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 charges in a magnetic field is invariant under a gauge transformation, $\mathbf {A} \rightarrow \mathbf {A} +\nabla \chi .$ We will now show how this is expressed in quantum mechanics.

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

$-{\frac {\hbar ^{2}}{2m}}\left(\nabla +{\frac {ie}{\hbar c}}\mathbf {A} \right)^{2}\psi =E\psi .$

If we perform a gauge transformation on the magnetic field, then this equation becomes

$-{\frac {\hbar ^{2}}{2m}}\left(\nabla +{\frac {ie}{\hbar c}}\mathbf {A} +{\frac {ie}{\hbar c}}\nabla \chi \right)^{2}\psi =E\psi .$

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.

Usually, we use two gauges in magnetic field. One is the Laudau Gauge: $A(r)=(-yB,0,0)\!$, the other is the Symmetric Gauge: $A(r)={\frac {1}{2}}(-yB,xB,0)\!$.

We choose Laudau Gauge in the following calculation.

Motion in electromagnetic field

The Hamiltonian of a particle of charge $e\!$ and mass $m\!$ in an external electromagnetic field, which may be time-dependent, is given as follows:

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

where ${\mathbf {A({\mathbf {r}},t)}}\!$ is the vector potential and ${\phi ({\mathbf {r}},t)}\!$ is the Coulomb potential of the electromagnetic field. In a problem, if there is a momentum operator ${\mathbf {p}}\!$, it must be replaced by

$\left({\mathbf {p}}-{\frac {e}{c}}{\mathbf {A}}({\mathbf {r}},t)\right)$

if a particle is under the influence of an electromagnetic field.

Let's find out the Heisenberg equations of motion for the position and velocity operators. For position operator${\mathbf {r}}\!$, we have:

${\begin{aligned}{\frac {d{\mathbf {r}}}{dt}}&={\frac {1}{i\hbar }}\left[{\mathbf {r}},H\right]\\&={\frac {1}{i\hbar }}\left[{\mathbf {r}},{\frac {1}{2m}}\left({\mathbf {p}}-{\frac {e}{c}}{\mathbf {A}}({\mathbf {r}},t)\right)^{2}+e\phi ({\mathbf {r}},t)\right]\\&={\frac {1}{2im\hbar }}\left[{\mathbf {r}},\left({\mathbf {p}}-{\frac {e}{c}}{\mathbf {A}}({\mathbf {r}},t)\right)^{2}\right]\\&={\frac {1}{2im\hbar }}\left[{\mathbf {r}},\left({\mathbf {p}}-{\frac {e}{c}}{\mathbf {A}}({\mathbf {r}},t)\right)\right]\left({\mathbf {p}}-{\frac {e}{c}}{\mathbf {A}}({\mathbf {r}},t)\right)+{\frac {1}{2im\hbar }}\left({\mathbf {p}}-{\frac {e}{c}}{\mathbf {A}}({\mathbf {r}},t)\right)\left[{\mathbf {r}},\left({\mathbf {p}}-{\frac {e}{c}}{\mathbf {A}}({\mathbf {r}},t)\right)\right]\\&={\frac {1}{2im\hbar }}\left[{\mathbf {r}},{\mathbf {p}}\right]\left({\mathbf {p}}-{\frac {e}{c}}{\mathbf {A}}({\mathbf {r}},t)\right)+{\frac {1}{2im\hbar }}\left({\mathbf {p}}-{\frac {e}{c}}{\mathbf {A}}({\mathbf {r}},t)\right)\left[{\mathbf {r}},{\mathbf {p}}\right]\\&={\frac {1}{2im\hbar }}i\hbar \left({\mathbf {p}}-{\frac {e}{c}}{\mathbf {A}}({\mathbf {r}},t)\right)+{\frac {1}{2im\hbar }}\left({\mathbf {p}}-{\frac {e}{c}}{\mathbf {A}}({\mathbf {r}},t)\right)i\hbar \\&={\frac {1}{m}}\left({\mathbf {p}}-{\frac {e}{c}}{\mathbf {A}}({\mathbf {r}},t)\right),\end{aligned}}$

where (${\mathbf {r}}\!$ does not depend on $t\!$ explicitly) is the equation of motion for the position operator ${\mathbf {r}}$. This equation also defines the velocity operator ${\mathbf {v}}$:

${\mathbf {v}}={\frac {1}{m}}\left({\mathbf {p}}-{\frac {e}{c}}{\mathbf {A}}({\mathbf {r}},t)\right)$

The Hamiltonian can be rewritten as:

$H={\frac {m}{2}}{\mathbf {v}}\cdot {\mathbf {v}}+e\phi$

Therefore, the Heisenberg equation of motion for the velocity operator is:

