Connecting with our theme of symmetries, here we will study the peculiar symmetries of quantum mechanics in a magnetic field. It turns out that quantum electrodynamics (or even non-relativistic quantum mechanics in electromagnatic fields) possesses a peculiar ``gauge" symmetry. We mentioned this term last week when discussing the standard model. Here we will learn what it means.
The other purpose for combining electromagnetism and quantum mechanics is to describe some really interesting physics which occurs in electric and magnetic fields. Things like Hall effects, and de Haas van Allen effects.
This and the next lecture are a wee bit abstract.
Schrodinger's equation in an electromagnetic field
I'll start with the punch line, then try to explain a bit about it.
The Schrodinger equation for a particle of charge $q$ in an electric and magnetic field is
\begin{equation}\label{schro}
(i\partial_t-q \phi)\psi=-\frac{1}{2m}\left(\nabla-iq{\bf A}\right)^2 \psi,
\end{equation}
where $\phi$ is the scalar potential and ${\bf A}$ is the vector potential, defined by
\begin{eqnarray}\label{con1}
{\bf E}&=&-\nabla \phi -\partial_t {\bf A}\\
{\bf B}&=&\nabla\times A.
\end{eqnarray}
This should seem very bizarre. You should have learned that $\phi$ and $\bf A$ are not physical -- however they appear in the Schrodinger Equation.
Next week you will numerically see that a wavepacket evolves under this equation just like a classical charged particle.
Plausibility Argument
Now I can't prove Eq.~(\ref{schro}) any more than I can prove that the gravitational force follows an inverse square law. I can however give some plausibility about this form of the Hamiltonian. The most straightforward is to look at the time dependence of the expectation values
\begin{eqnarray}
{\bf R}(t)&=& \int \psi^*({\bf r},t) {\bf r} \psi({\bf r},t)\\
{\bf \Pi}(t) &=& \int \psi^*({\bf r},t) \left(-i\nabla-q {\bf A}\right) \psi({\bf r},t).
\end{eqnarray}
If I take the time derivative of the first equation, I get two terms -- one where I take the derivative of $\psi^*$, and the other where I take the derrivitive of $\psi$. These two terms are complex conjugates of eachother, so I get
\begin{equation}
\partial_t {\bf R}(t)=\frac{1}{i}\int \psi^*({\bf r},t) {\bf r}\left[-\frac{1}{2m}\left[\nabla-iq{\bf A}\right)^2+q \phi\right] \psi({\bf r},t)+{\rm c.c}
\end{equation}
Upon integrating by parts, and adding the complex conjugates one finds
\begin{equation}
\partial_t {\bf R}(t)= \frac{{\bf \Pi}(t)}{m}.
\end{equation}
Thus $\Pi$ can be identified with the expectation of the momentum. Doing the same procedure with $\Pi$ yields something morally equivalent to
\begin{equation}
\partial_t \Pi=q\nabla\phi-q{\bf B\times R}.
\end{equation}
In homework you will work through the mathematics, using Heisenberg equations of motion to make this more formal.
Scalar and Vector Potentials as a 4-vector
In your introductory class you probably learned about the scalar and vector potential from electrostatics and magnetostatics. First one notes that in the absence of magnetic field $\nabla\times \bf E=0$ -- or in integral language
\begin{equation}
\oint {\bf E\cdot} d{\bf \ell}=0 \quad \mbox{when ${\bf B}=0$}
\end{equation}
which implies $\bf E$ is the gradient of a field $\phi$. Later one notes that there are no magnetic monopoles, hence $\nabla\cdot \bf B=0$ -- or in integral language
\begin{equation}
\oint {\bf B}\cdot d{\bf S}=0
\end{equation}
This implies that $\bf B$ is the curl of something -- called $\bf A$. Finally one considers time dependent magnetic fields, and you learn Faraday's law -- which states that a time dependent magnetic field leads to a voltage, $\nabla\times {\bf E}=-\partial_t {\bf B}$, or in integral language
\begin{equation}
\oint {\bf E\cdot} d{\bf \ell}= -\frac{d \Phi}{dt},
\end{equation}
where $\Phi$ is the magnetic flux through the loop. This equation is equivalent to Eq.~(\ref{con1}).
In this standard story, the important quantities are $\bf E$ and $\bf B$ -- the potentials are just tricks. Apparently in quantum mechanics they are real. Despite this, all physically measurable quantities are {\em gauge invariant}.
One way to get more insight into this structure is to write out the components of the electric and magnetic fields. For example
\begin{equation}
B_z= \partial_x A_y-\partial_y A_x.
\end{equation}
That is $B_z$ involves some sort of loops in the x-y plane. Makes sense -- a current loop will generate a magnetic field.
Maybe a more concrete representation is to note that the magnetic flux is related to a line integral of $A$:
\begin{equation}
\int\!\!\int {\bf B}\cdot d{\bf a}=\oint {\bf A}\cdot d{\bf s}.
\end{equation}
What is more surprising is that the electric
field can also be written this way. If we let $A_t=-\phi$, then
\begin{equation}
E_z = \partial_z A_t-\partial_t A_z.
\end{equation}
The electric field in the $z$ direction involves loops in the $z-t$ plane. Things can be made notationally more consistent if we instead write
\begin{equation}
F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu.
\end{equation}
The electric and magnetic fields are just components of this tensor.
Gauge Theory
Equation~(\ref{schro}) is a Gauge theory, meaning that is is invariant under the following transformation
\begin{eqnarray}
A&\to& A+\nabla\Lambda\\
\phi&\to& \phi+\partial_t\Lambda\\
\psi&\to&\psi e^{i\Lambda}.
