4.3.2 Landau Quantization

4.3.2 Landau Quantization#

Prompts

  • How does Bohr-Sommerfeld quantization apply to cyclotron orbits? Why does it yield \(E_n = (n + 1/2) \hbar \omega_c\)?

  • In the Landau gauge with \(\hat{\boldsymbol{A}} = (0, B\hat{x}, 0)\), why is \(\hat{p}_y\) conserved? How does this reduce the problem to a harmonic oscillator?

  • What are guiding center coordinates \(\hat{X}\) and \(\hat{Y}\), and why is their commutator \([\hat{X}, \hat{Y}] = \mathrm{i}\ell_B^2\) profound for quantum mechanics?

  • What do wavefunctions in the lowest Landau level look like in complex coordinates? Why are they holomorphic polynomials times a Gaussian?

  • How does the degeneracy of each Landau level, \(N_\phi = BA/\Phi_0\), scale with magnetic field strength?

Lecture Notes#

Overview#

Landau quantization describes the remarkable fate of a charged particle in a uniform magnetic field: continuous kinetic energy becomes quantized into discrete Landau levels, each with enormous degeneracy. The classical cyclotron orbit becomes a quantum harmonic oscillator. This is the foundation for understanding the quantum Hall effect and topological quantum phenomena.

Semiclassical Quantization of Cyclotron Orbits#

Before solving the full quantum problem, we can gain insight from Bohr-Sommerfeld quantization (§3.3). A classical charged particle in a magnetic field \(\boldsymbol{B} = B\hat{z}\) executes circular orbits at the cyclotron frequency:

\[ \omega_c = \frac{eB}{m} \]

The action integral around a closed orbit is:

\[ \oint \boldsymbol{p} \cdot \mathrm{d}\boldsymbol{r} = 2\pi m \omega_c r^2 = 2\pi \hbar(n + \tfrac{1}{2}) \]

where \(r\) is the orbit radius. This gives quantized energies:

\[ E_n = \frac{1}{2}m\omega_c^2 r^2 = (n + \tfrac{1}{2})\hbar\omega_c \]

Bohr-Sommerfeld Quantization of Cyclotron Motion

\[ E_n = \left(n + \tfrac{1}{2}\right)\hbar\omega_c \quad (n = 0, 1, 2, \ldots) \]

This semiclassical result matches the exact quantum answer. The zero-point energy \(E_0 = \frac{1}{2}\hbar\omega_c\) is purely quantum mechanical.

Quantum Hamiltonian in the Landau Gauge#

The quantum Hamiltonian for a charged particle in a uniform magnetic field is:

\[ \hat{H} = \frac{1}{2m}(\hat{\boldsymbol{p}} - e\hat{\boldsymbol{A}})^2 \]

(In SI units; in Gaussian units use \(q/c\) instead of \(e\).)

For \(\boldsymbol{B} = B\hat{z}\), we choose the Landau gauge:

\[ \hat{\boldsymbol{A}} = (0, B\hat{x}, 0) \]

This gauge breaks spatial translational symmetry in \(x\) but reveals a crucial simplification: \(\hat{p}_y\) commutes with the Hamiltonian and is a conserved quantum number.

The kinetic momentum components become:

\[ \hat{\pi}_x = \hat{p}_x, \quad \hat{\pi}_y = \hat{p}_y - eB\hat{x} \]

The Hamiltonian is:

\[ \hat{H} = \frac{\hat{p}_x^2}{2m} + \frac{(\hat{p}_y - eB\hat{x})^2}{2m} \]

Using \([\hat{p}_x, \hat{x}] = -\mathrm{i}\hbar\), we can shift the origin: let \(\hat{X} = \hat{x} + \hat{p}_y/(eB)\). Then:

\[ \hat{H} = \frac{\hat{p}_x^2}{2m} + \frac{1}{2}m\omega_c^2 \hat{X}^2 \]

This is a harmonic oscillator in the \(x\)-direction with frequency \(\omega_c\). The variable \(y\) (or equivalently \(\hat{p}_y\)) is a spectator: motion in \(y\) is free.

