4.1: Infinite Potential Well

Consider a particle of mass \(m\) and energy \(E\) moving in the following simple potential: \[V(x) = \left\{\begin{array}{lcl} 0&\hspace{1cm}&\mbox{for }0\leq x\leq a\\[0.5ex] \infty&&\mbox{otherwise} \end{array}\right..\] It follows from Equation ([e5.2]) that if \(d^{\,2}\psi/d x^{\,2}\) (and, hence, \(\psi\)) is to remain finite then \(\psi\) must go to zero in regions where the potential is infinite. Hence, \(\psi=0\) in the regions \(x\leq 0\) and \(x\geq a\). Evidently, the problem is equivalent to that of a particle trapped in a one-dimensional box of length \(a\). The boundary conditions on \(\psi\) in the region \(0<x<a\) are \[\label{e5.4} \psi(0) = \psi(a) = 0.\] Furthermore, it follows from Equation ([e5.2]) that \(\psi\) satisfies \[\label{e5.5} \frac{d^{\,2} \psi}{d x^{\,2}} = - k^{\,2}\,\psi\] in this region, where \[\label{e5.6} k^{\,2} = \frac{2\,m\,E}{\hbar^{\,2}}.\] Here, we are assuming that \(E>0\). It is easily demonstrated that there are no solutions with \(E<0\) which are capable of satisfying the boundary conditions ([e5.4]).

The solution to Equation ([e5.5]), subject to the boundary conditions ([e5.4]), is \[\psi_n(x) = A_n\,\sin(k_n\,x),\] where the \(A_n\) are arbitrary (real) constants, and

\[\label{e5.8} k_n = \frac{n\,\pi}{a},\] for \(n=1,2,3,\cdots\). Now, it can be seen from Equations ([e5.6]) and ([e5.8]) that the energy \(E\) is only allowed to take certain discrete values: that is, \[\label{eenergy} E_n = \frac{n^{\,2}\,\pi^{\,2}\,\hbar^{\,2}}{2\,m\,a^{\,2}}.\]

In other words, the eigenvalues of the energy operator are discrete. This is a general feature of bounded solutions: that is, solutions for which \(|\psi|\rightarrow 0\) as \(|x|\rightarrow\infty\). According to the discussion in Section [sstat], we expect the stationary eigenfunctions \(\psi_n(x)\) to satisfy the orthonormality constraint

\[\int_0^a \psi_n(x)\,\psi_m(x)\,dx = \delta_{nm}.\] It is easily demonstrated that this is the case, provided \(A_n = \sqrt{2/a}\). Hence,

\[\label{e5.11} \psi_n(x) = \sqrt{\frac{2}{a}}\,\sin\left(n\,\pi\,\frac{x}{a}\right)\] for \(n=1,2,3,\cdots\).

Finally, again from Section [sstat], the general time-dependent solution can be written as a linear superposition of stationary solutions:

\[\psi(x,t) = \sum_{n=0,\infty} c_n\,\psi_n(x)\,{\rm e}^{-{\rm i}\,E_n\,t/\hbar},\] where

\[\label{e5.13} c_n = \int_0^a\psi_n(x)\,\psi(x,0)\,dx.\]

Contributors and Attributions

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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