\end{eqnarray}
All physically measurable quantities are invariant under this transformation. For example, the probability of having a particle at position $x$ is
$|\psi(x)|^2$, which is gauge invariant.
I don't know if this is true, but I was taught that this terminology has roots in railroad track sizes (which are referred to as ``gauges"). Imagine you have a train which fits on one size track, and you want to scale it to fit on another. The transformation you need to make to your locomotive is a ``gauge transformation."
In that context, one imagines that we rescale a function $f(x)$ via $g(x)=f(x)/\lambda(x)$. Straightforward algebra yields
\begin{equation}
\partial_x f(x)=\lambda(x) \left[\partial_x +\frac{\lambda^\prime}{\lambda}\right] g(x).
\end{equation}
Thus a rescaling of a function modifies the derivative. Here we are rescaling by a complex phase, but it is the same general idea.
The other point of contact is differential geometry. Imagine you are doing calculus on a curved manifold -- say the surface of a sphere of radius 1. We know what the divergence on a sphere is:
\begin{eqnarray}
\nabla\cdot {\bf v}&=&\frac{1}{sin(\theta)}\partial_\theta\left(v_\theta \sin(\theta)\right)+\frac{1}{\sin(\theta)}\partial_\phi v_\phi\\
&=&\left(\partial_\theta+\cot(\theta)\right) v_\theta + \frac{1}{\sin(\theta)}\partial_\phi v_\phi.
\end{eqnarray}
Thus on a curved surface, the geometric quantities like the laplacian or {\em div, grad, curl} involve derivatives which are scaled and shifted. There is something {\em geometric} about the form of Eq.~(\ref{schro}).
As with any other symmetry in quantum mechanics, gauge invariance is associated with a conservation law. Here the conserved quantity is probability. I can illustrate this by throwing away space and just looking at time. Imagine I have a differential equation
\begin{equation}
i\partial_t \psi=\psi^*.
\end{equation}
The term on the right breaks gauge invariance. If we write $\phi=u+i v$, we find
\begin{eqnarray}
- \partial_t v&=& u\\
\partial_t u &=& -v,
\end{eqnarray}
which exhibits exponential growth. On the other hand the equation
\begin{equation}
i\partial_t \psi=\psi,
\end{equation}
which is a special case of Eq.~(\ref{schro}), with $\phi=-1$. This leads to the equation
\begin{eqnarray}
- \partial_t v&=& u\\
\partial_t u &=& v,
\end{eqnarray}
which preserves norm. The gauge symmetry guarantees this conservation of probability.
Lattice Gauge Theory
The last bit of formalism I need is the finite difference approximation to Eq.~(\ref{schro}). This is valuable conceptually -- as we can understand what the equation means then. We also need it for our next recitation.
Lets begin by thinking about the
the finite difference approximation to
\begin{equation}
D_x \psi(x)=\left[\partial_x-i q A_x\right]\psi(x).
\end{equation}
The simplest approximation we can make is something like
\begin{equation}\label{euler}
D_x \psi(x)\approx \frac{\psi(x+dx)-\psi(x-dx)}{2 dx} - i q A_x \psi(x).
\end{equation}
This is OK, but it breaks the gauge invariance. If I take $A_x\to A_x+\partial_x \Lambda$ and $\psi\to \psi e^{i\Lambda}$, I would like to have
\begin{equation}
D_x \psi(x)\to e^{i \Lambda} D_x \psi.
\end{equation}
The key to preserving this relationship is similar to what we used on Friday. We use a ``split step" method. We alternate the derivative with the vector potential, approximating
\begin{equation}\label{cov}
D_x\psi(x)\approx \frac{ e^{i \Phi(x+dx/2)}\psi(x+dx)-e^{-i\Phi(x-dx/2)}\psi(x-dx)}{2 dx}
\end{equation}
where the phases are the average of the vector potentials
\begin{eqnarray}
\Phi(x+dx/2)&=&\int_{x}^{x+dx} q A_x(s) ds\\
&\approx& q A_x(x+dx/2) dx.
\end{eqnarray}
If I expand Eq.~(\ref{cov}) to second order in $dx$ I recover the simple ``Euler" approximation in Eq.~(\ref{euler}), but the higher order terms maintain the symmetry. [You can readily verify that if you set $\Phi=0$ and go to $1D$ we recover our standard finite difference approximation to the Laplacian.]
The physics here is that the $x$-component of the vector potential lives on the $x$-bonds. The $y,z$-components live on the $y$ and $z$ bonds, and $\phi$ lives on the $t$-bonds. The Laplacian will be
\begin{equation}
\left(\nabla-iq{\bf A}\right)^2 \psi({\bf r})
\approx
\sum_{\hat e} \frac{e^{i \Phi({\bf r}+\delta \hat e/2)}\psi({\bf r}+\delta \hat e)-\psi({\bf r})}{\delta}
\end{equation}
where $\hat e$ is summed over all the spatial bonds, $\hat x,-\hat x,\hat y,-\hat y,\hat z,-\hat z$.
In this lattice picture the wavefunction lives on sites. The vector and scalar potentials live on bonds, and the electric and magnetic fields live on the center of the plaquettes. The z-component of the magnetic field is related to the adding up the potentials as you go around a plaquette in the x-y plane, while the z-component of the electric field comes from adding potentials as you go around in the $t-z$ plane.
