Charged Particles in an Electromagnetic Field

Quantum Mechanics A

Schrödinger Equation The most fundamental equation of quantum mechanics; given a Hamiltonian ${\mathcal {H}}$, it describes how a state $|\Psi \rangle$ evolves in time.

Physical Basis of Quantum Mechanics

Basic Concepts and Theory of Motion UV Catastrophe (Black-Body Radiation) Photoelectric Effect Stability of Matter Double Slit Experiment Stern-Gerlach Experiment The Principle of Complementarity The Correspondence Principle The Philosophy of Quantum Theory

Schrödinger Equation

Brief Derivation of Schrödinger Equation Relation Between the Wave Function and Probability Density Stationary States Heisenberg Uncertainty Principle Some Consequences of the Uncertainty Principle

Operators, Eigenfunctions, and Symmetry

Linear Vector Spaces and Operators Commutation Relations and Simultaneous Eigenvalues The Schrödinger Equation in Dirac Notation Transformations of Operators and Symmetry Time Evolution of Expectation Values and Ehrenfest's Theorem

Motion in One Dimension

One-Dimensional Bound States Oscillation Theorem The Dirac Delta Function Potential Scattering States, Transmission and Reflection Motion in a Periodic Potential Summary of One-Dimensional Systems

Discrete Eigenvalues and Bound States

Harmonic Oscillator Spectrum and Eigenstates Analytical Method for Solving the Simple Harmonic Oscillator Coherent States Charged Particles in an Electromagnetic Field WKB Approximation

Time Evolution and the Pictures of Quantum Mechanics

The Heisenberg Picture: Equations of Motion for Operators The Interaction Picture The Virial Theorem

Angular Momentum

Commutation Relations Angular Momentum as a Generator of Rotations in 3D Spherical Coordinates Eigenvalue Quantization Orbital Angular Momentum Eigenfunctions

Central Forces

General Formalism Free Particle in Spherical Coordinates Spherical Well Isotropic Harmonic Oscillator Hydrogen Atom WKB in Spherical Coordinates

The Path Integral Formulation of Quantum Mechanics

Feynman Path Integrals The Free-Particle Propagator Propagator for the Harmonic Oscillator

Continuous Eigenvalues and Collision Theory

Differential Cross Section and the Green's Function Formulation of Scattering Central Potential Scattering and Phase Shifts Coulomb Potential Scattering

A problem with some relation to the harmonic oscillator is that of the motion of a charged particle in a constant and uniform magnetic field. In classical mechanics, we know that the Hamiltonian for this system is

$H={\frac {1}{2m}}\left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)^{2},$

where $e\!$ is the charge of the particle and $\mathbf {A}$ is the vector potential. In fact, to obtain the Hamiltonian for any system in the presence of a magnetic field, we simply make the replacement, $\mathbf {p} \rightarrow \mathbf {p} -{\frac {e}{c}}\mathbf {A} .$ In quantum mechanics, we introduce the magnetic field in the same way; this process is referred to as minimal coupling.

Gauge Invariance in Quantum Mechanics

We know from Maxwell's equations that the classical physics of a charged particle in an electromagnetic field is invariant under a gauge transformation, $\Phi \rightarrow \Phi -{\frac {1}{c}}{\frac {\partial \chi }{\partial t}}$ and $\mathbf {A} \rightarrow \mathbf {A} +\nabla \chi ,$ where $\Phi \!$ is the scalar potential and $\chi (\mathbf {r} ,t)\!$ is a single-valued real function. We will now show how this is expressed in quantum mechanics.

In the position basis, the Schrödinger equation for a charged particle in an electromagnetic field is

$-i\hbar {\frac {\partial \Psi }{\partial t}}-e\Phi \Psi =-{\frac {\hbar ^{2}}{2m}}\left(\nabla +{\frac {ie}{\hbar c}}\mathbf {A} \right)^{2}\Psi .$

If we now perform the above gauge transformation on the electromagnetic field, then this equation becomes

$-i\hbar {\frac {\partial \Psi }{\partial t}}-e\Phi \Psi +{\frac {e}{c}}{\frac {\partial \chi }{\partial t}}\Psi -{\frac {\hbar ^{2}}{2m}}\left(\nabla +{\frac {ie}{\hbar c}}\mathbf {A} +{\frac {ie}{\hbar c}}\nabla \chi \right)^{2}\Psi .$

If we make the substitution, $\Psi \rightarrow e^{-ie\chi /\hbar c}\Psi ,$ then we recover the original equation. Therefore, a gauge transformation of the magnetic field effectively introduces a phase factor to the wave function. This does result in a change in the canonical momentum, but it will have no effect on, for example, the probability density for finding the particle at a given position or, as we will see later, on the expectation value of the position or velocity of the particle.

We see that, in quantum mechanics, gauge invariance is expressed as follows. If one introduces a single-valued phase factor into the wave function, then it may be "canceled out" by a corresponding change in the electromagnetic potentials that the particle is subject to.

For a constant and uniform magnetic field $\mathbf {B} =B{\hat {\mathbf {z} }},$ we typically work with one of two gauges. One of these is the Laudau gauge,

$\mathbf {A} (\mathbf {r} )=-yB{\hat {\mathbf {x} }}$ or $xB{\hat {\mathbf {y} }}.$

The other is the symmetric gauge,

$\mathbf {A} (\mathbf {r} )=-{\tfrac {1}{2}}yB{\hat {\mathbf {x} }}+{\tfrac {1}{2}}xB{\hat {\mathbf {y} }}.$

Eigenstates of a Charged Particle in a Static and Uniform Magnetic Field

Let us now find the eigenstates of a charged particle in a static and uniform magnetic field. We will be working in the Landau gauge,

$\mathbf {A} =xB{\hat {\mathbf {y} }}.$

The Schrödinger equation for this system is

$-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}\psi }{\partial x^{2}}}-{\frac {\hbar ^{2}}{2m}}\left({\frac {\partial }{\partial y}}+{\frac {ie}{\hbar c}}Bx\right)^{2}\psi -{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}\psi }{\partial z^{2}}}\psi =E\psi .$

In this gauge, the Hamiltonian is translationally invariant along the $y\!$ and $z\!$ axes. Therefore, our wave function will have the form,

$\psi (x,y,z)=e^{i(k_{y}y+k_{z}z)}f(x).\!$

Substituting this form into the equation, we obtain

$-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}f}{dx^{2}}}+{\frac {\hbar ^{2}}{2m}}\left(k_{y}+{\frac {e}{\hbar c}}Bx\right)^{2}f+{\frac {\hbar ^{2}k_{z}^{2}}{2m}}f=Ef,$

or

$-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}f}{dx^{2}}}+{\frac {e^{2}B^{2}}{2mc^{2}}}\left(x+{\frac {\hbar c}{eB}}k_{y}\right)^{2}f=\left(E-{\frac {\hbar ^{2}k_{z}^{2}}{2m}}\right)f.$

If we now introduce the shifted position coordinate $x'=x+{\frac {\hbar c}{eB}}k_{y}$ and the shifted energy $E'=E-{\frac {\hbar ^{2}k_{z}^{2}}{2m}},$ this becomes

$-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}f}{dx'^{2}}}+{\frac {e^{2}B^{2}}{2mc^{2}}}x'^{2}f=E'f.$

This is just the equation for a harmonic oscillator with frequency

$\omega ={\frac {eB}{mc}}.$

We recognize this as the cyclotron frequency of the particle. We may immediately write down the full eigenfunctions and energy levels of the system. The wave functions are

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Problems about Motion in electromagnetic field

Problem 1

Problem 2