Landau quantization
Encyclopedia
Landau quantization in quantum mechanics
is the quantization of the cyclotron orbits of charged particles in magnetic fields. As a result, the charged particles can only occupy orbits with discrete energy values, called Landau levels. The Landau levels are degenerate
, with the number of electrons per level directly proportional to the strength of the applied magnetic field. Landau quantization is directly responsible for oscillations in electronic properties of materials as a function of the applied magnetic field. It is named after the Soviet physicist Lev Landau
.
and spin 
confined to an area 
in the x-y plane. Apply a uniform magnetic field 
along the z-axis. Using CGS units, the Hamiltonian
of this system is
Here,
is the canonical
momentum operator
and
is the electromagnetic vector potential
, which is related to the magnetic field
by
There is some freedom in the choice of vector potential for a given magnetic field. However, the Hamiltonian is gauge invariant, which means that adding the gradient of a scalar field
to A changes the overall phase of the wave function by an amount corresponding to the scalar field. Physical properties are not influenced by the specific choice of gauge. For simplicity in calculation, choose the Landau gauge, which is
where
and 
is one component of the position operator. In this gauge the Hamiltonian is
The operator
commutes with this Hamiltonian since the operator 
is absent due to the choice of gauge. Then the operator 
can be replaced by its eigenvalue 
. The Hamiltonian can also be written more simply by noting that the cyclotron frequency is 
, giving
This is exactly the Hamiltonian for the quantum harmonic oscillator
, except shifted in coordinate space by
.
To find the energies, note that translating the harmonic oscillator potential left or right does not change the energies. The energies of this system are identical to those of the quantum harmonic oscillator:
The energy does not depend on the quantum number
, so there will be degeneracies.
For the wave functions, recall that
commutes with the Hamiltonian. Then the wave function factors into a product of momentum eigenstates in the 
direction and harmonic oscillator eigenstates 
shifted by an amount 
in the 
direction:
In sum, the state of the electron is characterized by two quantum numbers,
and 
.
is called a Landau level. Effects of Landau levels are 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 due to the second quantum number
. If periodic boundary conditions are assumed, 
can take the values 
, where 
is an integer. The allowed values of 
are further restricted by the condition that the center of the oscillator 
must physically lie within the system, 
. This gives the following range for 
:
For particles with charge
, the upper bound on 
can be simply written as a ratio of fluxes
:
where
is the fundamental quantum of flux and 
is the flux through the system (with area 
). Thus for particles with spin 
, the maximum number 
of particles per Landau level is
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
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.
The apparent "oscillator center"
is in
this sense spurious, as the system has no reference point on the x-axis. It is, however, an indication of the very real issue of translational symmetry breaking: orbits in a magnetic field are circles, so how to choose their centers? Related issues on a lattice have also been discussed at length.
Generally, Landau levels are observed in electronic systems, where
and 
. 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 will have 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.
Additionally, 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. This assumption is not essential for the obtained results. If electrons are free to move along the z direction, the wave function acquires an additional multiplicative term
; the energy corresponding to this free motion, 
, has to be added to E. This term fills the separation in energy of the different Landau levels, blurring the effect of the quantization. In any case, the motion in the x-y-plane, perpendicular to the magnetic field, is quantized.
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
is the quantization of the cyclotron orbits of charged particles in magnetic fields. As a result, the charged particles can only occupy orbits with discrete energy values, called Landau levels. The Landau levels are degenerate
Degenerate energy level
In physics, two or more different quantum states are said to be degenerate if they are all at the same energy level. Statistically this means that they are all equally probable of being filled, and in Quantum Mechanics it is represented mathematically by the Hamiltonian for the system having more...
, with the number of electrons per level directly proportional to the strength of the applied magnetic field. Landau quantization is directly responsible for oscillations in electronic properties of materials as a function of the applied magnetic field. It is named after the Soviet physicist Lev Landau
Lev Landau
Lev Davidovich Landau was a prominent Soviet physicist who made fundamental contributions to many areas of theoretical physics...
.
Derivation
Consider a two-dimensional system of non-interacting particles with charge and spin confined to an area in the x-y plane. Apply a uniform magnetic field along the z-axis. Using CGS units, the HamiltonianHamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
of this system is
Here,
is the canonicalCanonical coordinates
In mathematics and classical mechanics, canonical coordinates are particular sets of coordinates on the phase space, or equivalently, on the cotangent manifold of a manifold. Canonical coordinates arise naturally in physics in the study of Hamiltonian mechanics...
momentum operator
Momentum operator
In quantum mechanics, momentum is defined as an operator on the wave function. The Heisenberg uncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once...
and
is the electromagnetic vector potentialMagnetic potential
The term magnetic potential can be used for either of two quantities in classical electromagnetism: the magnetic vector potential, A, and the magnetic scalar potential, ψ...