Landau Gauge Reduction

In the Landau gauge, the Hamiltonian becomes a shifted harmonic oscillator:

\[ \hat{H} = \frac{\hat{p}_x^2}{2m} + \frac{1}{2}m\omega_c^2\left(\hat{x} + \frac{\hat{p}_y}{eB}\right)^2 \]

The conserved momentum \(\hat{p}_y\) labels degenerate states within each Landau level. The quantum number \(n\) from the harmonic oscillator labels the Landau level.

Landau Level Energies#

The energy eigenvalues are determined by the harmonic oscillator quantum number \(n\):

Landau Level Energies and Degeneracy

\[ E_n = \left(n + \tfrac{1}{2}\right)\hbar\omega_c \quad (n = 0, 1, 2, \ldots) \]

All states with the same \(n\) but different \(\hat{p}_y\) belong to the \(n\)-th Landau level. There is no energy splitting within a level (to leading order).

Guiding Center Coordinates#

A key insight is that quantum mechanics forbids simultaneously specifying both the \(x\) and \(y\) positions of a particle in a strong magnetic field. We can define the guiding center coordinates:

\[ \hat{X} = \hat{x} + \frac{\hat{p}_y}{eB}, \quad \hat{Y} = \hat{y} - \frac{\hat{p}_x}{eB} \]

These represent the classical center of the cyclotron orbit. Their commutator is:

\[ [\hat{X}, \hat{Y}] = -\mathrm{i}\frac{\hbar}{eB} = -\mathrm{i}\ell_B^2 \]

Guiding Center Noncommutativity

\[ [\hat{X}, \hat{Y}] = -\mathrm{i}\ell_B^2 \quad \text{where} \quad \ell_B = \sqrt{\frac{\hbar}{eB}} \]

This noncommutativity means that guiding center positions cannot be simultaneously localized to better than the magnetic length \(\ell_B\). A quantum state occupies a minimum area of \(\sim \ell_B^2\) in \((X, Y)\) space.

Ladder Operators and Two Towers of States#

The Landau level structure involves two independent sets of ladder operators:

  1. Landau level ladder operators \(\hat{a}, \hat{a}^\dagger\): These raise and lower the Landau level index \(n\).

  2. Guiding center ladder operators \(\hat{b}, \hat{b}^\dagger\): These label distinct states within a Landau level (different guiding centers).

Two Sets of Ladder Operators

Define:

\[ \hat{a} = \sqrt{\frac{m\omega_c}{2\hbar}}\hat{X} + \mathrm{i}\sqrt{\frac{1}{2\hbar m\omega_c}}\hat{p}_x \]
\[ \hat{b} = \sqrt{\frac{m\omega_c}{2\hbar}}\hat{Y} + \mathrm{i}\sqrt{\frac{1}{2\hbar m\omega_c}}\hat{p}_y \]

Both satisfy \([\hat{a}, \hat{a}^\dagger] = 1\) and \([\hat{b}, \hat{b}^\dagger] = 1\), but they commute with each other:

\[ [\hat{a}, \hat{b}] = [\hat{a}, \hat{b}^\dagger] = 0 \]

The Hamiltonian is:

\[ \hat{H} = \hbar\omega_c\left(\hat{a}^\dagger\hat{a} + \tfrac{1}{2}\right) \]

States are labeled \(\vert n, m \rangle\) where \(\hat{a}^\dagger\hat{a}\vert n, m \rangle = n\vert n, m \rangle\) (Landau level) and \(\hat{b}^\dagger\hat{b}\vert n, m \rangle = m\vert n, m \rangle\) (degeneracy index within the level).

Interpretation

The appearance of two commuting sets of oscillators is profound:

  • Quantum mechanics generates an internal structure (the \(\hat{b}\) oscillator) from the noncommutativity of guiding center positions.

  • States cannot be labeled by classical position \((X, Y)\); instead we need quantum numbers \(n\) and \(m\).

  • This structure underlies the topological properties of the quantum Hall effect.

