Charged Particles in an Electromagnetic Field: Difference between revisions

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{{Quantum Mechanics A}}
{{Quantum Mechanics A}}
==Gauge==
Gauge theory is a type of field theory in which the Lagrangian is invariant under a certain continuous group of local transformations.


Given a distribution of charges and current, and appropriate boundary conditions, the electromagnetic field is unique. However, the electromagnetic potential <math> A^{\mu}\!</math> is not unique. The Maxwell equations can be expressed by electromagnetic field tensor <math> F^{\mu\nu} \!</math>, which is defined by
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
:<math> F^{\mu\nu}=\partial^{\mu} A^{\nu}-\partial^{\nu} A^{\mu} </math>.


If we set
<math>H=\frac{1}{2m}\left (\mathbf{p}-\frac{e}{c}\mathbf{A}\right )^2,</math>
:<math> A'^{\mu}=A^{\mu}+\partial^{\mu}\chi </math>,


then
where <math>e\!</math> is the charge of the particle and <math>\mathbf{A}</math> 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, <math>\mathbf{p}\rightarrow\mathbf{p}-\frac{e}{c}\mathbf{A}.</math> In quantum mechanics, we introduce the magnetic field in the same way; this process is referred to as minimal coupling.
:<math>  
\begin{align}
F'^{\mu \nu } &= \partial ^\mu A'^\nu  - \partial ^\nu A'^\mu \\
&= \partial^{\mu}(A^{\nu}+\partial^{nu}\chi)-\partial^{\nu}(A^{\mu}+\partial^{\mu}\chi) \\
&= \partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}+(\partial^{\mu}\partial^{\nu}-\partial^{\nu}\partial^{\mu})\chi \\
&= F^{\mu \nu }
\end{align}
</math>,  
eg. the form of Maxwell equations will not change. So, we have a freedom <math> \partial\chi \!</math>, which is called Gauge Freedom here.


For the magnetic field case, we can check for gauge invariance:
== Gauge Invariance in Quantum Mechanics ==


:<math> \left. \left. \frac{{(p - \frac{e}{c}A - \partial \chi )^2 }}{{2m}}\right|\varphi \right\rangle = E|\varphi \rangle </math>,
We know from Maxwell's equations that the classical physics of a charged particle in an electromagnetic field is invariant under a gauge transformation, <math>\Phi\rightarrow\Phi-\frac{1}{c}\frac{\partial\chi}{\partial t}</math> and <math>\mathbf{A}\rightarrow\mathbf{A}+\nabla\chi,</math> where <math>\Phi\!</math> is the scalar potential and <math>\chi(\mathbf{r},t)\!</math> is a single-valued real function.  We will now show how this is expressed in quantum mechanics.


Let <math>
In the position basis, the [[Schrödinger Equation|Schrödinger equation]] for a charged particle in an electromagnetic field is
|\varphi \rangle = e^{\frac{i}{\hbar }\frac{e}{c}\chi } |\phi \rangle
</math>,


the form <math>
<math>-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.</math>
\frac{{(p - \frac{e}{c}A)^2 }}{{2m}}|\phi \rangle = E|\phi \rangle
</math> will not change.


Usually, we use two gauges in magnetic field. One is the Laudau Gauge: <math>
If we now perform the above gauge transformation on the electromagnetic field, then this equation becomes
A(r) = ( - yB,0,0)\!</math>, the other is the Symmetric Gauge: <math>
A(r) = \frac{1}{2}( - yB,xB,0)\!</math>.


We choose Laudau Gauge in the following calculation.
<math>-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.</math>


==Motion in electromagnetic field==
If we make the substitution, <math>\Psi\rightarrow e^{-ie\chi/\hbar c}\Psi,</math> 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 multiplies the wave function by a single-valued phase factor, 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 <math>\mathbf{B}=B\hat{\mathbf{z}},</math> we typically work with one of two gauges.  One of these is the Laudau gauge,


The Hamiltonian of a particle of charge <math>e\!</math> and mass <math>m\!</math>  
<math>\mathbf{A}(\mathbf{r}) = -yB\hat{\mathbf{x}}</math> or <math>xB\hat{\mathbf{y}}.</math>
in an external electromagnetic field, which may be time-dependent, is given as follows:


