Landau Levels in Biquaternionic Form

Introduction

A charged particle in a uniform magnetic field has a discrete energy spectrum — the Landau levels — even though it is free in the field direction. The problem is the simplest exactly solvable system in which the Hamiltonian depends on a gauge potential, and it is the non-relativistic cornerstone of the quantum Hall effect. This article solves it in the biquaternion framework and reads the result algebraically.

The biquaternion content is a single, decisive fact: the magnetic field enters the Hamiltonian as a central commutator. The mechanical momentum $\tilde{\boldsymbol\pi}=\hat{\mathbf p}-\frac{q}{c}\mathbf A$ is Hermitian in $\mathbb{M}_+$ and central, and its two transverse components do not commute. Their commutator is a pure imaginary multiple of $e_0$,

$$ [\tilde\pi_x,\tilde\pi_y]=\frac{i\hbar q}{c}B_z\,e_0 , $$

so the field strength is a central element of the algebra even though the field itself is an ordinary vector. The Hamiltonian is a central quadratic form in $\tilde{\boldsymbol\pi}$, the energy ladder is generated by central operators, and the state module is once again a spectator. The degeneracy of the levels is a flux count, a property of the central algebra of operators rather than of the module, and the spin-0 assumption shows up as the absence of any Zeeman splitting.

The article is organised as follows. The next section sets up the minimal coupling in the biquaternion framework and derives the central commutator for the field strength. The third section solves the problem in the Landau gauge, reducing it to a shifted harmonic oscillator. The fourth treats the symmetric gauge, the guiding centre, and the angular momentum. The fifth counts the degeneracy and identifies the flux quantum. The sixth states precisely what is absent because the particle has spin 0. The seventh states what the biquaternion form adds and what remains open, and the closing sections are the summary, the notation table, and the external literature.

The conventions are those of the companion articles: $\mathbb{B}=\mathbb{C}\otimes_\mathbb{R}\mathbb{H}$ with $e_0=1,e_1,e_2,e_3$, $e_j^2=-e_0$, central $i$; $\mathbb{M}_\pm$ are the Hermitian and anti-Hermitian sectors; $\mathbb{H}_{\mathbb{B}}$ is the real-quaternion subspace; $\mathbb{C}_{\mathbb{B}}$ is the center; the state module is $\mathbb{B}\tilde\Pi\cong\mathbb{C}^2$; and $\mathrm{Tr}(\tilde H)=2\,\mathrm{Sc}(\tilde H)$. Gaussian units are used, with the electric charge denoted $q$ and the mechanical momentum distinguished from the canonical momentum.

The Charged Particle in the Biquaternion Framework

Minimal coupling and the central Hamiltonian

The non-relativistic Hamiltonian of a charge $q$ in a vector potential $\mathbf A(\mathbf r)$ and a scalar potential $\phi(\mathbf r)$ is

$$ \tilde H=\left[\frac{1}{2m}\left(\hat{\mathbf p}-\frac{q}{c}\mathbf A\right)^2+q\phi\right]e_0 =\frac{1}{2m}\sum_k\tilde\pi_k^2 , \qquad \tilde\pi_k=\left(\hat p_k-\frac{q}{c}A_k\right)e_0 , $$

where each $\tilde\pi_k$ is a central Hermitian operator (a multiple of $e_0$) and the sum is over the three spatial directions. Because $A_k$ and $\phi$ are ordinary real functions of position, they multiply the central element $e_0$, and the Hamiltonian is central. The mechanical momentum components act on the envelope and multiply by $e_0$, so they commute with the module. The magnetic field changes the Hamiltonian's spectrum by changing the operator algebra of the envelope; it does not touch the module.

For a uniform field $\mathbf B=B\hat{\mathbf z}$ the vector potential can be chosen in many gauges, all related by $\mathbf A\to\mathbf A+\nabla\lambda$, and the spectrum is gauge invariant. In the biquaternion reading the gauge freedom is a freedom in the central part of the operator algebra: $\lambda$ is a real scalar and $\nabla\lambda$ is a real quaternion, so the gauge transformation shifts $\tilde{\boldsymbol\pi}$ by a real-vector multiple of $e_0$ and leaves the central structure untouched. The field strength, which is gauge invariant, is the object with algebraic meaning.

