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In [2] the following vector potential

\begin{equation}\label{eqn:solenoidConstantField:20}

\BA = \frac{B \rho_a^2}{2 \rho} \phicap,

\end{equation}

is introduced in a discussion on the Aharonov-Bohm effect, for configurations where the interior field of a solenoid is either a constant \( \BB \) or zero.

I wasn’t able to make sense of this since the field I was calculating was zero for all \( \rho \ne 0 \)

\begin{equation}\label{eqn:solenoidConstantField:40}

\begin{aligned}

\BB

&= \spacegrad \cross \BA \\

&= \lr{ \rhocap \partial_\rho + \zcap \partial_z + \frac{\phicap}{\rho}

\partial_\phi } \cross \frac{B \rho_a^2}{2 \rho} \phicap \\

&= \lr{ \rhocap \partial_\rho + \frac{\phicap}{\rho} \partial_\phi } \cross

\frac{B \rho_a^2}{2 \rho} \phicap \\

&=

\frac{B \rho_a^2}{2}

\rhocap \cross \phicap \partial_\rho \lr{ \inv{\rho} }

+

\frac{B \rho_a^2}{2 \rho}

\frac{\phicap}{\rho} \cross \partial_\phi \phicap \\

&=

\frac{B \rho_a^2}{2 \rho^2} \lr{ -\zcap + \phicap \cross \partial_\phi \phicap}.

\end{aligned}

\end{equation}

Note that the \( \rho \) partial requires that \( \rho \ne 0 \). To expand the cross product in the second term let \( j = \Be_1 \Be_2 \), and expand using a Geometric Algebra representation of the unit vector

\begin{equation}\label{eqn:solenoidConstantField:60}

\begin{aligned}

\phicap \cross \partial_\phi \phicap

&=

\Be_2 e^{j \phi} \cross \lr{ \Be_2 \Be_1 \Be_2 e^{j \phi} } \\

&=

– \Be_1 \Be_2 \Be_3

\gpgradetwo{

\Be_2 e^{j \phi} (-\Be_1) e^{j \phi}

} \\

&=

\Be_1 \Be_2 \Be_3 \Be_2 \Be_1 \\

&= \Be_3 \\

&= \zcap.

\end{aligned}

\end{equation}

So, provided \( \rho \ne 0 \), \( \BB = 0 \).

The errata [1] provides the clarification, showing that a \( \rho > \rho_a \) constraint is required for this potential to produce the desired results. Continuity at \( \rho = \rho_a \) means that in the interior (or at least on the boundary) we must have one of

\begin{equation}\label{eqn:solenoidConstantField:80}

\BA = \frac{B \rho_a}{2} \phicap,

\end{equation}

or

\begin{equation}\label{eqn:solenoidConstantField:100}

\BA = \frac{B \rho}{2} \phicap.

\end{equation}

The first doesn’t work, but the second does

\begin{equation}\label{eqn:solenoidConstantField:120}

\begin{aligned}

\BB

&= \spacegrad \cross \BA \\

&= \lr{ \rhocap \partial_\rho + \zcap \partial_z + \frac{\phicap}{\rho}

\partial_\phi } \cross \frac{B \rho}{2 } \phicap \\

&=

\frac{B }{2 } \rhocap \cross \phicap

+

\frac{B \rho}{2 }

\frac{\phicap}{\rho} \cross \partial_\phi \phicap \\

&= B \zcap.

\end{aligned}

\end{equation}

So the vector potential that we want for a constant \( B \zcap \) field in the interior \( \rho < \rho_a \) of a cylindrical space, we need

\begin{equation}\label{eqn:solenoidConstantField:140}

\BA =

\left\{

\begin{array}{l l}

\frac{B \rho_a^2}{2 \rho} \phicap & \quad \mbox{if \( \rho \ge \rho_a \) } \\

\frac{B \rho}{2} \phicap & \quad \mbox{if \( \rho \le \rho_a \).}

\end{array}

\right.

\end{equation}

An example of the magnitude of potential is graphed in fig. 1.

# References

[1] Jun John Sakurai and Jim J Napolitano. \emph{Errata: Typographical Errors, Mistakes, and Comments, Modern Quantum Mechanics, 2nd Edition}, 2013. URL http://www.rpi.edu/dept/phys/Courses/PHYS6520/Spring2015/ErrataMQM.pdf.

[2] Jun John Sakurai and Jim J Napolitano. *Modern quantum mechanics*. Pearson Higher Ed, 2014.