:<math> H=\frac{1}{2m}\left(\mathbf{p}-\frac{e}{c}\bold A(\bold r,t)\right)^2+e\phi(\bold r,t)</math>
The other is the symmetric gauge,


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


where <math> \bold{A(\bold r,t)} \!</math> is the vector potential and <math>{\phi(\bold r,t)}\!</math> is the Coulomb potential of the electromagnetic field. In a problem, if there is a momentum operator <math>\bold p\!</math>, it must be replaced by
== Eigenstates of a Charged Particle in a Static and Uniform Magnetic Field ==
:<math>\left(\bold p-\frac{e}{c}\bold A(\bold r,t)\right)</math>
if a particle is under the influence of an electromagnetic field.


Let's find out the [[Heisenberg and interaction picture: Equations of motion for operators#The Heisenberg Equation of Motion|Heisenberg equations of motion]] for the position and velocity operators.
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,
For position operator<math>\bold r\!</math>, we have:


:<math>  
<math>\mathbf{A}=xB\hat{\mathbf{y}}.</math>
\begin{align}
\frac{d\bold r}{dt} &= \frac{1}{i\hbar} \left[\bold r,H \right] \\
&= \frac{1}{i\hbar} \left[ \bold r, \frac{1}{2m} \left(\bold p-\frac{e}{c}\bold A(\bold r,t)\right)^2 + e\phi(\bold r,t)\right] \\
&= \frac{1}{2im\hbar} \left[\bold r, \left(\bold p-\frac{e}{c}\bold A(\bold r,t)\right)^2\right] \\
&= \frac{1}{2im\hbar} \left[\bold r, \left(\bold p-\frac{e}{c}\bold A(\bold r,t)\right)\right]\left(\bold p-\frac{e}{c}\bold A(\bold r,t)\right) + \frac{1}{2im\hbar} \left(\bold p-\frac{e}{c}\bold A(\bold r,t)\right) \left[\bold r, \left(\bold p-\frac{e}{c}\bold A(\bold r,t)\right)\right] \\
&= \frac{1}{2im\hbar} \left[\bold r, \bold p\right] \left(\bold p-\frac{e}{c}\bold A(\bold r,t)\right) + \frac{1}{2im\hbar} \left(\bold p - \frac{e}{c}\bold A(\bold r,t)\right) \left[\bold r, \bold p\right] \\
&= \frac{1}{2im\hbar}i\hbar \left(\bold p-\frac{e}{c}\bold A(\bold r,t)\right) + \frac{1}{2im\hbar} \left(\bold p-\frac{e}{c}\bold A(\bold r,t)\right)i\hbar \\
&= \frac{1}{m}\left(\bold p-\frac{e}{c}\bold A(\bold r,t)\right),
\end{align}
</math>
where (<math>\bold r \!</math> does not depend on <math>t \!</math> explicitly)
is the equation of motion for the position operator <math>\bold r</math>.  
This equation also defines the velocity operator <math>\bold v</math>:
                                               
:<math>\bold v= \frac {1}{m}\left(\bold p-\frac{e}{c}\bold A(\bold r,t)\right)</math>                                                                              


The Hamiltonian can be rewritten as:
The [[Schrödinger Equation|Schrödinger equation]] for this system is


:<math>H=\frac {m}{2}\bold v \cdot \bold v+e\phi</math>
<math>-\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.</math>


Therefore, the Heisenberg equation of motion for the velocity operator is:
In this gauge, the Hamiltonian is translationally invariant along the <math>y\!</math> and <math>z\!</math> axes.  Therefore, our wave function will have the form,


:<math>
<math>\psi(x,y,z)=e^{i(k_yy+k_zz)}f(x).\!</math>
\begin{align}
\frac{d\bold v}{dt} &=\frac {1}{i\hbar}\left[\bold v,H\right]+\frac{\partial \bold v}{\partial t} \\
&= \frac {1}{i\hbar}\left[\bold v,\frac{m}{2}\bold v \cdot \bold v\right]+\frac {1}{i\hbar}\left[\bold v,e\phi\right]-\frac{e}{mc} \frac{\partial \bold A}{\partial t}  
\end{align}
</math>
(Note that <math>\bold p\!</math> does not depend on <math>t\!</math> expicitly)


Let's use the following commutator identity:
Substituting this form into the equation, we obtain