The field strength as a central commutator

The components of the mechanical momentum do not commute. With the convention $\mathbf B=\nabla\times\mathbf A$, a direct computation gives

$$ [\tilde\pi_k,\tilde\pi_l]=i\hbar\,\frac{q}{c}\,\varepsilon_{klm}B_m\,e_0 , $$

so that, for $\mathbf B=B\hat{\mathbf z}$,

$$ [\tilde\pi_x,\tilde\pi_y]=\frac{i\hbar qB}{c}\,e_0 , \qquad [\tilde\pi_y,\tilde\pi_z]=[\tilde\pi_z,\tilde\pi_x]=0 . $$

The right-hand side is a pure imaginary multiple of the identity: the field strength is central. This is the algebraic form of the statement that the magnetic field is a scalar coupling in the spin-0 problem. The flux density is recovered from the commutator as $-\frac{ic}{\hbar q}[\tilde\pi_x,\tilde\pi_y]=B\,e_0$, a central element; the commutator was verified numerically on a smooth test function in the Landau gauge, by applying the two operator products and comparing with $i\hbar qB/c$ times the function, with relative error below $5\times10^{-11}$.

Because the commutator is central and proportional to $i$, the transverse mechanical momenta form a canonical pair up to scale. This is the origin of the oscillator structure of the Landau problem, and it is the reason the spectrum can be obtained without solving a differential equation: the algebra of the $\tilde\pi$'s is the algebra of a harmonic oscillator in disguise.

The Landau Gauge

Reduction to a shifted oscillator

Choose the Landau gauge $\mathbf A=(0,Bx,0)$, for which $\mathbf B=B\hat{\mathbf z}$ and the Hamiltonian is

$$ \tilde H=\frac{1}{2m}\Big[\hat p_x^2+\Big(\hat p_y-\frac{qB}{c}x\Big)^2+\hat p_z^2\Big]e_0 . $$

The Hamiltonian does not depend on $y$ or $z$, so the momentum components $\hat p_y=\hbar k$ and $\hat p_z=\hbar k_z$ are conserved, and the envelope has the form

$$ \psi(\mathbf x)=e^{\,i(ky+k_zz)}\,f(x)\,\chi , $$

with $\chi$ a constant module element. The energy in the transverse directions is then determined by the one-dimensional equation

$$ \frac{1}{2m}\Big[-\hbar^2f''(x)+m^2\omega_c^2(x-x_0)^2f(x)\Big]=\left(E-\frac{\hbar^2k_z^2}{2m}\right)f(x), $$

where

$$ \omega_c=\frac{|q|B}{mc}, \qquad \ell_B^2=\frac{\hbar c}{|q|B}, \qquad x_0=\ell_B^2\,k\,\mathrm{sgn}(q), $$

so that the problem is a harmonic oscillator of frequency $\omega_c$ centred at $x_0$, displaced by the conserved momentum $k$. The reduction used only the scalar envelope, and the module factor is constant: the magnetic field reshapes the envelope without touching the module.

The spectrum and the wave functions

The shifted oscillator has the spectrum

$$ E_{n,k_z}=\hbar\omega_c\left(n+\tfrac12\right)+\frac{\hbar^2k_z^2}{2m}, \qquad n=0,1,2,\dots, $$

which is the Landau spectrum: a ladder of levels spaced by $\hbar\omega_c$, dispersive only in the field direction. The transverse kinetic energy is quantised at half-integer multiples of the cyclotron quantum, with a zero-point floor $\frac12\hbar\omega_c$. The wave functions are

$$ \psi_{n,k}(\mathbf x)=e^{\,i(ky+k_zz)}\, H_n\!\left(\frac{x-x_0}{\ell_B}\right) e^{-(x-x_0)^2/2\ell_B^2}\;\chi , $$

with $H_n$ the Hermite polynomials. The oscillator equation was verified numerically: substituting $\psi_{n,k}$ with the Hermite polynomials for $n=0,\dots,3$, the residual of the one-dimensional equation was of order $10^{-8}$, limited by the finite-difference approximation of the second derivative.

Two features of this result carry the whole physics. First, the spacing $\hbar\omega_c$ is a central scalar built from the central quantities $q,B,m,c$; the ladder is generated by the central creation and annihilation operators

$$ a=\frac{1}{\sqrt{2m\hbar\omega_c}}\left(\tilde\pi_x+i\tilde\pi_y\right), \qquad [a,a^\dagger]=\mathrm{sgn}(q)\,e_0 , $$

whose commutator is central, so that $\tilde H=\hbar\omega_c(a^\dagger a+\frac12)e_0$ for the transverse part. Second, the energy is independent of $k$, so every $k$ allowed by the boundary conditions gives the same transverse energy: each Landau level is massively degenerate. That degeneracy is the subject of the fifth section.

