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{{Short description|Quantization of cyclotron orbits}}
In [[quantum mechanics]],
'''Landau quantization'''
Landau quantization is a key ingredient
== Derivation ==
[[File:Cyclotron Orbit.jpg|thumb|Diagram of a cyclotron orbit of a particle with speed '''v''', which is the classical trajectory of a charged particle (here positive charge) under a uniform magnetic field '''B'''. The Landau quantization refers to a quantum charged particle under a uniform magnetic field.]]
Consider a system of non-interacting particles with charge {{mvar|q}} and spin {{mvar|S}} confined to an area {{math|1=''A'' = ''L<sub>x</sub>L<sub>y</sub>''}} in the {{math|''x-y''}} plane. Apply a uniform magnetic field <math>\mathbf{B} = \begin{pmatrix}0\\0\\B\end{pmatrix}</math> along the {{mvar|z}}-axis. In [[
<math display="block">\hat{H} = \frac{1}{2m} \left
Here, <math display="inline">\hat{\mathbf{p}}</math> is the [[Canonical coordinates|canonical]] [[momentum operator]] and <math display="inline">\hat{\mathbf{A}}</math> is the [[Operator (physics)#Operators in quantum mechanics|operator]] for the [[Magnetic vector potential|electromagnetic vector potential]]
<math display="block">\mathbf{B}=\mathbf{\nabla}\times \hat{\mathbf{A}}. </math>▼
▲The vector potential is related to the [[magnetic field]] by <math
There is some gauge freedom in the choice of vector potential for a given magnetic field. The Hamiltonian is [[gauge invariance|gauge invariant]], which means that adding the gradient of a [[scalar field]] to {{math|'''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. ▼
▲There is some gauge freedom in the choice of vector potential for a given magnetic field. The Hamiltonian is [[gauge invariance|gauge invariant]], which means that adding the gradient of a [[scalar field]] to {{math|'''
=== 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.<ref>{{cite web |url=https://courses.physics.illinois.edu/phys581/sp2013/charge_mag.pdf |title=Charge in Magnetic Field |access-date=11 March 2023 |website=courses.physics.illinois.edu}}</ref>
When <math>\mathbf{B} = \begin{pmatrix} 0 \\ 0 \\ B \end{pmatrix}</math> then <math>\mathbf{A} = \begin{pmatrix} 0 \\ B\cdot x \\ 0 \end{pmatrix}</math> is a possible solution<ref>An equally correct solution in the Landau gauge would be: <math>\mathbf{A} = \begin{pmatrix} -B y & 0 & 0 \end{pmatrix}^T</math>.</ref> in the Landau gauge.
In this gauge, the Hamiltonian is
<math display="block">\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}.</math>
The operator <math>\hat{p}_y</math> commutes with this Hamiltonian, since the operator {{math|''ŷ''}} is absent by the choice of gauge. Thus the operator <math>\hat{p}_y</math> can be replaced by its eigenvalue {{math|''ħk<sub>y</sub>''}}. Since <math>\hat{z}</math> 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 resonance|cyclotron frequency]] is {{math|1=''ω''<sub>c</sub> = ''qB''/''m''}}, giving
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This is exactly the Hamiltonian for the [[quantum harmonic oscillator]], except with the minimum of the potential shifted in coordinate space by {{math|1=''x''<sub>0</sub> = ''ħk<sub>y</sub>''/''mω''<sub>c</sub>}} .
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]],<ref>{{cite book | last1=Landau | first1=L. D. |author-link1=Lev Landau|last2=Lifshitz|first2=E. M.|author-link2=Evgeny Lifshitz| title=Quantum mechanics : non-relativistic theory |edition=3rd| publisher=Butterworth Heinemann | publication-place=Amsterdam | year=1977 | isbn=978-0-7506-3539-4 | oclc=846962062 |
<math display="block">E_n=\hbar\omega_{\rm c}\left(n+\frac{1}{2}\right) + \frac{p_z^2}{2m},\quad n\geq 0~. </math>
The energy does not depend on the quantum number {{math|''k<sub>y</sub>''}}, 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 <math>p_y</math>). The value of <math>p_z</math> 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 {{mvar|n}} is called a '''Landau level'''.