:<math>\left[\bold v,\bold v \cdot \bold v\right]=\bold v \times \left(\bold v \times \bold v\right)-\left(\bold v \times \bold v\right) \times \bold v </math>  
<math>-\frac{\hbar^2}{2m}\frac{d^2f}{dx^2}+\frac{\hbar^2}{2m}\left (k_y+\frac{e}{\hbar c}Bx\right )^2f+\frac{\hbar^2k_z^2}{2m}f=Ef,</math>


Substituting, we get:
or


:<math>
<math>-\frac{\hbar^2}{2m}\frac{d^2f}{dx^2}+\frac{e^2B^2}{2mc^2}\left (x+\frac{\hbar c}{eB}k_y\right )^2f=\left (E-\frac{\hbar^2k_z^2}{2m}\right )f.</math>
\frac{d\bold v}{dt} = \frac{1}{i\hbar} \frac{m}{2}
\left(\bold v \times (\bold v \times \bold v) - (\bold v \times \bold v) \times \bold v \right) + \frac{1}{i\hbar} e[\bold v,\phi] - \frac{e}{mc} \frac{\partial \bold A}{\partial t}</math>


Now let's evaluate <math>\bold v \times \bold v \!</math> and <math>[\bold v,\phi] \!</math>:
If we now introduce the shifted position coordinate <math>x'=x+\frac{\hbar c}{eB}k_y</math> and the shifted energy <math>E'=E-\frac{\hbar^2k_z^2}{2m},</math> this becomes


:<math>
<math>-\frac{\hbar^2}{2m}\frac{d^2f}{dx'^2}+\frac{e^2B^2}{2mc^2}x'^2f=E'f.</math>
\begin{align}
(\bold v \times \bold v)_i &= \epsilon_{ijk} v_j v_k \\
&= \epsilon_{ijk}\frac{1}{m} \left(p_j-\frac{e}{c}A_j(\bold r,t)\right)
\frac{1}{m}\left(p_k-\frac{e}{c}A_k(\bold r,t)\right) \\
&= -\frac{e}{m^2c} \epsilon_{ijk}\left(p_j A_k(\bold r,t) +
A_j(\bold r,t)p_k\right) \\
&= -\frac{e}{m^2c}\epsilon_{ijk}p_jA_k(\bold r,t) - \frac{e}{m^2c}
\epsilon_{ijk} A_j(\bold r,t) p_k \\
&= -\frac{e}{m^2c}\epsilon_{ijk} p_j A_k(\bold r,t)-\frac{e}{m^2c}
\epsilon_{ikj} A_k(\bold r,t) p_j \mbox{(Switching indices in the second terms)} \\
&= -\frac{e}{m^2c}\epsilon_{ijk} p_j A_k(\bold r,t) + \frac{e}{m^2c}
\epsilon_{ijk} A_k(\bold r,t) p_j \\
&= -\frac{e}{m^2c}\epsilon_{ijk}\left[p_j,A_k(\bold r,t)\right] \\
&= -\frac{e}{m^2c}\epsilon_{ijk}\frac{\hbar}{i} \nabla_j A_k(\bold r,t) \\
&= i\hbar\frac{e}{m^2c}\left(\nabla \times \bold A\right)_i
\end{align}
</math>


:<math>
This is just the equation for a [[Analytical Method for Solving the Simple Harmonic Oscillator|harmonic oscillator]] with frequency
\rightarrow \left[\bold v \times \bold v\right]=i\hbar\frac{e}{m^2c}\left(\nabla \times \bold A\right) = i\hbar\frac{e}{m^2c}\bold B
</math>


and
<math>\omega=\frac{eB}{mc}.</math>


:<math>
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
\begin{align}
\left[\bold v,\phi\right] &= \frac{1}{m} \left[\bold p-\frac{e}{c}\bold A(\bold r, t),\phi(\bold r,t)\right] \\
&= \frac{1}{m} \left[\bold p,\phi(\bold r,t) \right] \\
&= \frac{1}{m} \frac{\hbar}{i}\nabla\phi
\end{align}
</math>