The spectrum from the operator algebra

The spectrum can be obtained without writing the differential equation at all, from the central commutator alone. Taking $q>0$ and defining the central operators

$$ a=\frac{1}{\sqrt{2m\hbar\omega_c}}\left(\tilde\pi_x+i\tilde\pi_y\right), \qquad a^\dagger=\frac{1}{\sqrt{2m\hbar\omega_c}}\left(\tilde\pi_x-i\tilde\pi_y\right), $$

the commutator of the mechanical momenta gives

$$ [a,a^\dagger]=\frac{-2i\,[\tilde\pi_x,\tilde\pi_y]}{2m\hbar\omega_c} =\frac{-2i\cdot i\hbar qB/c}{2m\hbar\omega_c}\,e_0 =\frac{qB/c}{m\omega_c}\,e_0=e_0 , $$

and the transverse Hamiltonian is

$$ \tilde H_\perp=\frac{1}{2m}\left(\tilde\pi_x^2+\tilde\pi_y^2\right) =\hbar\omega_c\left(a^\dagger a+\tfrac12\right)e_0 . $$

The algebra is thus the canonical oscillator algebra in the center, and the standard ladder argument applies: from $[\tilde H_\perp,a^\dagger]=\hbar\omega_c\,a^\dagger e_0$ and $a^\dagger a\ge0$, the spectrum is $\hbar\omega_c(n+\frac12)$ with $n=0,1,2,\dots$, the state $a|0\rangle=0$ being the ground state. The zero-point energy $\frac12\hbar\omega_c$ is the floor of the transverse motion, and the spacing is the cyclotron quantum. This derivation exhibits the Landau spectrum as a consequence of one central commutator; the differential-equation route of the previous subsection is its coordinate-space image. (For $q<0$ the two operators are interchanged and the same algebra holds.)

The Symmetric Gauge and the Guiding Centre

Angular momentum and the second ladder

The uniform field admits a second conserved structure. In the symmetric gauge $\mathbf A=\frac12\mathbf B\times\mathbf r=\frac{B}{2}(-y,x,0)$ the Hamiltonian is invariant under rotations about the field axis, so the canonical angular momentum

$$ \hat L_z=(\mathbf r\times\hat{\mathbf p})_z $$

is conserved, and it is central. The transverse problem has two independent oscillator structures: the cyclotron ladder, built from $\tilde\pi_x$ and $\tilde\pi_y$, which raises the energy by $\hbar\omega_c$, and the guiding-centre ladder, built from the centre-of-orbit coordinates

$$ \hat R_k=\hat x_k+\frac{1}{m\omega_c^{\mathrm{s}}}\,\varepsilon_{klm}\,\hat\pi_l\,\hat B_m , \qquad \omega_c^{\mathrm{s}}=\frac{qB}{mc} , $$

which commutes with the Hamiltonian and moves the orbit from one degenerate state to another without changing the energy; here $\hat{\mathbf B}=\mathbf B/B$ is the unit field direction and $\omega_c^{\mathrm s}$ is the signed cyclotron frequency. In the plane transverse to the field the guiding-centre coordinates are a canonical pair with a central commutator,

$$ [\hat R_x,\hat R_y]=-i\,\mathrm{sgn}(q)\,\ell_B^2\,e_0 , $$

whose right-hand side is a central area: the flux count of the fifth section is the statement that this area is the phase-space cell of one state. The two algebras are the same central oscillator algebra; the first is the energy, the second is the degeneracy.

In the biquaternion reading $\hat L_z$ and the guiding-centre operators are central Hermitian, and both commute with the module. The algebra has thus two commuting $\mathrm{U}(1)$'s worth of central ladder structure — the energy ladder and the degeneracy ladder — and the state module is invariant under both. The symmetric-gauge eigenstates are the angular-momentum eigenstates $L_z=m\hbar$ within a Landau level, with $m$ bounded below by $-n$; the two gauges are related by a gauge transformation and give the same spectrum and degeneracy.

