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\exnumber{5666} 
\heading{A particle in a two dimensional box with a potential step}
\auname{Shir Ezra, Liad Shamir}

\textbf{The problem:}

\Dn

An electron with mass \textbf{\textit{M}} is in a two dimensional box $\vert x \vert<a$, $ \vert y \vert<b$.
Inside the box there is a step potential $V(x)=\dfrac{1}{2}V_0\cdot sgn(x)$    .\\
In this question you need to consider spin-orbit interaction. Units: $c=1$, $e=1$, $g\approx2$.
\\
\begin{enumerate}
  \item Write the Hamiltonian using the position and momentum operators and the spin operator $\sigma_z$.
  \item Write the unperturbed eigenstates $\vert n,m\rangle$ in the standard basis. 
  \item Write an expression for the matrix elements $\langle n,m\vert V(x)\vert 1,1\rangle$.
  \item Calculate the second order correction of the ground state energy which arise from the electrostatic interaction.
  \item Which states are not effected by the spin-orbit interaction?
  \item Repeat section 3 for spin-orbit interaction.
  \item Repeat section 4 for spin-orbit interaction. 
\end{enumerate}
TIP: Use the following symbols:
\\
$\mathrm{SIN}_\nu(\theta)=cos(\nu\theta) \quad \text{for} \quad \nu=1,3,\hdots$
\\
$\mathrm{SIN}_\nu(\theta)=sin(\nu\theta) \quad \text{for} \quad \nu=2,4,\hdots $
\\
$c_\nu=\int_{0}^{\pi\setminus2}sin(\nu\theta)cos(\theta)d\theta$
\\
\\
The spin state $\sigma_z$ is a "good quantum number"- you should only address it in your answer to section 6.
\\
The answers to sections 4 and 7 are series, clearly state the summation index.  



\Dn\Dn

\textbf{The solution:}

\Dn 
\begin{enumerate}
	\item \begin{align*}
	\mathcal{H}=\dfrac{1}{2M}\hat{p}^2+V(\hat{x})-\dfrac{1}{2M^2}(g-1)(\vec{\mathcal{E}}\times\vec{p})\vec{S}=\dfrac{1}{2M}\hat{p}^2+\dfrac{1}{2}V_0\cdot  \mathrm{sgn}(\hat{x})+\dfrac{V_0}{4M^2}\hat{p_y}\cdot\delta(\hat{x})\cdot\hat{\sigma_z}
	\end{align*}
	\item Boundary conditions: \begin{align*}\Psi(x=\pm a)=\Psi(y=\pm b)=0
	\end{align*}
	\begin{align*}\mid n,m\rangle=\dfrac{1}{\sqrt{ab}}\cdot  \mathrm{SIN}_n\left(\dfrac{\pi x}{2a}\right)\cdot  \mathrm{SIN}_m\left(\dfrac{\pi y}{2b}\right)
	\end{align*}
	eigenvalues: \begin{align*}\mathcal{E}_{n,m}=\dfrac{1}{2M}\left[\left(\dfrac{\pi n}{2a}\right)^2+\left(\dfrac{\pi m}{2b}\right)^2\right], \quad \text{for}\quad V_0=0
	\end{align*}
	\item \begin{align*}\langle n,m\vert V(x)\vert 1,1\rangle=\dfrac{V_0}{2ab}\int\limits_{-a}^{a} \mathrm{SIN}_n\left(\dfrac{\pi x}{2a}\right)cos\left(\dfrac{\pi x}{2a}\right)  \mathrm{sgn}(x)dx\cdot \int\limits_{-b}^{b} \mathrm{SIN}_m\left(\dfrac{\pi y}{2b}\right)cos\left(\dfrac{\pi y}{2b}\right) dy=
	\end{align*}
	\begin{align*}=\dfrac{2\delta_{1,m}\cdot V_0\cdot c_n}{\pi}, \quad	\text{for} \quad n=2,4,\hdots
\end{align*}
	\item \begin{align*} E_{11}^{[2]}=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty} \dfrac{(\vert\langle n,m\mid V\mid 1,1\rangle\vert)^2}{E_{11}-E_{n,m}}=-\dfrac{32MV_0^2a^2}{\pi^4}\sum_{n=1}^{\infty}\dfrac{c_{2n}^2}{4n^2-1} 
	\end{align*}
	\item The spin-orbit term of the Hamiltonian is:
	\begin{align*}\mathcal{H}_{SO}=\dfrac{V_0}{4M^2}\hat{p_y}\cdot\delta(\hat{x})\cdot\hat{\sigma_z}
	 \end{align*} 
	 Thus the eigenstates for which $\Psi(x=0)=0$ \hspace{0.01cm} are unaffected by the delta function. This condition applies only to even n.
	 
	\item \begin{align*}\langle n,m\vert \mathcal{H}_{SO}\vert 1,1\rangle=\dfrac{-iV_0\cdot\sigma_z}{4M^2ab}\int\limits_{-a}^{a} \mathrm{SIN}_n\left(\dfrac{\pi x}{2a}\right)cos\left(\dfrac{\pi x}{2a}\right) \delta(x) dx\cdot \int\limits_{-b}^{b} \mathrm{SIN}_m\left(\dfrac{\pi y}{2b}\right)\frac{\partial}{\partial y}cos\left(\dfrac{\pi y}{2b}\right)dy=
	\end{align*}
	\begin{align*}=\dfrac{iV_0\cdot\sigma_z\cdot c_m}{2M^2ab}  \quad	\text{for} \quad m=2,4,\hdots\hspace{0.1cm} ;\hspace{0.1cm} n=1,3,\hdots
	\end{align*}
	\item \begin{align*}E_{11}^{[2]}=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty} \dfrac{(\vert\langle n,m\mid \mathcal{H}_{SO}\mid 1,1\rangle\vert)^2}{E_{11}-E_{n,m}}=-\dfrac{2V_0^2}{M^3\pi^2a^2b^2}\sum_{n=1,3,\hdots}^{\infty}\sum_{n=2,4,\hdots}^{\infty}\dfrac{c_{m}^2}{\dfrac{n^2-1}{a^2}+\dfrac{m^2-1}{b^2}} 
	\end{align*}
	
	
	
\end{enumerate}
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