In quantum mechanics, the energies of cyclotron orbits of charged particles in a uniform magnetic field are quantized to discrete values, thus known as 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 who developed the theory in 1930. Landau quantization contributes towards magnetic 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 the De Haas–Van Alphen and Shubnikov–de Haas effects. Landau quantization is a key ingredient in explanation of the integer quantum Hall effect.
Derivation
Consider a system of non-interacting particles with charge q and spin S confined to an area A = LxLy in the x-y plane. Apply a uniform magnetic field B = ( 0 0 B ) {\displaystyle \mathbf {B} ={\begin{pmatrix}0\\0\\B\end{pmatrix}}} along the z-axis. In SI units, the Hamiltonian of this system (here, the effects of spin are neglected) is
H ^ = 1 2 m ( p ^ − q A ^ ) 2 . {\displaystyle {\hat {H}}={\frac {1}{2m}}\left({\hat {\mathbf {p} }}-q{\hat {\mathbf {A} }}\right)^{2}.}
Here, p ^ {\textstyle {\hat {\mathbf {p} }}} is the canonical momentum operator and A ^ {\textstyle {\hat {\mathbf {A} }}} is the operator for the electromagnetic vector potential A {\textstyle \mathbf {A} } (in position space A ^ = A {\textstyle {\hat {\mathbf {A} }}=\mathbf {A} } ). The vector potential is related to the magnetic field by B = ∇ × A . {\displaystyle \mathbf {B} =\mathbf {\nabla } \times \mathbf {A} .}
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 A 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 From the possible solutions for A, a gauge fixing introduced by Lev Landau is often used for charged particles in a constant magnetic field. When B = ( 0 0 B ) {\displaystyle \mathbf {B} ={\begin{pmatrix}0\\0\\B\end{pmatrix}}} then A = ( 0 B ⋅ x 0 ) {\displaystyle \mathbf {A} ={\begin{pmatrix}0\\B\cdot x\\0\end{pmatrix}}} is a possible solution in the Landau gauge (not to be mixed up with the Landau R ξ {\displaystyle R_{\xi }} gauge). In this gauge, the Hamiltonian is
H ^ = p ^ x 2 2 m + 1 2 m ( p ^ y − q B x ^ ) 2 + p ^ z 2 2 m . {\displaystyle {\hat {H}}={\frac {{\hat {p}}_{x}^{2}}{2m}}+{\frac {1}{2m}}\left({\hat {p}}_{y}-qB{\hat {x}}\right)^{2}+{\frac {{\hat {p}}_{z}^{2}}{2m}}.}
The operator p ^ y {\displaystyle {\hat {p}}_{y}} commutes with this Hamiltonian, since the operator y ^ {\displaystyle {\hat {y}}} is absent for this choice of gauge. Thus the operator p ^ y {\displaystyle {\hat {p}}_{y}} can be replaced by its eigenvalue ℏ k y {\displaystyle \hbar k_{y}} . Since z ^ {\displaystyle {\hat {z}}} 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 = q B / m {\displaystyle \omega _{c}=qB/m} , giving
H ^ = p ^ x 2 2 m + 1 2 m ω c 2 ( x ^ − ℏ k y m ω c ) 2 + p ^ z 2 2 m . {\displaystyle {\hat {H}}={\frac {{\hat {p}}_{x}^{2}}{2m}}+{\frac {1}{2}}m\omega _{\rm {c}}^{2}\left({\hat {x}}-{\frac {\hbar k_{y}}{m\omega _{\rm {c}}}}\right)^{2}+{\frac {{\hat {p}}_{z}^{2}}{2m}}.}
This is exactly the Hamiltonian for the quantum harmonic oscillator, except with the minimum of the potential shifted in coordinate space by x 0 = ℏ k y / m ω c {\displaystyle x_{0}=\hbar k_{y}/m\omega _{c}} . 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,
E n = ℏ ω c ( n + 1 2 ) + p z 2 2 m , n ≥ 0. {\displaystyle E_{n}=\hbar \omega _{\rm {c}}\left(n+{\frac {1}{2}}\right)+{\frac {p_{z}^{2}}{2m}},\quad n\geq 0.}
The energy does not depend on the quantum number k y {\displaystyle 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 p y {\displaystyle p_{y}} ). The value of p z {\displaystyle p_{z}} 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 n {\displaystyle n} is called a Landau level. For the wave functions, recall that p ^ y {\displaystyle {\hat {p}}_{y}} commutes with the Hamiltonian. Then the wave function factors into a product of momentum eigenstates in the y {\displaystyle y} direction and harmonic oscillator eigenstates | ϕ n ⟩ {\displaystyle |\phi _{n}\rangle } shifted by an amount x 0 {\displaystyle x_{0}} in the x {\displaystyle x} direction:
Ψ ( x , y , z ) = e i ( k y y + k z z ) ϕ n ( x − x 0 ) {\displaystyle \Psi (x,y,z)=e^{i(k_{y}y+k_{z}z)}\phi _{n}(x-x_{0})}
where k z = p z / ℏ {\displaystyle k_{z}=p_{z}/\hbar } and ϕ n ( x − x 0 ) {\displaystyle \phi _{n}(x-x_{0})} is the n-th state for the quantum harmonic oscilator. In sum, the state of the electron is characterized by the quantum numbers, n {\displaystyle n} , k y {\displaystyle k_{y}} and k z {\displaystyle k_{z}} .