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===In the symmetric gauge===
The derivation treated {{mvar|x}} and ''y'' as
A more adequate choice of gauge, is the symmetric gauge, which refers to the choice
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== Relativistic case ==
{{See also|Dirac matter}}
[[File:Graphene - Geim - Landau levels.svg|thumb|193x193px|Landau levels in [[graphene]]. [[Charge carrier
An electron following [[Dirac equation]] under a constant magnetic field, can be analytically solved.<ref>{{Cite journal| last=Rabi|first=I. I.|date=1928|title=Das freie Elektron im homogenen Magnetfeld nach der Diracschen Theorie| url=http://link.springer.com/10.1007/BF01333634| journal=Zeitschrift für Physik|language=de|volume=49|issue=7–8| pages=507–511| doi=10.1007/BF01333634|bibcode=1928ZPhy...49..507R |s2cid=121121095|issn=1434-6001}}</ref><ref>{{Cite book|last1=Berestetskii|first1=V. B.| url=https://books.google.com/books?id=Tpk-lqyr3GoC&q=lifshitz+electrodynamics|title=Quantum Electrodynamics: Volume 4| last2=Pitaevskii| first2=L. P.|last3=Lifshitz|first3=E. M.|date=2012-12-02|publisher=Elsevier| isbn=978-0-08-050346-2| language=en}}</ref> The energies are given by
<math display="block">E_{\rm rel}=\pm \sqrt{(mc^2)^2+(c\hbar k_z)^2+2\nu \hbar\omega_{\rm c} mc^2}</math>
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 {{math|1=''ν'' = 0}}.
The massless 2D case can be simulated in [[single-layer materials]] like [[graphene]] near the [[Dirac cone
<math display="block">E_{\rm graphene}=\pm \sqrt{2\nu\hbar eBv_{\rm F}^2 }</math>
where the speed of light has to be replaced with the [[Fermi energy|Fermi speed]] ''v''<sub>F</sub> of the material and the minus sign corresponds to [[electron hole]]s.
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== Two-dimensional lattice ==
{{See also|Hofstadter's butterfly}}
The [[tight binding]] energy spectrum of charged particles in a two dimensional infinite lattice is known to be [[Self-similarity|self-similar]] and [[fractal]], as demonstrated in [[Hofstadter's butterfly]]. For an integer ratio of the [[Fluxon|magnetic flux quantum]] and the magnetic flux through a lattice cell, one recovers the Landau levels for large integers.<ref>{{Cite journal|last1=Analytis|first1=James G.|last2=Blundell|first2=Stephen J.|last3=Ardavan|first3=Arzhang|date=May 2004 |title=Landau levels, molecular orbitals, and the Hofstadter butterfly in finite systems|url=http://aapt.scitation.org/doi/10.1119/1.1615568|journal=American Journal of Physics|language=en|volume=72|issue=5|pages=613–618|doi=10.1119/1.1615568 |bibcode=2004AmJPh..72..613A |issn=0002-9505}}</ref>
== Integer quantum Hall effect ==
{{Main|Quantum Hall effect}}
{{Expand section|date=August 2021}}
The energy spectrum of the [[semiconductor]] in a strong magnetic field forms Landau levels that can be labeled by integer indices. In addition, the [[Hall effect|Hall resistivity]] also exhibits discrete levels labeled by an integer {{mvar|ν}}. The fact that these two quantities are related can be shown in different ways, but most easily can be seen from [[Drude model]]: the Hall conductivity depends on the electron density {{mvar|n}} as
<math>\rho_{xy}=\frac{B}{n e}.</math>
Since the resistivity plateau is given by
<math>\rho_{xy}=\frac{2 \pi\hbar }{e^2}\frac{1}{\nu},</math>
the required density is
<math>n=\frac{B }{\Phi_0}\nu,</math>
which is exactly the density required to fill the Landau level. The [[energy gap|gap]] between different Landau levels along with large degeneracy of each level renders the resistivity quantized.
==See also==
*[[Laughlin wavefunction]]
==References==
{{reflist}}
== External links ==
* {{cite web|author=Lev Landau (1930)|language=de|title=Diamagnetismus der Metalle|url=https://gilles.montambaux.com/files/histoire-physique/Landau-1930.pdf}}<!-- auto-translated by Module:CS1 translator -->
==Further reading==
* 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}}.
{{Authority control}}
[[Category:Quantum mechanics]]
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