Complex Coordinates and Lowest Landau Level Wavefunctions#

In the lowest Landau level (\(n = 0\)), wavefunctions have a special structure in complex coordinates \(z = x + \mathrm{i}y\).

The lowest Landau level condition \(\hat{a}\vert\psi\rangle = 0\) implies that the wavefunction must be holomorphic (analytic in \(z\)). The general form is:

\[ \psi_m(z, \bar{z}) = P_m(z) \exp\left(-\frac{\vert z\vert^2}{4\ell_B^2}\right) \]

where \(P_m(z)\) is a holomorphic polynomial of degree \(m\) in \(z\):

\[ P_m(z) = c_m z^m \]

The Gaussian factor ensures normalizability; the polynomial factor labels the \(m\) different degenerate states.

Lowest Landau Level Wavefunctions

\[ \psi_m(z, \bar{z}) = \frac{c_m}{\sqrt{\pi^{1/2}2^m m!\ell_B^{2m}}} z^m \exp\left(-\frac{\vert z\vert^2}{4\ell_B^2}\right) \]

Each monomial \(z^m\) corresponds to a distinct state within the \(n=0\) level. The Gaussian width is set by \(\ell_B\).

Why Holomorphic?

The lowest Landau level condition is \(\hat{a}\vert\psi\rangle = 0\), which expands to:

\[ \left[\sqrt{\frac{m\omega_c}{2\hbar}}z + \sqrt{\frac{1}{2\hbar m\omega_c}}\frac{\partial}{\partial z}\right]\psi = 0 \]

This is equivalent to \(\frac{\partial\psi}{\partial\bar{z}} = 0\) (after rescaling), which means \(\psi\) depends only on \(z\), not \(\bar{z}\). Thus \(\psi\) is holomorphic.

Degeneracy of Landau Levels#

The key fact is that many different guiding center positions (labeled by \(m = 0, 1, 2, \ldots\)) fit within the same Landau level. The number of distinct states in a Landau level is determined by the total magnetic flux through the sample.

For a sample of area \(A\) in a uniform field \(B\):

\[ \Phi_{\text{total}} = BA \]

Each state occupies an area \(\sim \ell_B^2\) in guiding center space. The degeneracy is:

\[ N_\phi = \frac{\Phi_{\text{total}}}{\Phi_0} = \frac{BA}{\Phi_0} \]

where \(\Phi_0 = h/e = 4.14 \times 10^{-15}\) T·m\(^2\) is the flux quantum.

Equivalently:

\[ N_\phi = \frac{A}{\pi\ell_B^2} \]

Degeneracy Formula

\[ N_\phi = \frac{BA}{\Phi_0} = \frac{A}{\pi\ell_B^2} \]

For a macroscopic sample (\(A \sim 1 \text{ mm}^2\)) at \(B \sim 1\) T, the degeneracy is \(N_\phi \sim 10^{10}\)—a macroscopic number. All these states have (nearly) identical energy.

Discussion

In the quantum Hall regime, the degeneracy within a Landau level vastly exceeds the number of electrons:

(a) Suppose you have a 2D electron system with area \(A = 1 \text{ mm}^2\) at \(B = 5\) T, containing \(N_e = 10^{11}\) electrons. What is the filling factor \(\nu = N_e/N_\phi\)? How many Landau levels are partially occupied?

(b) At very strong fields (say \(B = 30\) T), why does the system become “incompressible”—unable to absorb additional electrons at a fixed Fermi energy?

(c) If disorder lifts the degeneracy by \(\Delta E \sim \hbar\omega_c/1000\), does this destroy the quantum Hall effect?

Magnetic Length: The Fundamental Scale#

The magnetic length emerges naturally from the commutation relations:

\[ \ell_B = \sqrt{\frac{\hbar}{eB}} \]

It is the only length scale in the problem and determines the wavefunction size, the separation between lattice sites in guiding center space, and the energy scale of orbital physics.

