A charged matter wave moves in a uniform magnetic field: a localized packet executes a cyclotron orbit, while radial Landau states remain stationary with quantized energies (n+1/2) hbar omega_c.
This page describes the phenomenon using the current QPE preset and numerical modules as its primary source.
Cyclotron motion and Landau quantization are two faces of the same magnetic Hamiltonian. A localized packet shows the orbit; states projected onto Landau levels show energy quantization.
One Hamiltonian, two faces. Step 44 reuses the gauge-covariant solver introduced for Aharonov-Bohm, now with uniform B throughout the accessible region. No scalar force is used to bend the packet.
Cyclotron orbit mode. Press Play. The packet starts at x=-R with the correct tangential mechanical momentum and follows an almost circular orbit. Analysis -> Landau plots <x>,<y>. In general omega_c=|qB|/m=|b| hbar/m.
Why a special initial phase? In symmetric gauge A=(-By/2,Bx/2), canonical and mechanical momentum differ. QPE prepares the phase so the requested physical radius obeys R=p_mech/(qB).
Landau eigenstate mode. Choose Landau eigenstate in Dynamics -> Landau, select n=0,1,2... and Apply/Reset. Density remains stationary; Analysis compares the numerical energy with E_n=(n+1/2) hbar omega_c and shows survival fidelity.
Try this. Reverse the sign of b: orbit direction reverses while energies depend on |B|. Increase |b|: l_B=1/sqrt(|b|) shrinks, level spacing grows and cyclotron motion speeds up.
History. Lev Landau described orbital quantization in a uniform magnetic field in 1930. Landau levels underpin phenomena from orbital diamagnetism to the quantum Hall effect.
Model boundary. Spinless non-relativistic 2D particle, prescribed uniform external field, no synchrotron radiation, spin-Zeeman coupling or electromagnetic back-reaction.
Landau levels entered quantum theory through diamagnetism. In two-dimensional electron systems they form the microscopic basis of the quantum Hall effect.
| Representative platform | GaAs-like 2DEG in a uniform mesoscopic magnetic field |
|---|---|
| Mapping basis | effective-mass mapping (m*=0.067 m_e) |
| 1 QPE length unit | 50 nm |
| 1 QPE time unit | 1.45 ps |
| Energy scale E0=ħ/T0 | 0.455 meV |
| Equivalent frequency E0/h | 0.11 THz |
| Note | With guided dimensionless b=0.45 and L0=50 nm, the magnetic-field unit gives B about 0.12 T. The GaAs-like effective mass sets a time unit near 1.45 ps. |
Open Model & Constraints to see exactly what QPE computes and what is left outside the model. For current live values use Help → Current Experiment → Current Parameters.