In the symmetric gauge The derivation treated x and y as 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. A more adequate choice of gauge, is the symmetric gauge, which refers to the choice
A ^ = 1 2 B × r ^ = 1 2 ( − B y B x 0 ) . {\displaystyle {\hat {\mathbf {A} }}={\frac {1}{2}}\mathbf {B} \times {\hat {\mathbf {r} }}={\frac {1}{2}}{\begin{pmatrix}-By\\Bx\\0\end{pmatrix}}.}
In terms of dimensionless lengths and energies, the Hamiltonian can be expressed as
H ^ = 1 2 [ ( − i ∂ ∂ x + y 2 ) 2 + ( − i ∂ ∂ y − x 2 ) 2 ] {\displaystyle {\hat {H}}={\frac {1}{2}}\left[\left(-i{\frac {\partial }{\partial x}}+{\frac {y}{2}}\right)^{2}+\left(-i{\frac {\partial }{\partial y}}-{\frac {x}{2}}\right)^{2}\right]}
The correct units can be restored by introducing factors of q , ℏ , B {\displaystyle q,\hbar ,\mathbf {B} } and m {\displaystyle m} . Consider operators
a ^ = 1 2 [ ( x 2 + ∂ ∂ x ) − i ( y 2 + ∂ ∂ y ) ] a ^ † = 1 2 [ ( x 2 − ∂ ∂ x ) + i ( y 2 − ∂ ∂ y ) ] b ^ = 1 2 [ ( x 2 + ∂ ∂ x ) + i ( y 2 + ∂ ∂ y ) ] b ^ † = 1 2 [ ( x 2 − ∂ ∂ x ) − i ( y 2 − ∂ ∂ y ) ] {\displaystyle {\begin{aligned}{\hat {a}}&={\frac {1}{\sqrt {2}}}\left[\left({\frac {x}{2}}+{\frac {\partial }{\partial x}}\right)-i\left({\frac {y}{2}}+{\frac {\partial }{\partial y}}\right)\right]\\{\hat {a}}^{\dagger }&={\frac {1}{\sqrt {2}}}\left[\left({\frac {x}{2}}-{\frac {\partial }{\partial x}}\right)+i\left({\frac {y}{2}}-{\frac {\partial }{\partial y}}\right)\right]\\{\hat {b}}&={\frac {1}{\sqrt {2}}}\left[\left({\frac {x}{2}}+{\frac {\partial }{\partial x}}\right)+i\left({\frac {y}{2}}+{\frac {\partial }{\partial y}}\right)\right]\\{\hat {b}}^{\dagger }&={\frac {1}{\sqrt {2}}}\left[\left({\frac {x}{2}}-{\frac {\partial }{\partial x}}\right)-i\left({\frac {y}{2}}-{\frac {\partial }{\partial y}}\right)\right]\end{aligned}}}
These operators follow certain commutation relations
[ a ^ , a ^ † ] = [ b ^ , b ^ † ] = 1. {\displaystyle [{\hat {a}},{\hat {a}}^{\dagger }]=[{\hat {b}},{\hat {b}}^{\dagger }]=1.}
In terms of above operators the Hamiltonian can be written as
H ^ = ℏ ω c ( a ^ † a ^ + 1 2 ) , {\displaystyle {\hat {H}}=\hbar \omega _{\rm {c}}\left({\hat {a}}^{\dagger }{\hat {a}}+{\frac {1}{2}}\right),}
where we reintroduced the units back. The Landau level index n {\displaystyle n} is the eigenvalue of the operator N ^ = a ^ † a ^ {\displaystyle {\hat {N}}={\hat {a}}^{\dagger }{\hat {a}}} . The application of b ^ † {\displaystyle {\hat {b}}^{\dagger }} increases m z {\displaystyle m_{z}} by one unit while preserving n {\displaystyle n} , whereas a ^ † {\displaystyle {\hat {a}}^{\dagger }} application simultaneously increase n {\displaystyle n} and decreases m z {\displaystyle m_{z}} by one unit. The analogy to quantum harmonic oscillator provides solutions
