Talk:PlanetPhysics/Particle in a Potential

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: particle in a potential %%% Primary Category Code: 03.65.-w %%% Filename: ParticleInAPotential.tex %%% Version: 1 %%% Owner: bloftin %%% Author(s): bloftin %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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\begin{document}

As a simple example of the calculation of eigenvalues and eigenfunctions, we shall consider the \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} of a particle in a potential well. Since the chief interest of this problem is simply that it provides an illustration of methods used in the solution of this example, we may assume a very simple dependence of the potential \htmladdnormallink{energy}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} on distance, Figure 1:

\begin{equation} V(x) =\left\{ \begin{array}{llll} V_0 & for & - \infty < x < 0 & (region \, I) \\ 0  & for & 0 < x < l &  (region \, II \\ V_0 & for & l < x < \infty &  (region \, III) \\ \end{array} \right. \end{equation}

\begin{figure}[h] \centering \includegraphics{Figure1.eps}

\centerline{Figure 1: The motion of a particle in a potential well} \label{Figure1} \end{figure}

In the potential well (region II), where $E > V = 0$, the Schr\"odinger equation takes the form

\begin{equation} \psi^{\prime \prime}_{II} - k^2 \psi_{II} = 0 \end{equation}

where

$$ \psi^{\prime \prime} = \frac{d^2\psi}{dx^2}$$

and

$$ k^2 = \frac{2m_0}{\hbar^2}E = \frac{p^2}{\hbar^2} > 0 $$

We note that the case $E < 0$ has no physical meaning in this problem. Since the general solution of Eq. (2) is oscillatory, we have

\begin{equation} \psi_{II} = B_{II} \cos(kx) +A_{II} sin(kx) \end{equation}

In regions I and III, the Schr\"odinger equation has the form

$$ \psi^{\prime \prime} + \frac{2 m_0}{\hbar^2} (E - V_0) \psi = 0 $$

Here two cases must be distinguised. In the first case $(E > V_0)$, the solution for these regions is also oscillatory in character (an equation of the elliptic \htmladdnormallink{type}{http://planetphysics.us/encyclopedia/Bijective.html}). It is given by Eq. (3), the value of $k$ being

\begin{equation} k = \frac{1}{\hbar} \sqrt{2 m_0 (E - V_0)} \end{equation}

No restrictions need to be imposed on the \htmladdnormallink{wave}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} at infinity. Therefore, the energy $E$ can assume any value in a continous \htmladdnormallink{spectrum}{http://planetphysics.us/encyclopedia/CohomologyTheoryOnCWComplexes.html} of energies. It is better, however, not to investigate the case of a contiuous spectrum on the basis of this example, but rather on the basis of the motion of a free \htmladdnormallink{particle}{http://planetphysics.us/encyclopedia/Particle.html}. The potential well only adds to the mathematical difficulties of the problem, without changing the general character of the solution.

In the second case, namely, the case of a potential barrier $(E < V_0)$, the solution of Eq. (3) is exponential in character (an equation of the hyperbolic type). The general solution can be written in the form

$$ \psi_{I, III} = A_{I, III} e^{kx} + B_{I, III} e^{-kx} $$

where

$$k^2 = \frac{1}{\hbar^2} 2 m_0 (V_0 - E) = \frac{|p|^2}{\hbar^2} > 0 $$

If the energy can assume any value without restriction, the wave function inside the potential barrier $(0 < E < V_0)$ will contain both an exponentially increasing part and an exponentally decreasing part (see Fig. 2). Therefore, we must choose only those values of $E$ for which exponentially increasing solutinos do not exist inside the potential barrier.

\begin{figure}[h] \centering \includegraphics{Figure2.eps}

\centerline{Figure 2: Wave function for a given value of $E$.} \label{Figure2} \end{figure}

Accordingly, we require that the coefficient $B_I = 0$ in region I$(x < 0)$, and the coefficient $A_{III} = 0$ in region III$(x > l)$. We then have

$$ \psi_I = A_I e^{kx} = Ae^{-k |x|} $$ \begin{equation} \psi_{III} = B_{III} e^{-kx} = B e^{-k(x-l)} \end{equation}

where, for the sake of simplicity, we have made

$$B_{III} = B e^{kl}$$

By joining the solutions at the \htmladdnormallink{boundary}{http://planetphysics.us/encyclopedia/GenericityInOpenSystems.html} of regions I and II $(x = 0)$, and also at the boundary of regions IIand III $(x = l)$, and making use of the requirement that the exponentially increasing solution vanish, we obtain the equation for the eigenvalues of the energy $E$.

