leads to oscillations in electronic properties of materials  as a function of the applied magnetic field known as De Haas–Van Alphen and Shubnikov–de Haas effects. Landau quantization is a key ingredient to explain the integer quantum Hall effect. Derivation. Consider a system of non-interacting particles with charge and spin confined to an area in the plane. Apply a uniform magnetic field formula_1 along the -axis. In CGS units, the Hamiltonian of this system (here, the effects of spin are neglected) is formula_2 Here, formula_3 is the canonical momentum operator and formula_4 is the electromagnetic vector potential, which is related to the magnetic field by formula_5  There is some gauge freedom in the choice of vector potential for a given magnetic field. The Hamiltonian is gauge invariant, which means that adding the gradient of a scalar field to changes the overall phase of the wave function by an amount corresponding to the scalar field. But physical properties are not influenced by the specific choice of gauge. In the Landau gauge. For simplicity in calculation, choose the Landau gauge, which is formula_6 where and is the component of the position operator. In this gauge, the Hamiltonian is formula_7 The operator formula_8 commutes with this Hamiltonian, since the  operator is absent by the choice of gauge. Thus the operator formula_8 can be replaced by its eigenvalue . Since formula_10 does not appear in the Hamiltonian and only the z-momentum appears in the kinetic energy, this motion along the z-direction is a free motion. The Hamiltonian can also be written more simply by noting that the cyclotron frequency is , giving formula_11 This is exactly the Hamiltonian for the quantum harmonic oscillator, except with the minimum of the potential shifted in coordinate space by . To find the energies, note that translating the harmonic oscillator potential does not affect the  energies. The energies of this system are thus identical to those of the standard quantum harmonic oscillator, formula_12 The energy does not depend on the quantum number , so there will be a finite number of degeneracies (If the particle is placed in an unconfined space, this degeneracy will correspond to a continuous sequence of formula_13). The value of formula_14 is continuous if the particle is unconfined in the z-direction and discrete if the particle is bounded in the z-direction also. Each set of wave functions with the same value of is called a Landau level. For the wave  functions, recall that formula_8 commutes with the Hamiltonian. Then the wave function factors into a product of momentum eigenstates in the direction and harmonic oscillator eigenstates formula_16 shifted by an amount in the direction: formula_17 where formula_18. In sum, the state of the electron is characterized by the quantum numbers, , and . In the symmetric gauge. The derivation treated and "y" as slightly asymmetric. However, by the symmetry of the system, there is no physical quantity which distinguishes these coordinates. The same result could have been obtained with an appropriate interchange of and . A more adequate choice of gauge, is the  symmetric gauge, which refers to the choice formula_19 In terms of dimensionless lengths and energies, the Hamiltonian can be expressed as formula_20 The correct units can be restored by introducing factors of formula_21 and formula_22. Consider operators formula_23 These operators follow certain commutation relations formula_24 In terms of above operators the Hamiltonian can be written as formula_25 where we reintroduced the units back. The Landau level index formula_26 is the eigenvalue of the operator formula_27. The application of formula_28 increases formula_29 by one unit while preserving formula_26, whereas formula_31 application simultaneously increase formula_26 and decreases formula_29 by one unit. The  analogy to quantum harmonic oscillator provides solutions formula_34 where formula_35 and formula_36 One may verify that the above states correspond to choosing wavefunctions proportional to formula_37 where formula_38. In particular, the lowest Landau level formula_39 consists of arbitrary analytic functions multiplying a Gaussian, formula_40. Degeneracy of the Landau levels. In the Landau gauge. The effects of Landau levels may only observed when the mean thermal energy is smaller than the energy level separation, , meaning low temperatures and strong magnetic fields. Each Landau level is degenerate because of the second quantum number , which can take the values formula_41, where is an  integer. The allowed values of are further restricted by the condition that the center of force of the oscillator, , must physically lie within the system, . This gives the following range for , formula_42 For particles with charge , the upper bound on can be simply written as a ratio of fluxes, formula_43 where is the fundamental magnetic flux quantum and is the flux through the system (with area ). Thus, for particles with spin , the maximum number of particles per Landau level is formula_44 which for electrons (where and ) gives , two available states for each flux quantum that penetrates the system. The above  gives only a rough idea of the effects of finite-size geometry. Strictly speaking, using the standard solution of the harmonic oscillator is only valid for systems unbounded in the -direction (infinite strips). If the size is finite, boundary conditions in that direction give rise to non-standard quantization conditions on the magnetic field, involving (in principle) both solutions to the Hermite equation. The filling of these levels with many electrons is still an active area of research. In general, Landau levels are observed in electronic systems. As the magnetic field is increased, more and more electrons can fit into  a given Landau level. The occupation of the highest Landau level ranges from completely full to entirely empty, leading to oscillations in various electronic properties (see De Haas–Van Alphen effect and Shubnikov–de Haas effect). If Zeeman splitting is included, each Landau level splits into a pair, one for spin up electrons and the other for spin down electrons. Then the occupation of each spin Landau level is just the ratio of fluxes . Zeeman splitting has a significant effect on the Landau levels because their energy scales are the same, . However, the Fermi energy and ground state energy stay  roughly the same in a system with many filled levels, since pairs of split energy levels cancel each other out when summed. Moreover, the above derivation in the Landau gauge assumed an electron confined in the -direction, which is a relevant experimental situation — found in two-dimensional electron gases, for instance. Still, this assumption is not essential for the results. If electrons are free to move along the direction, the wave function acquires an additional multiplicative term ; the energy corresponding to this free motion, , is added to the discussed. This term then fills in the separation in energy of the  different Landau levels, blurring the effect of the quantization. Nevertheless, the motion in the --plane, perpendicular to the magnetic field, is still quantized. In the symmetric gauge. Each Landau level has degenerate orbitals labeled by the quantum numbers formula_29 in symmetric gauge. The degeneracy per unit area is the same in each Landau level. The "z" component of angular momentum is formula_46 Exploiting the property formula_47 we chose eigenfunctions which diagonalize formula_48 and formula_49, The eigenvalue of formula_49 is denoted by formula_51, where it is clear that formula_52 in the formula_26th Landau level. However, it may be arbitrarily large, which  is necessary to obtain the infinite degeneracy (or finite degeneracy per unit area) exhibited by the system. Relativistic case. An electron following Dirac equation under a constant magnetic field, can be analytically solved. The energies are given by formula_54 where "c" is the speed of light, the sign depends on the particle-antiparticle component and "ν" is a non-negative integer. Due to spin, all levels are degenerate except for the ground state at . The massless 2D case can be simulated in single-layer materials like graphene near the Dirac cones, where the eigenergies are given by formula_55 where the  speed of light has to be replaced with the Fermi speed "v" of the material and the minus sign corresponds to electron holes. Magnetic susceptibility of a Fermi gas. The Fermi gas (an ensemble of non-interacting fermions) is part of the basis for understanding of the thermodynamic properties of metals. In 1930 Landau derived an estimate for the magnetic susceptibility of a Fermi gas, known as Landau susceptibility, which is constant for small magnetic fields. Landau also noticed that the susceptibility oscillates with high frequency for large magnetic fields, this physical phenomenon is known as the De Haas–Van  Alphen effect. Two-dimensional lattice. The tight binding energy spectrum of charged particles in a two dimensional infinite lattice is known to be self-similar and fractal, as demonstrated in Hofstadter's butterfly. For an integer ratio of the magnetic flux quantum and the magnetic flux through a lattice cell, one recovers the Landau levels for large integers.