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The Classical Hall Effect Pauli Paramagnetism Landau Quantization The Aharonov-Bohm Phase Cyclotron Resonance The Quantu

The Classical Hall Effect Pauli Paramagnetism Landau Quantization The Aharonov-Bohm Phase Cyclotron Resonance The Quantum Hall Effect. Section 1.7: Magnetic Properties of a Free Electron Fermi Gas. A Free Electron in a Magnetic Field. neglect spin. : vector potential.

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The Classical Hall Effect Pauli Paramagnetism Landau Quantization The Aharonov-Bohm Phase Cyclotron Resonance The Quantu

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  1. The Classical Hall Effect Pauli Paramagnetism Landau Quantization The Aharonov-Bohm Phase Cyclotron Resonance The Quantum Hall Effect Section 1.7: Magnetic Properties of a Free Electron Fermi Gas

  2. A Free Electron in a Magnetic Field neglect spin : vector potential Different ways to solve this problem: • Classical mechanics • 1a) Newtonian formalism • 1b) Hamiltonian formalism • Quantum mechanics • 2a) Wavefunction formalism • 2b) Operator formalism

  3. A Free Electron in a Magnetic Field Classical mechanics (newtonian formalism) ( in this section) Cyclotron (spiral) motion

  4. A Free Electron in a Magnetic Field Classical mechanics (hamiltonian formalism) canonical momemtum kinematic momemtum Fields & potentials: Gauge transformation  nothing changes

  5. A Free Electron in a Magnetic Field Classical mechanics (hamiltonian formalism) Landau gauge  This is independent of y & z cyclic coordinates  py and pz are constants of motion EOM: Same results as newtonian Relative coordinates

  6. Simple Harmonic Oscillator : parabolic

  7. A Free Electron in a Magnetic Field Q.M.  Landau levels Landau-Peierls substitution : SHO!!  Landau gauge 

  8. Density of States in a Magnetic Field Landau ladder 4 3 2 : cyclotron energy … 1 n = 0 1 2 3 4 5 n = 0 … DOS DOS Energy Energy 2D system + B 0D system 3D system + B 1D system

  9. Operator Algebra for Landau Levels : Hamiltonian operator Raising & lowering operators for LL

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