H ^ | n , m z ⟩ = E n | n , m z ⟩ , {\displaystyle {\hat {H}}|n,m_{z}\rangle =E_{n}|n,m_{z}\rangle ,}
where
E n = ℏ ω c ( n + 1 2 ) {\displaystyle E_{n}=\hbar \omega _{\rm {c}}\left(n+{\frac {1}{2}}\right)}
and
| n , m z ⟩ = ( b ^ † ) m z + n ( m z + n ) ! ( a ^ † ) n n ! | 0 , 0 ⟩ . {\displaystyle |n,m_{z}\rangle ={\frac {({\hat {b}}^{\dagger })^{m_{z}+n}}{\sqrt {(m_{z}+n)!}}}{\frac {({\hat {a}}^{\dagger })^{n}}{\sqrt {n!}}}|0,0\rangle .}
One may verify that the above states correspond to choosing wavefunctions proportional to
ψ n , m z ( x , y ) = ( ∂ ∂ w − w ¯ 4 ) n w n + m z e − | w | 2 / 4 {\displaystyle \psi _{n,m_{z}}(x,y)=\left({\frac {\partial }{\partial w}}-{\frac {\bar {w}}{4}}\right)^{n}w^{n+m_{z}}e^{-|w|^{2}/4}}
where w = x − i y {\displaystyle w=x-iy} . In particular, the lowest Landau level n = 0 {\displaystyle n=0} consists of arbitrary analytic functions multiplying a Gaussian, ψ ( x , y ) = f ( w ) e − | w | 2 / 4 {\displaystyle \psi (x,y)=f(w)e^{-|w|^{2}/4}} .
Degeneracy of the Landau levels
In the Landau gauge The effects of Landau levels may only be observed when the mean thermal energy kT is smaller than the energy level separation, k T ≪ ℏ ω c {\displaystyle kT\ll \hbar \omega _{c}} , meaning low temperatures and strong magnetic fields. Each Landau level is degenerate because of the second quantum number k y {\displaystyle k_{y}} , which can take the values
k y = 2 π N L y , {\displaystyle k_{y}={\frac {2\pi N}{L_{y}}},}
where N {\displaystyle N} is an integer. The allowed values of N {\displaystyle N} are further restricted by the condition that the center of force of the oscillator, x 0 {\displaystyle x_{0}} , must physically lie within the system, 0 ≤ x 0 < L x {\displaystyle 0\leq x_{0}<L_{x}} . This gives the following range for N {\displaystyle N} ,
0 ≤ N < m ω c L x L y 2 π ℏ . {\displaystyle 0\leq N<{\frac {m\omega _{\rm {c}}L_{x}L_{y}}{2\pi \hbar }}.}
For particles with charge q = Z e {\displaystyle q=Ze} , the upper bound on N {\displaystyle N} can be simply written as a ratio of fluxes,
Z B L x L y ( h / e ) = Z Φ Φ 0 , {\displaystyle {\frac {ZBL_{x}L_{y}}{(h/e)}}=Z{\frac {\Phi }{\Phi _{0}}},}
where Φ 0 = h / e {\displaystyle \Phi _{0}=h/e} is the fundamental magnetic flux quantum and Φ = B A {\displaystyle \Phi =BA} is the flux through the system (with area A = L x L y {\displaystyle A=L_{x}L_{y}} ). Thus, for particles with spin S {\displaystyle S} , the maximum number D {\displaystyle D} of particles per Landau level is
D = Z ( 2 S + 1 ) Φ