Magnetic Length (Recap)

\[ \ell_B = \sqrt{\frac{\hbar}{eB}} \approx 25.66 \text{ nm} \times (1 \text{ T}/B)^{1/2} \]

For \(B = 1\) T: \(\ell_B \approx 25.66\) nm
For \(B = 10\) T: \(\ell_B \approx 8.1\) nm

Summary#

  • Bohr-Sommerfeld quantization of cyclotron orbits yields \(E_n = (n + 1/2)\hbar\omega_c\), which matches the exact quantum result.

  • Landau gauge \(\hat{\boldsymbol{A}} = (0, B\hat{x}, 0)\) reduces the problem to a shifted harmonic oscillator with \(\hat{p}_y\) conserved.

  • Landau level energies are \(E_n = (n + 1/2)\hbar\omega_c\), independent of the guiding center position (to leading order).

  • Guiding center coordinates satisfy \([\hat{X}, \hat{Y}] = -\mathrm{i}\ell_B^2\): positions are fundamentally fuzzy at the quantum scale.

  • Two ladder operators: \(\hat{a}, \hat{a}^\dagger\) label Landau levels; \(\hat{b}, \hat{b}^\dagger\) label degenerate states within a level.

  • Lowest Landau level wavefunctions are holomorphic polynomials \(z^m\) times a Gaussian, reflecting the quantum constraint.

  • Degeneracy \(N_\phi = BA/\Phi_0 \sim 10^{10}\) for macroscopic samples: enormous and crucial for quantum Hall physics.

  • Magnetic length \(\ell_B = \sqrt{\hbar/(eB)}\) is the universal length scale characterizing quantum orbital motion.

Homework#

1. A charged particle in a uniform magnetic field \(\boldsymbol{B} = B\hat{z}\) executes circular orbits at the classical cyclotron frequency \(\omega_c = eB/m\). Use Bohr-Sommerfeld quantization to show that the quantized energies are:

\[E_n = \left(n + \frac{1}{2}\right)\hbar\omega_c\]

(Hint: The action integral is \(\oint \boldsymbol{p} \cdot \mathrm{d}\boldsymbol{r} = 2\pi m\omega_c r^2\), where \(r\) is the orbit radius. Quantize this using \(\oint \boldsymbol{p} \cdot \mathrm{d}\boldsymbol{r} = 2\pi\hbar(n + 1/2)\).)

2. In the Landau gauge \(\hat{\boldsymbol{A}} = (0, B\hat{x}, 0)\), the Hamiltonian is:

\[\hat{H} = \frac{\hat{p}_x^2}{2m} + \frac{(\hat{p}_y - eB\hat{x})^2}{2m}\]

(a) Show that \([\hat{H}, \hat{p}_y] = 0\). Why does this mean \(\hat{p}_y\) is a conserved quantum number?

(b) Use a coordinate shift \(\tilde{x} = \hat{x} + \hat{p}_y/(eB)\) to show that the Hamiltonian reduces to:

\[\hat{H} = \frac{\hat{p}_x^2}{2m} + \frac{1}{2}m\omega_c^2\tilde{x}^2\]

Interpret this as a harmonic oscillator.

3. The guiding center coordinates are:

\[\hat{X} = \hat{x} + \frac{\hat{p}_y}{eB}, \quad \hat{Y} = \hat{y} - \frac{\hat{p}_x}{eB}\]

Calculate the commutator \([\hat{X}, \hat{Y}]\) using \([x, \hat{p}_x] = \mathrm{i}\hbar\), \([y, \hat{p}_y] = \mathrm{i}\hbar\), and show that:

\[[\hat{X}, \hat{Y}] = -\mathrm{i}\ell_B^2 \quad \text{where} \quad \ell_B = \sqrt{\frac{\hbar}{eB}}\]

What is the physical meaning of this nonzero commutator?

4. Define the ladder operators (in the Landau gauge):

\[\hat{a} = \sqrt{\frac{m\omega_c}{2\hbar}}\hat{X} + \mathrm{i}\sqrt{\frac{1}{2\hbar m\omega_c}}\hat{p}_x\]
\[\hat{b} = \sqrt{\frac{m\omega_c}{2\hbar}}\hat{Y} + \mathrm{i}\sqrt{\frac{1}{2\hbar m\omega_c}}\hat{p}_y\]

(a) Verify that \([\hat{a}, \hat{a}^\dagger] = 1\) and \([\hat{b}, \hat{b}^\dagger] = 1\).

