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.^{ [1] }

- Derivation
- Landau levels
- Landau levels in symmetric gauge
- Effects of gauge transformation
- Relativistic case
- Magnetic susceptibility of a Fermi gas
- Two-dimensional lattice
- See also
- References
- Further reading

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.

Consider a system of non-interacting particles with charge q and spin S confined to an area *A* = *L _{x}L_{y}* in the

Here, ** is the canonical momentum operator and **** is the electromagnetic vector potential, which is related to the magnetic field by**

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 **Â** 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. For simplicity in calculation, choose the Landau gauge, which is

where B=|**B**| and *x̂* is the x component of the position operator.

In this gauge, the Hamiltonian is

The operator commutes with this Hamiltonian, since the operator *ŷ* is absent by the choice of gauge. Thus the operator can be replaced by its eigenvalue *ħk _{y}* . Since 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 *ω*_{c} = *qB/m*, giving

This is exactly the Hamiltonian for the quantum harmonic oscillator, except with the minimum of the potential shifted in coordinate space by *x*_{0} = *ħk _{y}/mω*

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,^{ [2] }

The energy does not depend on the quantum number *k _{y}*, 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 ). The value of is continuous if the particle is unconfined in the z-direction and discrete if the particle is bounded in the z-direction also.

For the wave functions, recall that commutes with the Hamiltonian. Then the wave function factors into a product of momentum eigenstates in the y direction and harmonic oscillator eigenstates shifted by an amount x_{0} in the x direction:

where . In sum, the state of the electron is characterized by the quantum numbers, n, *k _{y}* and

Each set of wave functions with the same value of n is called a Landau level. Effects of Landau levels are only observed when the mean thermal energy *kT* is smaller than the energy level separation, *kT ≪ ħω*_{c}, meaning low temperatures and strong magnetic fields.

Each Landau level is degenerate because of the second quantum number *k _{y}*, which can take the values

- ,

where N is an integer. The allowed values of N are further restricted by the condition that the center of force of the oscillator, *x _{0}*, must physically lie within the system, 0 ≤

For particles with charge *q* = *Ze*, the upper bound on N can be simply written as a ratio of fluxes,

where * Φ _{0} = h/e* is the fundamental magnetic flux quantum and

Thus, for particles with spin S, the maximum number D of particles per Landau level is

which for electrons (where Z=1 and S=1/2) gives *D = 2Φ/Φ _{0}*, 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 x-direction (infinite strips). If the size *L _{x}* 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

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 D = *Φ/Φ _{0}*. Zeeman splitting has a significant effect on the Landau levels because their energy scales are the same, 2

This derivation treats x 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 x and y.

Moreover, the above derivation assumed an electron confined in the z-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 z direction, the wave function acquires an additional multiplicative term exp(*ik _{z}z*); the energy corresponding to this free motion, (

The symmetric gauge refers to the choice

In terms of dimensionless lengths and energies, the Hamiltonian can be expressed as

The correct units can be restored by introducing factors of and

Consider operators

These operators follow certain commutation relations

- .

In terms of above operators the Hamiltonian can be written as

Landau Level index is the eigenvalue of

The z component of angular momentum is

Exploiting the property we chose eigenfunctions which diagonalize and , The eigenvalue of is denoted by , where it is clear that in the th 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.

The application of increases by one unit while preserving , whereas application simultaneously increase and decreases by one unit. The analogy to quantum harmonic oscillator provides solutions

Each Landau level has degenerate orbitals labeled by the quantum numbers *k _{y}* and in the Landau and symmetric gauges respectively. The degeneracy per unit area is the same in each Landau level.

One may verify that the above states correspond to choosing wavefunctions proportional to

where .

In particular, the lowest Landau level consists of arbitrary analytic functions multiplying a Gaussian, .

The definition for kinematical momenta is

where are the canonical momenta. The Hamiltonian is a gauge invariant so and will remain invariant under gauge transformations but will depend upon gauge. For observing the effect of gauge transformation on the quantum state of the particle, consider the state with **A** and **A**' as vector Potential, with states and .

