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\subsubsection*{3.3.2 Beam interpretation}

The idea: ignore normalisation problem and take
$\chi_k = e^{ikx}$ as eigenfunction of $\hat{H}$.
Take
\[ \chi_k(x) = A e^{ikx} \quad A \in \CC \]
\[ \psi_k(x, t) = A e^{ikx} e^{-i \frac{\hbar^2
k^2}{2m}t} \]
but instead of $\chi_n(x)$ describing a single
particle they describe a beam of particles with
\[ p_k = \hbar k \]
\[ E_k = \frac{\hbar^2 k^2}{2m} \]
with probability density
\[ \rho_k(x, t) = |A|^2 \]
representing constant average density of
particles. Compute probability current
\[ j_k(x, t) = -\frac{i\hbar}{2m} \left( \psi_k^*
\pfrac{\psi}{x} - \psi_k \pfrac{\psi_k^*}{x} \right) \]
\[ \left[ \pfrac{\rho}{t} + \pfrac{j}{x} = 0 \right] \]
(lecture 3)
In this case taking ($*$)
\[ j_k(x, t) = |A|^2 \frac{\hbar^k}{m} = |A|^2
\frac{p}{m} = \text{average flux of particles} \]

\subsection{Scattering states}

What happens if we have an unbound potential
$U(x)$ and throw a particle on it
\begin{center}
    \includegraphics[width=0.6\linewidth]
    {images/5c4cc83e5f6111ed.png}
\end{center}
\begin{definition*}
    Probability for particle to be reflected is
    given by the reflection coefficient
    \[ R = \lim_{t \to \infty} \int{-\infty}^0
    |\psi_{\text{GP}}(x, t)|^2 \dd x \]
\end{definition*}

\begin{definition*}
    Probability for particle to be transmitted is
    given by the transmission coefficient
    \[ T = \lim_{t \to \infty} \int_0^\infty
    |\psi_{\text{GP}}(x, t)|^2 \dd x \]
\end{definition*}

Clearly $T + R = 1$. Solving scattering problems
using beam interpretation gives some results for
$R$ and $T$, so we will use it.

