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The \dv{(\Delta x(t))}{t} = \Delta v(t = 0) = \frac{\hbar}{m} \Delta k(t = 0) \] changes proportionally to the initial impulse uncertainty. + +\subsection{Potential Step} + +We are still considering the particles from the previous example, however we introduce a potential step at $x = 0$. This means we are considering the potential +\[ + \phi(x) = \begin{cases} + 0 & x < 0 \\ + \phi_0 & x \ge 0 + \end{cases} +\] +This means the particles are still moving in direction $+x$ and are free ($\epot = 0$) if their position is $x < 0$. However at position $x = 0$ they enter into an area of higher potential $\phi(x \ge 0) = \phi_0 > 0$. +The potential energy in this area is still constant $\epot = \energy{0}$. Thus, at $x = 0$ we have a potential jump $\Delta E = \energy{0}$. +This problem has an equivalent in classic optics: a planar lightwave encountering a boundary between vacuum and material (e.g.\ a glass surface). + +We divide the domain $-\infty < x < +\infty$ into two areas I and II\@. For area I with $\epot = 0$ we still have the equation~\eqref{eq:onedschroedinger} with the solution~\eqref{eq:onedsolution} +for the location part of the wave function +\[ + \psi_{\text{I}}(x) = Ae^{ikx} + Be^{-ikx} +\] +where $A$ is the amplitude of the incidental wave, and $B$ the amplitude of the wave reflected from the potential step. + +\textbf{Note:} The complete solution is~\eqref{eq:onedsolutioncomplete}. The temporal part of the soltuion is often omitted, because it has no influence in the stationary problems considered here. + +In area II, the Schrödinger equation becomes +\begin{equation} + \dv[2]{\psi}{x} + \frac{2m}{\hbar^2}(E - \energy{0})\psi = 0 +\end{equation} +If we use the shorthand $\alpha = \sqrt{2m(\energy{0} - E)} / \hbar$ we can reduce the equation to +\begin{equation}\label{eq:potentialstep} + \dv[2]{\psi}{x} - \alpha^2\psi = 0 +\end{equation} +This equation has the solution +\begin{equation}\label{eq:potentialstepsolution} + \psi_{\text{II}} = Ce^{+\alpha x} + De^{-\alpha x} +\end{equation} +If +\[ + \psi(x) = \begin{cases} + \psi_{\text{I}} & x < 0 \\ + \psi_{\text{II}} & x \ge 0 + \end{cases} +\] +is a solution to the Schrödinger equation~\eqref{eq:potentialstep} on the entire domain $-\infty < x < +\infty$, then $\psi$ has to be continuously differentiable at every point, +or else the second derivative $\dd^2\psi / \dd x^2$ is not defined, and thus the Schrödinger equation is not applicable. +Using~\eqref{eq:onedsolution} and~\eqref{eq:potentialstepsolution} this results in the boundary conditions +\begin{subequations}\label{eq:boundaryconditions} + \begin{equation} + \begin{split} + \psi_{\text{I}}(x = 0) &= \psi_{\text{II}}(x = 0) \\ + &\implies A + B = C + D + \end{split} + \end{equation} + + \begin{equation} + \begin{split} + \eval{\dv{\psi_{\text{I}}}{x}}_0 &= \eval{\dv{\psi_{\text{II}}}{x}}_0 \\ + &\implies ik(A - B) = \alpha (C - D) + \end{split} + \end{equation} +\end{subequations} +We can now investigate the two cases where the energy $\ekin = E$ of the incoming particle is smaller or larger than the potential step. + +\subsubsection{(a) $E < \energy{0}$} +In this case, $\alpha$ is real valued and the coefficient $C$ in~\eqref{eq:potentialstepsolution} must be zero, because otherwise +\[ + \psi_{\text{II}} \xrightarrow{x \rightarrow +\infty} \pm\infty +\] +If this happens, the wave function is not normalizable. With the above boundary conditions this yields +\begin{align} + B = \frac{ik + \alpha}{ik - \alpha}A && \text{and} && D = \frac{2ik}{ik - \alpha} A +\end{align} +Thus the wave function for $x < 0$ becomes +\begin{equation} + \psi_{\text{I}}(x) = A \left[ e^{ikx} + \frac{ik + \alpha}{ik - \alpha}e^{-ikx} \right] +\end{equation} +The fraction of reflected particles is calculated as +\begin{equation} + R = \frac{\abs{Be^{-ikx}}^2}{\abs{Ae^{ikx}}^2} = \frac{\abs{B}^2}{\abs{A}^2} = \abs{\frac{ik + \alpha}{ik - \alpha}}^2 = 1 +\end{equation} +which means that \textit{all} particles are being reflected if $E < \energy{0}$. This corresponds to the expected classical behaviour or particles. +However there is a notable difference to classic particle mechanics: +\begin{tcolorbox} + The particles are not being reflected at $x = 0$, but instead penetrate the domain $x > 0$ where $\epot = \energy{0} > \ekin$ before returning, + even if their energy $\ekin < \energy{0}$ should not be enough to do so in a classical model. +\end{tcolorbox} +The probability $P(x)$ of finding a particle in $x > 0$ is +\begin{equation} + P(x) = \abs{\psi_{\text{II}}}^2 = \abs{De^{-\alpha x}}^2 = \frac{4k^2}{\alpha^2 + k^2} \abs{A}^2 e^{-2\alpha x} = \frac{4k^2}{k_0^2} \abs{A}^2 e^{-2\alpha x} +\end{equation} +where $k_0^2 = 2m\energy{0} / \hbar^2$. After a distance $x = 1/(2\alpha)$, the penetration probability is reduced to $1/e$ of its value at $x = 0$. + +We already know this result from wave optics. Even if waves are reflected totally at a boundary with refraction index $n = n' - i\kappa$, the wave penetrates the surface of the medium before returning, and the intensity +of the penetrating wave sinks to $1/e$ of its initial value after a distance $x = 1/(2k\kappa) = \lambda/(4\pi\kappa)$. +\begin{tcolorbox} + Particles with energy $E$ can penetrate into potential areas $\energy{0} > E$ with a certain probability, even if they shouldn't be able to according to classic particle physics. +\end{tcolorbox} +Once we accept that particles are described by waves, we can come to the conclusion that particles are allowed to exist in \textit{classically forbidden} locations. \end{document} \ No newline at end of file diff --git a/script.pdf b/script.pdf index c9063d332947a6d3adca8794cc8ad25f680205c8..ece4377061665b77ae62860581acc360f63a2162 100644 GIT binary patch delta 157113 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\usepackage{tikz, pgfplots} +\usepackage[parfill]{parskip} \usepackage{kbordermatrix} -\usepackage{fancyhdr} \usepackage[most]{tcolorbox} \usepackage{pdfpages} +\usepackage{fancyhdr} \usepackage[arrowdel]{physics} \usetikzlibrary{calc,trees,decorations.markings,positioning,arrows,fit,shapes,angles,patterns} \DeclareDocumentCommand\vnabla{}{\vectorarrow{\nabla}} @@ -68,6 +69,8 @@ \dominitoc \tableofcontents +\pagestyle{headings} + \hypersetup{ colorlinks, citecolor=black,