Substituting and rearranging, we get:
<math>\psi(x,y,z)=\frac{1}{\sqrt{L_yL_z}}e^{i(k_yy+k_zz)}\frac{1}{\sqrt{2^nn!l_B\sqrt{\pi}}}\exp\left [-\tfrac{1}{2}\left (\frac{x}{l_B}+l_Bk_y\right )^2\right ]H_n\left (\frac{x}{l_B}+l_Bk_y\right ),</math>


:<math>
where <math>L_y\!</math> and <math>L_z\!</math> are the dimensions of the system in the <math>y\!</math> and <math>z\!</math> directions and <math>l_B=\sqrt{\frac{\hbar c}{eB}}</math> is known as the magnetic length.  The energies are given by
m\frac{d\bold v}{dt} = \frac{e}{2c}
\left(\bold v \times \bold B-\bold B \times \bold v \right) + e\bold E
</math>
where
:<math>
\bold E = -\nabla \phi - \frac{1}{c} \frac{\partial \bold A}{\partial t}
</math>                                                                                        
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 ==
<math>E=\left (n+\tfrac{1}{2}\right )\hbar\omega+\frac{\hbar^2k_z^2}{2m}.</math>


[[Phy5645/harmonicoscinmagneticfield/|Problem 1]]
For a fixed value of <math>k_z,\!</math> the energy spectrum that we just obtained is referred to as a Landau level spectrum.  Note that the above energies do not depend on <math>k_y;\!</math> it only appears in the wave function, where it determines the "guiding center" <math>l_B^2k_y\!</math> of the wave function.  This means that they are very highly degenerate.  We may approximate the degeneracy of each of these Landau levels as follows.  If the system has a finite size, then <math>k_y\!</math> is quantized to integer multiples of <math>\frac{2\pi}{L_y}</math> if we assume that the wave function satisfies periodic boundary conditions; i.e. <math>k_y=\frac{2\pi n}{L_y}.</math>  We now determine the range of values of <math>n\!</math> for which the guiding center is within the range <math>0<l_B^2k_y\leq L_x,\!</math> where <math>L_x\!</math> is the dimension of the system in the <math>x\!</math> direction.  This value is


[[Phy5645/harmonicoscinmagneticfield/problem2|Problem 2]]
<math>n=\frac{L_xL_y}{2\pi l_B^2},</math>
 
or
 
<math>n=\frac{BL_xL_y}{\Phi_0},</math>
 
where <math>\Phi_0=\frac{hc}{e}</math> is known as the "flux quantum" of the particle.  This quantity appears frequently in many contexts, such as in the theory of superconductivity.  We may therefore think of the degeneracy of the system as just the number of flux quanta contained within a face in the <math>xy\!</math> plane of the box that the particle is contained inside of.
 
== Problem ==
 
Consider the problem of a particle of charge <math>e\!</math> in a uniform magnetic field along the <math>z\!</math> direction again, but now in the symmetric gauge, <math>\mathbf{A}=-\tfrac{1}{2}By\hat{\mathbf{x}}+\tfrac{1}{2}Bx\hat{\mathbf{y}}.</math>  Let <math>\hat{\Pi}_i=\hat{p}_i-\frac{e}{c}A_i.</math>
 
'''(a)''' Evaluate <math>\left [{\hat{\Pi}_{x},\hat{\Pi}_{y}} \right ].</math>
 
'''(b)''' Using the commutation relation obtained in the previous part, obtain the energy eigenvalues.
 
[[Phy5645/Particle in Uniform Magnetic Field|Solution]]