\subsubsection*{3.4.1 Scattering off potential
step}

\[ U(x) = \begin{cases}
    0 & x \le 0 \\
    U_0 & x > 0
\end{cases} \quad U_0 \in \RR^+ \]
\begin{center}
    \includegraphics[width=0.6\linewidth]
    {images/eb3bc0045f6111ed.png}
\end{center}
To find $\chi_k(x)$, solve TISE
\[ -\frac{\hbar^2}{2m}\chi''_n(x) + U(x) \chi_n(x)
= E \chi_n(x) \]
Region I, $x \le 0$, $U(x) = 0$.
\[ \chi''_n(x) + k^2 \chi_n(x) = 0 \quad k =
\sqrt{\frac{2mE}{\hbar^2}} > 0 \]
\[ \chi_n(x) = A e^{ikx} + B e^{-ikx} \]
($A$ part is the beam of incident particles, $B$
part is the beam of reflected particles). \\
Region II, $x > 0$, $U(x) = U_0$.
\[ \chi''_{\ol{k}}(x) + \ol{k}^2 \chi_{\ol{k}}(x)
= 0 \]
\[ \ol{k} = \sqrt{\frac{2m(E - U_0)}{\hbar^2}} \]
$\ol{k}$ real for $E \ge U_0$, and imaginary for
$E < U_0$.
\begin{itemize}
    \item For $E \ge U_0$,
        \[ \alpha_{\ol{k}}(x) = C e^{i\ol{k}x} + D
        e^{-i\ol{k}x} \]
        (the $C$ term is the transmitted beam, and
        the $D$ term is the incident beam from
        $\infty$). $D = 0$ due to initial
        condition.
    \item For $E > U_0$,
        \[ \chi_{\ol{k}}(x) = Ce^{-\eta x} + D
        e^{\eta x} \]
        where $\eta = \sqrt{\frac{2m(U_0 -
        E)}{\hbar^2}}$. $D = 0$ otherwise
        $\chi_{\ol{k}}$ diverges at $\infty$.
\end{itemize}
Putting I and II:
\[ \chi_{n, \ol{k}}(x) = \begin{cases}
    A e^{in x} + B e^{-inx} & x \le 0 \\
    C e^{i\ol{k} x} & x > 0
\end{cases} \]
Impose continuity of $\chi(x), \chi'(x)$ at $x =
0$ and get
\[ A + B = C \]
\[ ikA - ikB = i\ol{k}C \]
\[ \implies B = \frac{k - \ol{k}}{k + \ol{k}} A \]
\[ C = \frac{2k}{k + \ol{k}}A \]
We can view these in terms of particle flux
\[ J(x, t = -\frac{i\hbar}{2m} \left( \chi^*
\pfrac{\chi}{x} - \chi \pfrac{\chi^*}{x} \right) \]
Compute for
\begin{itemize}
    \item $E > U_0$
        \[ J(x, t) = \begin{cases}
            \frac{\hbar k}{m} (|A|^2 - |B|^2) & x
            < 0 \\
            \frac{\hbar \ol{k}}{m} |C|^2 & x \ge 0
        \end{cases} \]
        \[ J_{inc}(x, t) = \frac{\hbar x}{m} |A|^2 \]
        \[ J_{ref}(x,t) \frac{\hbar k}{m}|B|^2 \]
        \[ J_{trans}(x, t) = \frac{\hbar
        \ol{k}}{m} |C|^2 \]
        \[  R = \frac{J_{refl}}{J_{inc}} =
        \frac{|B|^2}{|A|^2} = \left( \frac{k -
        \ol{k}}{k + \ol{k}} \right)^2 \]
        \[ T = \frac{J_{trans}}{J_{inc}} =
        \frac{|C|^2}{|A|^2} \frac{\ol{k}}{k} =
        \frac{4k\ol{k}}{(k + \ol{k})^2} \]
        Interpretation:
        \begin{itemize}
            \item $R + T = 1$
            \item $E \to U_0$, $\ol{k} \to 0$, $T
                \to 0$, $R \to 1$.
            \item $E \to \infty$, $T \to 1$, $R
                \to 0$.
        \end{itemize}
    \item $E < U_0$.
        \[ J_{inc}(x, t) = \frac{\hbar k}{m} |A|^2 \]
        \[ J_{ref}(x, t) = \frac{\hbar k}{m} |B|^2 \]
        \[ J_{trans}(x, t) = 0 \]
        $R = 1$, $T = 0$ but $\chi_{\ol{k}}(x)
        \neq 0$ from $x > 0$.
\end{itemize}

\subsubsection*{Scattering off potential barrier}

\[ U(x) = \begin{cases}
    0 & x \le 0, x \ge a \\
    U_0 & 0 < x < a
\end{cases} \]
Consider $E < U_0$.
\[ k = \sqrt{\frac{2mE}{\hbar^2}} > 0 \]
\[ \eta = \sqrt{\frac{2m(U_0 - E)}{\hbar^2}} > 0 \]
Solution of TISE
\[ \chi(x) = \begin{cases}
    e^{ikx} + A e^{iikx} & x \le 0 \\
    B e^{-\eta x} + C e^{\eta x} & 0 < x < a \\
    D e^{ikx} + \ub{E e^{-ikx}}_{=0} & x \ge a
\end{cases} \]
4 free coefficients with 4 boundary conditions
given by continuity of $\chi(X)$ and $\chi'(x)$ at
$x = 0$ and $x = a$.
\[ 1 + A = B + C \]
\[ ik - ikA = -\eta B + \eta C \]
\[ B e^{-\eta a} + C e^{\eta a} = D e^{ika} \]
\[ -\eta B e^{-\eta a} + \eta C e^{\eta a} = ikD
e^{ika} \]
Find
\[ D = -\frac{4\eta k}{(\eta - ik)^2 \exp[(\eta +
ik)a] - (\eta + ik)^2 \exp[-(\eta - ik) a]} \]
\[ \implies T = |D|^2 = 4k^2 \eta^2 \]
Take limit $U_0 \gg E \implies \eta a \gg 1$
\[ T \to \frac{16k^2 \eta^2}{(\eta^2 + k^2)^2}
\ub{e^{-2m
a}}_{e^{-\frac{2a}{\hbar}\sqrt{2m(U_0 - E)}}} \]
