(a) Show that the Hamiltonian of a free electron in a...

(a) Show that the Hamiltonian of a free electron in a uniform timeindependent magnetic field $\mathscr{S}_{3}=9_{\xi} \hat{\mathbf{z}}$ is given by $H=H_{x y}+H_{u}$, with $$ H_{x y}=\frac{1}{2 m}\left(p_{x}^{2}+p_{y}^{2}\right)+\frac{1}{2 m} \omega_{L}^{2}\left(x^{2}+y^{2}\right) $$ and $$ H_{z}=\frac{1}{2 m} p_{z}^{2}+\omega_{L}\left(L_{z}+2 S_{z}\right) $$ where $\omega_{\mathrm{L}}=\left(\mu_{\mathrm{B}} / \hbar\right) \mathscr{B}_{z}=2 \pi v_{\mathrm{L}}$ is the Larmor angular frequency. (b) Using the fact that $H$ can be written as a square, $H=(\mathbf{p}+e \mathbf{A})^{2} / 2 m$, and that the Hamiltonian $H_{x y}$ of the harmonic motion in the $X Y$ plane is invariant under the reflection $x \rightarrow-x, y \rightarrow-y$, show that the energy eigenvalues are given by $$ E=\frac{h^{2} k^{2}}{2 m}+\hbar \omega_{L}\left(2 r+2 m_{s}+1\right) $$ where $-\infty<k<+\infty, r=0,1,2, \ldots$ and $m_{s}=\pm 1 / 2$. For given $k$ and $m_{s}$ the discrete energy levels labelled by the quantum number $r$ are called Landau levels. (c) In neutron stars magnetic fields of the order of $10^{8} \mathrm{~T}$ may occur. Find the energy separation between the adjacent Landau levels. What is the

(a) Show that the Hamiltonian of a free electron in a uniform timeindependent magnetic field $\mathscr{S}_{3}=9_{\xi} \hat{\mathbf{z}}$ is given by $H=H_{x y}+H_{u}$, with
$$
H_{x y}=\frac{1}{2 m}\left(p_{x}^{2}+p_{y}^{2}\right)+\frac{1}{2 m} \omega_{L}^{2}\left(x^{2}+y^{2}\right)
$$
and
$$
H_{z}=\frac{1}{2 m} p_{z}^{2}+\omega_{L}\left(L_{z}+2 S_{z}\right)
$$
where $\omega_{\mathrm{L}}=\left(\mu_{\mathrm{B}} / \hbar\right) \mathscr{B}_{z}=2 \pi v_{\mathrm{L}}$ is the Larmor angular frequency.
(b) Using the fact that $H$ can be written as a square, $H=(\mathbf{p}+e \mathbf{A})^{2} / 2 m$, and that the Hamiltonian $H_{x y}$ of the harmonic motion in the $X Y$ plane is invariant under the reflection $x \rightarrow-x, y \rightarrow-y$, show that the energy eigenvalues are given by
$$
E=\frac{h^{2} k^{2}}{2 m}+\hbar \omega_{L}\left(2 r+2 m_{s}+1\right)
$$
where $-\infty<k<+\infty, r=0,1,2, \ldots$ and $m_{s}=\pm 1 / 2$. For given $k$ and $m_{s}$ the discrete energy levels labelled by the quantum number $r$ are called Landau levels.
(c) In neutron stars magnetic fields of the order of $10^{8} \mathrm{~T}$ may occur. Find the energy separation between the adjacent Landau levels. What is the

Key Concepts

Hamiltonian in a Magnetic Field

This concept involves the formulation of the total energy of a charged particle, like an electron, in the presence of a magnetic field. The Hamiltonian is modified to account for the magnetic interaction by including terms that represent the coupling between the electron’s motion and the external field, leading to altered kinetic and potential energy contributions.

Minimal Coupling

Minimal coupling is the fundamental procedure used in quantum mechanics to include electromagnetic interactions. It involves replacing the canonical momentum p with (p + eA), where A is the vector potential corresponding to the magnetic field. This substitution introduces extra terms in the Hamiltonian that capture how charged particles respond to electromagnetic fields.

Landau Quantization

Landau quantization refers to the phenomenon where the energy levels of an electron moving in a uniform magnetic field become discrete in the plane perpendicular to the field. These quantized levels, known as Landau levels, arise from the formation of an effective harmonic oscillator potential due to the magnetic field’s influence on the electron's motion.

Larmor Frequency

Larmor frequency is the rate at which the magnetic moment of a charged particle precesses in a magnetic field. It sets the energy scale for the cyclotron motion of the electron and quantifies the splitting of energy levels associated with the orbital and spin magnetic moments in the presence of the magnetic field.

Quantum Harmonic Oscillator

The quantum harmonic oscillator model appears in the analysis of the electron’s motion in a magnetic field, particularly in the plane perpendicular to the field. The effective quadratic potential that arises due to the magnetic field leads to quantized energy levels similar to those of a standard harmonic oscillator, making it a key model for understanding Landau levels.

Zeeman Effect

The Zeeman effect describes the splitting of energy levels when an external magnetic field interacts with the magnetic moments of electrons arising from both their orbital angular momentum and intrinsic spin. This interaction results in additional energy terms in the Hamiltonian, which modify the spectrum of the system by lifting degeneracies and producing finer energetic structures.

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