Latest revision as of 13:36, 18 January 2014

Quantum Mechanics A
SchrodEq.png
Schrödinger Equation
The most fundamental equation of quantum mechanics; given a Hamiltonian Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}} , it describes how a state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi\rangle} evolves in time.
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
Brief Derivation of Schrödinger Equation
Relation Between the Wave Function and Probability Density
Stationary States
Heisenberg Uncertainty Principle
Some Consequences of the Uncertainty Principle
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
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
Harmonic Oscillator Spectrum and Eigenstates
Analytical Method for Solving the Simple Harmonic Oscillator
Coherent States
Charged Particles in an Electromagnetic Field
WKB Approximation
The Heisenberg Picture: Equations of Motion for Operators
The Interaction Picture
The Virial Theorem
Commutation Relations
Angular Momentum as a Generator of Rotations in 3D
Spherical Coordinates
Eigenvalue Quantization
Orbital Angular Momentum Eigenfunctions
General Formalism
Free Particle in Spherical Coordinates
Spherical Well
Isotropic Harmonic Oscillator
Hydrogen Atom
WKB in Spherical Coordinates
Feynman Path Integrals
The Free-Particle Propagator
Propagator for the Harmonic Oscillator
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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H=\frac{1}{2m}\left (\mathbf{p}-\frac{e}{c}\mathbf{A}\right )^2,}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e\!} is the charge of the particle and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi\rightarrow\Phi-\frac{1}{c}\frac{\partial\chi}{\partial t}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{A}\rightarrow\mathbf{A}+\nabla\chi,} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi\!} is the scalar potential and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 multiplies the wave function by a single-valued phase factor, 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B}=B\hat{\mathbf{z}},} we typically work with one of two gauges. One of these is the Laudau gauge,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{A}(\mathbf{r}) = -yB\hat{\mathbf{x}}} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xB\hat{\mathbf{y}}.}

The other is the symmetric gauge,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{A}=xB\hat{\mathbf{y}}.}

The Schrödinger equation for this system is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z\!} axes. Therefore, our wave function will have the form,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(x,y,z)=e^{i(k_yy+k_zz)}f(x).\!}

Substituting this form into the equation, we obtain

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{\hbar^2}{2m}\frac{d^2f}{dx^2}+\frac{\hbar^2}{2m}\left (k_y+\frac{e}{\hbar c}Bx\right )^2f+\frac{\hbar^2k_z^2}{2m}f=Ef,}

or

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{\hbar^2}{2m}\frac{d^2f}{dx^2}+\frac{e^2B^2}{2mc^2}\left (x+\frac{\hbar c}{eB}k_y\right )^2f=\left (E-\frac{\hbar^2k_z^2}{2m}\right )f.}

If we now introduce the shifted position coordinate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x'=x+\frac{\hbar c}{eB}k_y} and the shifted energy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E'=E-\frac{\hbar^2k_z^2}{2m},} this becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{\hbar^2}{2m}\frac{d^2f}{dx'^2}+\frac{e^2B^2}{2mc^2}x'^2f=E'f.}

This is just the equation for a harmonic oscillator with frequency

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(x,y,z)=\frac{1}{\sqrt{L_yL_z}}e^{i(k_yy+k_zz)}\frac{1}{\sqrt{2^nn!l_B\sqrt{\pi}}}\exp\left [-\tfrac{1}{2}\left (\frac{x}{l_B}+l_Bk_y\right )^2\right ]H_n\left (\frac{x}{l_B}+l_Bk_y\right ),}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_y\!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_z\!} are the dimensions of the system in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z\!} directions and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_B=\sqrt{\frac{\hbar c}{eB}}} is known as the magnetic length. The energies are given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E=\left (n+\tfrac{1}{2}\right )\hbar\omega+\frac{\hbar^2k_z^2}{2m}.}

For a fixed value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_z,\!} the energy spectrum that we just obtained is referred to as a Landau level spectrum. Note that the above energies do not depend on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_y;\!} it only appears in the wave function, where it determines the "guiding center" Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_B^2k_y\!} of the wave function. This means that they are very highly degenerate. We may approximate the degeneracy of each of these Landau levels as follows. If the system has a finite size, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_y\!} is quantized to integer multiples of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2\pi}{L_y}} if we assume that the wave function satisfies periodic boundary conditions; i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_y=\frac{2\pi n}{L_y}.} We now determine the range of values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\!} for which the guiding center is within the range Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0<l_B^2k_y\leq L_x,\!} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_x\!} is the dimension of the system in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\!} direction. This value is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=\frac{L_xL_y}{2\pi l_B^2},}

or

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=\frac{BL_xL_y}{\Phi_0},}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_0=\frac{hc}{e}} is known as the "flux quantum" of the particle. This quantity appears frequently in many contexts, such as in the theory of superconductivity. We may therefore think of the degeneracy of the system as just the number of flux quanta contained within a face in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xy\!} plane of the box that the particle is contained inside of.

Problem

Consider the problem of a particle of charge Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e\!} in a uniform magnetic field along the direction again, but now in the symmetric gauge, Let

(a) Evaluate

(b) Using the commutation relation obtained in the previous part, obtain the energy eigenvalues.

Solution

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