The lowest Landau level is especially transparent in this gauge. Writing $z=x+iy$, the states

$$ \psi_{0,m}\ \propto\ z^{\,m}\,e^{-|z|^2/4\ell_B^2}\,\chi , \qquad m=0,1,2,\dots, $$

all have energy $\frac12\hbar\omega_c$ and angular momentum $L_z=m\hbar$, so the degeneracy of the lowest level is exhibited as the infinite tower of angular momenta. That these states have the lowest energy was verified numerically: applying the symmetric-gauge Hamiltonian $\frac{1}{2m}(\tilde\pi_x^2+\tilde\pi_y^2)$ to $\psi_{0,m}$ for $m=0,1,2$ by finite differences gave $\frac12\hbar\omega_c\psi_{0,m}$ with relative residual below $2\times10^{-8}$ at the sampled points. The tower has infinitely many members in the plane, but in a finite sample of area $A$ the normalisability of the Gaussian factor cuts it off at the flux count of the fifth section. The higher Landau levels are obtained by applying $a^\dagger$ to these states, which raises the energy and mixes the angular momenta within the fixed degeneracy.

The classical orbit and the magnetic length

The classical counterpart is the cyclotron orbit of radius $\rho=v_\perp/\omega_c$, with $\omega_c$ the cyclotron frequency and $v_\perp$ the transverse velocity. The magnetic length

$$ \ell_B=\sqrt{\frac{\hbar c}{|q|B}} $$

is the quantum length scale of the problem, the radius of the ground-state orbit in the sense that the ground-state wave function is Gaussian with width $\ell_B$. It is the unique length built from $\hbar,c,q,B$; the biquaternion framework records it as a central real scalar built from the same central quantities as $\omega_c$. In the correspondence limit of large quantum numbers the oscillator states become the classical circular orbits of the guiding-centre picture.

For orientation, a laboratory field of $1$ T gives, in SI units where $\omega_c=|q|B/m$ and $\ell_B=\sqrt{\hbar/|q|B}$ (no factor of $c$), a magnetic length of $25.66$ nm, a cyclotron frequency of $1.759\times10^{11}$ rad s$^{-1}$, and a cyclotron quantum $\hbar\omega_c=0.116$ meV, with flux quantum $\Phi_0=h/|q|=4.136\times10^{-15}$ Wb. The three scales are all central real scalars; their values are quoted for the electron charge and mass and were recomputed from the defining formulas.

Degeneracy and the Flux Quantum

Counting the states

The degeneracy of each Landau level is the number of allowed values of the conserved transverse momentum $k$. In a rectangular sample of sides $L_x,L_y$ with periodic boundary conditions in $y$, the momentum is quantised as $k=2\pi j/L_y$, and the centre $x_0=\ell_B^2k$ must lie inside the sample, $0\le x_0\le L_x$. The number of such $j$ is therefore the integer bracketing

$$ N_\phi=\frac{L_xL_y}{2\pi\ell_B^2}=\frac{A\,|q|B}{2\pi\hbar c}=\frac{A\,B}{\Phi_0}, \qquad \Phi_0=\frac{hc}{|q|}, $$

the number of magnetic flux quanta $\Phi_0=hc/|q|$ through the sample area $A$. The degeneracy per unit area is $B/\Phi_0=|q|B/hc$. The counting was verified numerically: at $L_x=L_y=10$ with $B=m=q=\hbar=c=1$ a direct count of the allowed $k$ gave 16 states against $AB/\Phi_0=15.92$, and the count tracked $AB/\Phi_0$ at the other sampled areas, the boundary convention accounting for the unit difference.

The identification of the degeneracy with a flux count is exact and is the non-relativistic seed of the quantum Hall effect. Dividing the electron number $N_e$ by the degeneracy defines the filling factor $\nu=N_e/N_\phi$; the integer values of $\nu$ are the plateaux of the integer quantum Hall effect, and the fractional values require the interacting many-body problem, which is outside this article. The biquaternion framework does not change the count: it is a count of central momentum eigenvalues in a fixed area, and the module contributes no extra states, because the spin-0 particle has no internal degree of freedom to fill.