, which is related to the magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
by
There is some freedom in the choice of vector potential for a given magnetic field. However, the Hamiltonian is gauge invariant, which means that adding the gradient of a scalar field
Scalar field
In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...
to A changes the overall phase of the wave function by an amount corresponding to the scalar field. Physical properties are not influenced by the specific choice of gauge. For simplicity in calculation, choose the Landau gauge, which is
where
and is one component of the position operator. In this gauge the Hamiltonian isThe operator
commutes with this Hamiltonian since the operator is absent due to the choice of gauge. Then the operator can be replaced by its eigenvalue . The Hamiltonian can also be written more simply by noting that the cyclotron frequency is , givingThis is exactly the Hamiltonian for the quantum harmonic oscillator
Quantum harmonic oscillator
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can 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...
, except shifted in coordinate space by
.To find the energies, note that translating the harmonic oscillator potential left or right does not change the energies. The energies of this system are identical to those of the quantum harmonic oscillator:
The energy does not depend on the quantum number
, so there will be degeneracies.For the wave functions, recall that
commutes with the Hamiltonian. Then the wave function factors into a product of momentum eigenstates in the direction and harmonic oscillator eigenstates shifted by an amount in the direction:In sum, the state of the electron is characterized by two quantum numbers,
and .Landau levels
Each set of wave functions with the same value of is called a Landau level. Effects of Landau levels are 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 due to the second quantum number
. If periodic boundary conditions are assumed, can take the values , where is an integer. The allowed values of are further restricted by the condition that the center of the oscillator must physically lie within the system, . This gives the following range for :For particles with charge
, the upper bound on can be simply written as a ratio of fluxesMagnetic flux
Magnetic flux , is a measure of the amount of magnetic B field passing through a given surface . The SI unit of magnetic flux is the weber...
:
where
is the fundamental quantum of flux and is the flux through the system (with area ). Thus for particles with spin , the maximum number of particles per Landau level isThe 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
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.The apparent "oscillator center"
is inthis sense spurious, as the system has no reference point on the x-axis. It is, however, an indication of the very real issue of translational symmetry breaking: orbits in a magnetic field are circles, so how to choose their centers? Related issues on a lattice have also been discussed at length.
Generally, Landau levels are observed in electronic systems, where
and . 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 effectDe Haas-van Alphen effect
The de Haas–van Alphen effect, often abbreviated to dHvA, is a quantum mechanical effect in which the magnetic moment of a pure metal crystal oscillates as the intensity of an applied magnetic field B is increased. Other quantities also oscillate, such as the resistivity , specific heat, and sound...
and Shubnikov-De Haas effect
Shubnikov-De Haas effect
An oscillation in the conductivity of a material that occurs at low temperatures in the presence of very intense magnetic fields, the Shubnikov–de Haas effect is a macroscopic manifestation of the inherent quantum mechanical nature of matter...
).
If Zeeman splitting
Zeeman effect
The Zeeman effect is the splitting of a spectral line into several components in the presence of a static magnetic field. It is analogous to the Stark effect, the splitting of a spectral line into several components in the presence of an electric field...
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 will have 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.Discussion
This derivation treats x and y as slightly asymmetric. However, because of the symmetry of the system, there is no physical quantity which differentiates these coordinates. The same result could have been obtained with an appropriate exchange of x and y.Additionally, 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. This assumption is not essential for the obtained results. If electrons are free to move along the z direction, the wave function acquires an additional multiplicative term
; the energy corresponding to this free motion, , has to be added to E. This term fills the separation in energy of the different Landau levels, blurring the effect of the quantization. In any case, the motion in the x-y-plane, perpendicular to the magnetic field, is quantized.See also
- De Haas–van Alphen effect
- Shubnikov–De Haas effect
- Quantum Hall effect
- Laughlin wavefunctionLaughlin wavefunctionIn condensed matter physics, the Laughlin wavefunction is an ansatz, proposed by Robert Laughlin for the ground state of a two-dimensional electron gas placed in a uniform background magnetic field in the presence of a uniform jellium background when the filling factor of the lowest Landau level is...
- Coulomb potential between two current loops embedded in a magnetic field