(b) Show that \([\hat{a}, \hat{b}] = 0\) (they commute). Why is this significant?

(c) Express the Hamiltonian in terms of \(\hat{a}, \hat{a}^\dagger\) and show that \(\hat{H} = \hbar\omega_c(\hat{a}^\dagger\hat{a} + 1/2)\).

5. The lowest Landau level (\(n=0\)) wavefunctions in complex coordinates \(z = x + \mathrm{i}y\) have the form:

\[\psi_m(z, \bar{z}) = C_m z^m \exp\left(-\frac{\vert z\vert^2}{4\ell_B^2}\right)\]

where \(m = 0, 1, 2, \ldots\) labels distinct degenerate states.

(a) Explain why the wavefunction must be holomorphic (depend only on \(z\), not \(\bar{z}\)). (Hint: What is the condition \(\hat{a}|\psi_0\rangle = 0\)?)

(b) Normalize the \(m=0\) state to find \(C_0\). (Note: \(\int_{\mathbb{C}} |\psi_0|^2 \mathrm{d}^2z = 1\).)

(c) How many independent states exist in the lowest Landau level for a sample of area \(A\) in magnetic field \(B\)?

6. The degeneracy of each Landau level is:

\[N_\phi = \frac{BA}{\Phi_0}\]

where \(\Phi_0 = h/e \approx 4.14 \times 10^{-15}\) T·m\(^2\) is the flux quantum.

(a) For a \(2 \times 2\) mm\(^2\) sample at \(B = 1\) T, calculate \(N_\phi\).

(b) If you increase the field to \(B = 10\) T (at fixed sample size), how much does \(N_\phi\) increase?

(c) Explain why this massive degeneracy is crucial for the quantum Hall effect.

7. The magnetic length \(\ell_B = \sqrt{\hbar/(eB)}\) sets the size of Landau level wavefunctions. Calculate \(\ell_B\) numerically for:

(a) \(B = 1\) T (b) \(B = 10\) T

(c) In a typical quantum Hall experiment, what happens to \(\ell_B\) and the density of states as you increase \(B\)?

8. In the symmetric gauge \(\hat{\boldsymbol{A}} = \frac{B}{2}(-\hat{y}, \hat{x}, 0)\), the kinetic momentum components are:

\[\hat{\pi}_x = \hat{p}_x + \frac{eB\hat{y}}{2}, \quad \hat{\pi}_y = \hat{p}_y - \frac{eB\hat{x}}{2}\]

Show that the Hamiltonian can be written as:

\[\hat{H} = \frac{1}{2m}(\hat{\pi}_x^2 + \hat{\pi}_y^2) = \hbar\omega_c\left(\hat{a}^\dagger\hat{a} + \frac{1}{2}\right)\]

where the ladder operators are constructed from \(\hat{\pi}_x\) and \(\hat{\pi}_y\). How does this compare to the Landau gauge result?

9. (Conceptual) In the quantum Hall regime, the filling factor \(\nu = N_e/N_\phi\) is the ratio of electron number to degeneracy. Explain why:

(a) At \(\nu = 1\) (one electron per Landau level state), the lowest Landau level is completely filled.

(b) The integer quantum Hall effect occurs when \(\nu\) is an integer.

(c) Fractional quantum Hall effect requires electron-electron interactions and occurs at non-integer \(\nu\).

10. Compare semiclassical (Bohr-Sommerfeld) and quantum treatments:

(a) In the semiclassical picture, what is the physical meaning of the orbital radius \(r\) in the Bohr-Sommerfeld condition?

(b) In the quantum picture, what prevents us from assigning a definite orbit radius?

(c) How does the zero-point energy \(E_0 = \frac{1}{2}\hbar\omega_c\) reveal the difference between classical and quantum mechanics?