We shall now further simplify our problem by requiring that $V_0$, together with $k$, go to infinity (see Fig.3). It is apparent from Eq. (5) that $ \psi_{I} = \psi_{III} = 0$, and therefore the boundary conditions for the solution (3) inside the potential well (region II) take the form

$$\psi_{II} = 0 $$

for $x = 0$, and

$$\psi_{II} = 0 $$

for $x = l$.

Applying these two equations to the general solution (3) in region II, we find that $B_{II} = 0$, and the eigenvalue are described by the equation

$$sin (kl) = 0$$

from which

$$kl = \pi n$$

where $n = 1, 2, 3, 4, ....$ We exclude the value $n = 0$ from further considerations, since the wave function in this case is identically equal to zero. It is not necessary to consider separately the negative values of $n$, since the wave functions for negative $n$ are equal to the wave functions for positive $n$, taken with the opposite sign.

\begin{figure}[h] \centering \includegraphics{Figure3.eps}

\centerline{Figure 3: Particle in a infinite potential well.} \label{Figure3} \end{figure}

Since $k^2 = \frac{2 m_0}{\hbar^2}E$, we obtain the following equation for the energy spectrum (the eigenvalues):

$$ E_n = \frac{\pi^2 \hbar^2 n^2}{2 m_0 l^2}$$

The wave functions corresponding to these values of energy (eigenfunctions) are

\begin{equation} \psi_n = A_I sin \left(\pi n \frac{x}{l} \right ) \end{equation}

The coefficient $A_n$ can be found from the normalization condition

$$ \int_0^l \psi_n^2 dx = A_n^2 \int_0^l sin^2 \left(\pi n \frac{x}{l}\right)dx = \frac{l}{2}A_n^2 = 1 $$

which gives

$$ A_n = \sqrt{\frac{2}{l}} $$

Substituting the expression for $A_n$ into Eq. (6), we finally obtain

\begin{equation} \psi_n = \sqrt{\frac{2}{l}} \sin \left(\pi n \frac{x}{l} \right) \end{equation}

Accoring to the general \htmladdnormallink{theorem}{http://planetphysics.us/encyclopedia/Formula.html} of eigenfunctions, the eigenfunctions (7) of the Schr\"odinger equation satisfy the orthogonality condition

$$ \int_0^l \psi_n^* \psi_n dx = 0 \,\,\,\,\, for \,\,\,\,\,\, n^{\prime} \ne  n $$

as can be readily seen by performing the direct integration after substituting Eq. (7) for $\psi_n$.

We shall now write down a few specific eigenvalues $E_n$ and eigenfunctions $\psi_n$, shown in Fig. 3:

\begin{equation} E_1 =\frac{\pi^2 \hbar^2}{2 m_0 l^2} \,\,\,\,\,\, \psi_1 = \sqrt{\frac{2}{l}} \sin \left(\frac{\pi x}{l} \right) \end{equation}

\begin{equation} E_2 =4E_1 \,\,\,\,\,\, \psi_2 = \sqrt{\frac{2}{l}} \sin \left(2\pi\frac{x}{l} \right) \end{equation}

\begin{equation} E_3 = 9 E_1 \,\,\,\,\,\, \psi_3 = \sqrt{\frac{2}{l}} \sin \left( 3 \pi \frac{x}{l} \right) \end{equation}

These solutions are very similar to the familiar standing-wave solutions for a vibrating string with fixed ends. The case $n=1$ (8) corresponds to the fundamental mode, the case $n=2$ (9), to the first harmonic, etc.


 * derivative of the Public domain work of [Sokolov].

\begin{thebibliography}{9} \bibitem{Sokolov} Sokolov, Quantum Mechanics \htmladdnormallink{Internet Archive }{http://www.archive.org/details/quantummeshanics032665mbp} \end{thebibliography}

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