${\begin{aligned}{\frac {d{\mathbf {v}}}{dt}}&={\frac {1}{i\hbar }}\left[{\mathbf {v}},H\right]+{\frac {\partial {\mathbf {v}}}{\partial t}}\\&={\frac {1}{i\hbar }}\left[{\mathbf {v}},{\frac {m}{2}}{\mathbf {v}}\cdot {\mathbf {v}}\right]+{\frac {1}{i\hbar }}\left[{\mathbf {v}},e\phi \right]-{\frac {e}{mc}}{\frac {\partial {\mathbf {A}}}{\partial t}}\end{aligned}}$

(Note that ${\mathbf {p}}\!$ does not depend on $t\!$ expicitly)

Let's use the following commutator identity:

$\left[{\mathbf {v}},{\mathbf {v}}\cdot {\mathbf {v}}\right]={\mathbf {v}}\times \left({\mathbf {v}}\times {\mathbf {v}}\right)-\left({\mathbf {v}}\times {\mathbf {v}}\right)\times {\mathbf {v}}$

Substituting, we get:

${\frac {d{\mathbf {v}}}{dt}}={\frac {1}{i\hbar }}{\frac {m}{2}}\left({\mathbf {v}}\times ({\mathbf {v}}\times {\mathbf {v}})-({\mathbf {v}}\times {\mathbf {v}})\times {\mathbf {v}}\right)+{\frac {1}{i\hbar }}e[{\mathbf {v}},\phi ]-{\frac {e}{mc}}{\frac {\partial {\mathbf {A}}}{\partial t}}$

Now let's evaluate ${\mathbf {v}}\times {\mathbf {v}}\!$ and $[{\mathbf {v}},\phi ]\!$:

${\begin{aligned}({\mathbf {v}}\times {\mathbf {v}})_{i}&=\epsilon _{ijk}v_{j}v_{k}\\&=\epsilon _{ijk}{\frac {1}{m}}\left(p_{j}-{\frac {e}{c}}A_{j}({\mathbf {r}},t)\right){\frac {1}{m}}\left(p_{k}-{\frac {e}{c}}A_{k}({\mathbf {r}},t)\right)\\&=-{\frac {e}{m^{2}c}}\epsilon _{ijk}\left(p_{j}A_{k}({\mathbf {r}},t)+A_{j}({\mathbf {r}},t)p_{k}\right)\\&=-{\frac {e}{m^{2}c}}\epsilon _{ijk}p_{j}A_{k}({\mathbf {r}},t)-{\frac {e}{m^{2}c}}\epsilon _{ijk}A_{j}({\mathbf {r}},t)p_{k}\\&=-{\frac {e}{m^{2}c}}\epsilon _{ijk}p_{j}A_{k}({\mathbf {r}},t)-{\frac {e}{m^{2}c}}\epsilon _{ikj}A_{k}({\mathbf {r}},t)p_{j}{\mbox{(Switching indices in the second terms)}}\\&=-{\frac {e}{m^{2}c}}\epsilon _{ijk}p_{j}A_{k}({\mathbf {r}},t)+{\frac {e}{m^{2}c}}\epsilon _{ijk}A_{k}({\mathbf {r}},t)p_{j}\\&=-{\frac {e}{m^{2}c}}\epsilon _{ijk}\left[p_{j},A_{k}({\mathbf {r}},t)\right]\\&=-{\frac {e}{m^{2}c}}\epsilon _{ijk}{\frac {\hbar }{i}}\nabla _{j}A_{k}({\mathbf {r}},t)\\&=i\hbar {\frac {e}{m^{2}c}}\left(\nabla \times {\mathbf {A}}\right)_{i}\end{aligned}}$

$\rightarrow \left[{\mathbf {v}}\times {\mathbf {v}}\right]=i\hbar {\frac {e}{m^{2}c}}\left(\nabla \times {\mathbf {A}}\right)=i\hbar {\frac {e}{m^{2}c}}{\mathbf {B}}$

and

${\begin{aligned}\left[{\mathbf {v}},\phi \right]&={\frac {1}{m}}\left[{\mathbf {p}}-{\frac {e}{c}}{\mathbf {A}}({\mathbf {r}},t),\phi ({\mathbf {r}},t)\right]\\&={\frac {1}{m}}\left[{\mathbf {p}},\phi ({\mathbf {r}},t)\right]\\&={\frac {1}{m}}{\frac {\hbar }{i}}\nabla \phi \end{aligned}}$

Substituting and rearranging, we get:

$m{\frac {d{\mathbf {v}}}{dt}}={\frac {e}{2c}}\left({\mathbf {v}}\times {\mathbf {B}}-{\mathbf {B}}\times {\mathbf {v}}\right)+e{\mathbf {E}}$

where

${\mathbf {E}}=-\nabla \phi -{\frac {1}{c}}{\frac {\partial {\mathbf {A}}}{\partial t}}$

Above is the quantum mechanical version of the equation for the acceleration of the particle in terms of the Lorentz force.

These results can also be deduced in Hamiltonian dynamics due to the similarity between the Hamiltonian dynamics and quantum mechanics.

Problems about Motion in electromagnetic field

Problem 1

Problem 2