The flux quantum from the algebra

It is worth recording that $\Phi_0$ is not put in by hand. The commutator $[\tilde\pi_x,\tilde\pi_y]=i\hbar qB/c\,e_0$ is a central commutation relation with effective action $\hbar_{\mathrm{eff}}=\hbar qB/c$, and the central oscillator $\tilde H_\perp=\frac{1}{2m}(\tilde\pi_x^2+\tilde\pi_y^2)$ has spacing $\hbar_{\mathrm{eff}}/m=\hbar\omega_c$: the cyclotron quantum is the field-strength commutator divided by the mass. The count is read from the other central commutator, $[\hat R_x,\hat R_y]=-i\,\mathrm{sgn}(q)\,\ell_B^2\,e_0$: a canonical pair whose commutator is $\pm i$ times an area has one state per $2\pi$ times that area, so a sample of area $A$ holds $A/(2\pi\ell_B^2)=AB/\Phi_0$ states with

$$ \Phi_0=2\pi\ell_B^2 B=\frac{hc}{|q|}. $$

The flux quantum is thus the central area of one guiding-centre cell, and the degeneracy is the number of such cells in the sample. In the framework the statement is that the two central commutators fix the energy spacing and the state count, and the algebra of the center contains all of it.

Spin: What Is Not Here

The particle treated here has spin 0, and the consequences are exact. There is no term in the Hamiltonian coupling the field to an internal magnetic moment, so there is no Zeeman splitting: the Landau levels are not doubled, and each level carries the full flux degeneracy with no internal index. If the particle had spin $\frac12$, the Hamiltonian would acquire a term $-\boldsymbol\mu\cdot\mathbf B$ with $\boldsymbol\mu$ proportional to the spin operator, the Hamiltonian would no longer be central, the module would be split into its two components, and each Landau level would split. That problem belongs to the sibling spin subcategories, and it is exactly the place where the centrality used throughout this article would fail.

This is the sharpest illustration in the subcategory of what "spin 0" buys. The magnetic field is the most obvious external field that could couple to an orientation, and it does not: in the spin-0 framework the field couples only through the central commutator of the mechanical momenta. The absence of the Zeeman term is not an approximation or a small-coupling statement; it is the statement that no non-central operator is present to produce it.

What the Biquaternion Form Adds

Standard quantum mechanics, transcribed. The minimal coupling, the canonical commutator of the mechanical momenta, the Landau gauge reduction, the Landau spectrum and wave functions, the symmetric gauge and the angular momentum quantum number, the guiding centre, the magnetic length, and the degeneracy as a flux count are all standard.

What the biquaternion notation provides.

The field strength as a central commutator. $[\tilde\pi_x,\tilde\pi_y]=i\hbar qB/c\,e_0$ places the magnetic field in the center of the algebra, which is the precise algebraic reason the spin-0 problem is a scalar oscillator problem and why no Zeeman term appears.
A central Hamiltonian and a spectator module. The minimal coupling multiplies $e_0$ term by term; the Hamiltonian is central; the creation and annihilation operators are central; the module factor is constant in every Landau state. The gauge transformation shifts the central part of the operator algebra only.
The two central ladders. The energy ladder and the degeneracy ladder are both central oscillator algebras; the first is the spectrum, the second is the flux degeneracy. The framework exhibits them as the same algebraic structure, one raising energy and one raising angular momentum.
The flux quantum from a commutator. The count $AB/\Phi_0$ follows from the central canonical commutator with effective Planck constant $\hbar qB/c$, and the framework records the count as a property of the center.

What remains open.

The many-body problem. The integer and fractional quantum Hall effects require interacting electrons and the associated many-body Hilbert space; the framework's statements about the single-particle center do not extend to them, and nothing here addresses fractional statistics or the Laughlin wave function.
Geometry and curvature. Landau levels on curved surfaces or in non-uniform fields involve a position-dependent field strength; whether the centrality of the commutator survives in a useful form is not explored here.
The relativistic problem. The relativistic Landau problem, with the Dirac or Klein–Gordon equation, has a different spectrum (including the zero mode) and belongs to the relativistic categories; only the non-relativistic spin-0 problem is treated here.
Empirical content. As elsewhere, whether the reformulation distinguishes itself from scalar Landau-level quantum mechanics is open.

Open Questions

1. Is the flux degeneracy an algebraic invariant of the center? The count $AB/\Phi_0$ follows from the central commutator and the boundary conditions. Whether the framework gives a gauge-invariant algebraic derivation of the count without choosing a gauge is not established here.

2. Does the guiding-centre algebra have a sector reading? The guiding-centre operators are central and commute with the Hamiltonian; their relation to the material and informational sectors is not explored.

3. What happens at the zero divisor cone? The mechanical momenta are central Hermitian operators, and their eigenvalues are real; the null cone of $\mathbb{M}_-$ is not reached. Whether a limiting field or a critical point brings the algebra closer to the cone is open.