As and is invariant under the gauge transformation we get

Consider an operator such that

from above relation we deduce that

from this we conclude

An electron following Dirac equation under a constant magnetic field, can be analytically solved.^{ [4] }^{ [5] } The energies are given by

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 *ν*=0.

The massless 2D case can be simulated in single-layer materials like graphene near the Dirac cones, where the eigenergies are given by^{ [6] }

where the speed of light has to be replaced with the Fermi speed *v*_{F} of the material and the minus sign corresponds to electron holes.

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,^{ [7] } this physical phenomenon is known as the De Haas–Van Alphen effect.

The tight binding energy spectrum of charged particles in a two dimensional infinite lattice is know 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.^{ [8] }

In quantum mechanics, the **Hamiltonian** of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's *energy spectrum* or its set of *energy eigenvalues*, is the set of possible outcomes obtainable from a measurement of the system's total energy. Due to its close relation to the energy spectrum and time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.

The **quantum harmonic oscillator** is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.

The **Schrödinger equation** is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

In physics, an **operator** is a function over a space of physical states onto another space of physical states. The simplest example of the utility of operators is the study of symmetry. Because of this, they are very useful tools in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory.

The **path integral formulation** is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.

**Creation and annihilation operators** are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator lowers the number of particles in a given state by one. A creation operator increases the number of particles in a given state by one, and it is the adjoint of the annihilation operator. In many subfields of physics and chemistry, the use of these operators instead of wavefunctions is known as second quantization.

The **adiabatic theorem** is a concept in quantum mechanics. Its original form, due to Max Born and Vladimir Fock (1928), was stated as follows:

In quantum mechanics, a **two-state system** is a quantum system that can exist in any quantum superposition of two independent quantum states. The Hilbert space describing such a system is two-dimensional. Therefore, a complete basis spanning the space will consist of two independent states. Any two-state system can also be seen as a qubit.

In quantum mechanics, the **Hellmann–Feynman theorem** relates the derivative of the total energy with respect to a parameter, to the expectation value of the derivative of the Hamiltonian with respect to that same parameter. According to the theorem, once the spatial distribution of the electrons has been determined by solving the Schrödinger equation, all the forces in the system can be calculated using classical electrostatics.

In quantum mechanics, an energy level is **degenerate** if it corresponds to two or more different measurable states of a quantum system. Conversely, two or more different states of a quantum mechanical system are said to be degenerate if they give the same value of energy upon measurement. The number of different states corresponding to a particular energy level is known as the degree of degeneracy of the level. It is represented mathematically by the Hamiltonian for the system having more than one linearly independent eigenstate with the same energy eigenvalue. When this is the case, energy alone is not enough to characterize what state the system is in, and other quantum numbers are needed to characterize the exact state when distinction is desired. In classical mechanics, this can be understood in terms of different possible trajectories corresponding to the same energy.

In quantum mechanics, the **angular momentum operator** is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum problems involving rotational symmetry. Such an operator is applied to a mathematical representation of the physical state of a system and yields an angular momentum value if the state has a definite value for it. In both classical and quantum mechanical systems, angular momentum is one of the three fundamental properties of motion.

The **Jaynes–Cummings model** is a theoretical model in quantum optics. It describes the system of a two-level atom interacting with a quantized mode of an optical cavity, with or without the presence of light. It was originally developed to study the interaction of atoms with the quantized electromagnetic field in order to investigate the phenomena of spontaneous emission and absorption of photons in a cavity.

**Photon polarization** is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.

The **theoretical and experimental justification for the Schrödinger equation** motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics described by Maxwell's equations, as an analogue for all types of particles.

In linear algebra, a **raising** or **lowering operator** is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raising operator is sometimes called the creation operator, and the lowering operator the annihilation operator. Well-known applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum.