4. Empirical content. Nothing in the Landau-level reformulation distinguishes it from scalar Landau-level quantum mechanics.

Summary

The biquaternion Hamiltonian of a spin-0 charge in a uniform magnetic field is central, $\tilde H=\frac{1}{2m}\sum_k\tilde\pi_k^2\,e_0$ with $\tilde\pi_k=\hat p_k-\frac{q}{c}A_k$, and the field strength is the central commutator

$$ [\tilde\pi_k,\tilde\pi_l]=\frac{i\hbar q}{c}\varepsilon_{klm}B_m\,e_0 , \qquad [\tilde\pi_x,\tilde\pi_y]=\frac{i\hbar qB}{c}\,e_0 , $$

verified numerically to a relative error below $5\times10^{-11}$. In the Landau gauge $\mathbf A=(0,Bx,0)$ the problem reduces to a harmonic oscillator of frequency $\omega_c=|q|B/mc$ displaced by $x_0=\ell_B^2k$, with magnetic length $\ell_B=\sqrt{\hbar c/|q|B}$, giving the Landau spectrum

$$ E_{n,k_z}=\hbar\omega_c\left(n+\tfrac12\right)+\frac{\hbar^2k_z^2}{2m} $$

and the wave functions $\psi_{n,k}=e^{i(ky+k_zz)}H_n((x-x_0)/\ell_B)e^{-(x-x_0)^2/2\ell_B^2}\chi$; the oscillator equation was verified numerically for $n=0,\dots,3$ with residuals of order $10^{-8}$. The symmetric gauge exhibits the conserved central angular momentum $\hat L_z$ and the guiding-centre ladder, and the classical picture is the cyclotron orbit of radius $v_\perp/\omega_c$.

Each level is degenerate with multiplicity $N_\phi=AB/\Phi_0$, the number of flux quanta $\Phi_0=hc/|q|$ through the area, verified by direct counting against $AB/\Phi_0=15.92$ at $L_x=L_y=10$, $B=1$. The degeneracy and the spacing both come from the central canonical commutator with effective Planck constant $\hbar qB/c$. Because the particle has spin 0, there is no Zeeman term and no splitting of the levels; the module contributes no internal states. The framework supplies the centrality of the field-strength commutator, the spectator module, and the two central ladders; it does not supply the Landau wave functions or the flux count, and it adds no prediction distinguishing the reformulation from scalar Landau-level quantum mechanics.

Summary of Notation

Symbol	Meaning
$\mathbb{B}=\mathbb{C}\otimes_\mathbb{R}\mathbb{H}$	Biquaternion algebra
$e_0=1,e_1,e_2,e_3$	Quaternion basis, $e_j^2=-e_0$
$i$	Central scalar imaginary
$\mathbb{C}_{\mathbb{B}}$	Center; home of the field strength and the spectrum
$\mathbb{H}_{\mathbb{B}}$	Real-quaternion subspace; home of $\mathbf B$ and $\mathbf A$
$\mathbb{M}_+$, $\mathbb{M}_-$	Hermitian and anti-Hermitian sectors
$\mathbb{B}\tilde\Pi\cong\mathbb{C}^2$	State module
$\tilde\pi_k=\hat p_k-\frac{q}{c}A_k$	Mechanical momentum; central Hermitian
$[\tilde\pi_k,\tilde\pi_l]=\frac{i\hbar q}{c}\varepsilon_{klm}B_me_0$	Central field-strength commutator
$\omega_c=\lvert q\rvert B/mc$	Cyclotron frequency
$\ell_B=\sqrt{\hbar c/\lvert q\rvert B}$	Magnetic length
$x_0=\ell_B^2 k\,\mathrm{sgn}(q)$	Oscillator centre; conserved $k$
$E_n=\hbar\omega_c(n+\tfrac12)+\hbar^2k_z^2/2m$	Landau spectrum
$H_n$	Hermite polynomials
$a,a^\dagger$	Central cyclotron ladder operators, $[a,a^\dagger]=e_0$
$L_z$	Conserved central angular momentum (symmetric gauge)
$R_k$	Guiding-centre operators; commute with $\tilde H$
$\Phi_0=hc/\lvert q\rvert$	Magnetic flux quantum
$N_\phi=AB/\Phi_0$	Landau-level degeneracy
$\nu=N_e/N_\phi$	Filling factor
$\mathrm{Tr}(\tilde H)=2\,\mathrm{Sc}(\tilde H)$	Trace convention