This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.

In pure and applied mathematics, quantum mechanics and computer graphics, a **tensor operator** generalizes the notion of operators which are scalars and vectors. A special class of these are **spherical tensor operators** which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. The coordinate-free generalization of a tensor operator is known as a **representation operator**.

**Magnetic resonance** is a quantum mechanical resonant effect that can appear when a magnetic dipole is exposed to a static magnetic field and perturbed with another, oscillating electromagnetic field. Due to the static field, the dipole can assume a number of discrete energy eigenstates, depending on the value of its angular momentum quantum number. The oscillating field can then make the dipole transit between its energy states with a certain probability and at a certain rate. The overall transition probability will depend on the field's frequency and the rate will depend on its amplitude. When the frequency of that field leads to the maximum possible transition probability between two states, a magnetic resonance has been achieved. In that case, the energy of the photons composing the oscillating field matches the energy difference between said states. If the dipole is tickled with a field oscillating far from resonance, it is unlikely to transition. That is analogous to other resonant effects, such as with the forced harmonic oscillator. The periodic transition between the different states is called Rabi cycle and the rate at which that happens is called Rabi frequency. The Rabi frequency should not be confused with the field's own frequency. Since many atomic nuclei species can behave as a magnetic dipole, this resonance technique is the basis of nuclear magnetic resonance, including nuclear magnetic resonance imaging and nuclear magnetic resonance spectroscopy.

The **Peierls substitution** method, named after the original work by Rudolf Peierls is a widely employed approximation for describing tightly-bound electrons in the presence of a slowly varying magnetic vector potential.

Different subfields of physics have different programs for determining the state of a physical system.

- ↑ Landau, L. D. (1930). Diamagnetism of Metals. Zeitschrift für Physik, 64(9-10), 629-637.
- ↑ Landau, L. D., & Lifshitz, E. M., (1981). Quantum Mechanics; Non-relativistic Theory. 3rd edition. Butterworth-Heinemann. pp. 424-426.
- ↑ Mikhailov, S. A. (2001). "A new approach to the ground state of quantum Hall systems. Basic principles".
*Physica B: Condensed Matter*.**299**: 6. arXiv: cond-mat/0008227 . Bibcode:2001PhyB..299....6M. doi:10.1016/S0921-4526(00)00769-9. - ↑ Rabi, I. I. (1928). "Das freie Elektron im homogenen Magnetfeld nach der Diracschen Theorie".
*Zeitschrift für Physik*(in German).**49**(7–8): 507–511. doi:10.1007/BF01333634. ISSN 1434-6001. - ↑ Berestetskii, V. B.; Pitaevskii, L. P.; Lifshitz, E. M. (2012-12-02).
*Quantum Electrodynamics: Volume 4*. Elsevier. ISBN 978-0-08-050346-2. - ↑ Yin, Long-Jing; Bai, Ke-Ke; Wang, Wen-Xiao; Li, Si-Yu; Zhang, Yu; He, Lin (2017). "Landau quantization of Dirac fermions in graphene and its multilayers".
*Frontiers of Physics*.**12**(4): 127208. doi: 10.1007/s11467-017-0655-0 . ISSN 2095-0462. - ↑ Landau, L. D.; Lifshitz, E. M. (22 October 2013).
*Statistical Physics: Volume 5*. Elsevier. p. 177. ISBN 978-0-08-057046-4. - ↑ Analytis, James G.; Blundell, Stephen J.; Ardavan, Arzhang (May 2004). "Landau levels, molecular orbitals, and the Hofstadter butterfly in finite systems".
*American Journal of Physics*.**72**(5): 613–618. doi:10.1119/1.1615568. ISSN 0002-9505.

- Landau, L. D.; and Lifschitz, E. M.; (1977).
*Quantum Mechanics: Non-relativistic Theory. Course of Theoretical Physics*. Vol. 3 (3rd ed. London: Pergamon Press). ISBN 0750635398.

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