8.2: Infinite Spherical Potential Well
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Consider a particle of mass \(m\) and energy \(E>0\) moving in the following simple central potential:
\[V(r) = \left\{\begin{array}{lcl} 0&\,&\mbox{for $0\leq r\leq a$}\\[0.5ex] \infty&&\mbox{otherwise} \end{array}\right..\]
Clearly, the wavefunction \(\psi\) is only non-zero in the region \(0\leq r \leq a\). Within this region, it is subject to the physical boundary conditions that it be well behaved (i.e., square-integrable) at \(r=0\), and that it be zero at \(r=a\). (See Section [s5.2].) Writing the wavefunction in the standard form
\[\label{e9.27} \psi(r,\theta,\phi) = R_{n,l}(r)\,Y_{l,m}(\theta,\phi),\]
we deduce (see the previous section) that the radial function \(R_{n,l}(r)\) satisfies
\[\frac{d^{\,2} R_{n,l}}{dr^{\,2}} + \frac{2}{r}\frac{dR_{n,l}}{dr} + \left[k^{\,2} - \frac{l\,(l+1)}{r^{\,2}}\right] R_{n,l} = 0\] in the region \(0\leq r \leq a\), where
\[\label{e9.29} k^{\,2} = \frac{2\,m\,E}{\hbar^{\,2}}.\]Defining the scaled radial variable \(z=k\,r\), the previous differential equation can
be transformed into the standard form
\[\frac{d^{\,2} R_{n,l}}{dz^{\,2}} + \frac{2}{z}\frac{dR_{n,l}}{dz} + \left[1 - \frac{l\,(l+1
)}{z^{\,2}}\right] R_{n,l} = 0.\]
The two independent solutions to this well-known second-order differential equation are called spherical Bessel functions, and can be written
\[\begin{aligned} j_l(z)&= z^{\,l}\left(-\frac{1}{z}\frac{d}{dz}\right)^l\left(\frac{\sin z}{z}\right),\\[0.5ex] y_l(z)&= -z^{\,l}\left(-\frac{1}{z}\frac{d}{dz}\right)^l\left(\frac{\cos z}{z}\right).\end{aligned}\]
Thus, the first few spherical Bessel functions take the form \[\begin{aligned} j_0(z) &= \frac{\sin z}{z},\\[0.5ex] j_1(z)&=\frac{\sin z}{z^{\,2}} - \frac{\cos z}{z},\\[0.5ex] y_0(z) &= - \frac{\cos z}{z},\\[0.5ex] y_1(z) &= - \frac{\cos z}{z^{\,2}} - \frac{\sin z}{z}.\end{aligned}\]
These functions are also plotted in Figure [sph]. It can be seen that the spherical Bessel functions are oscillatory in nature, passing through zero many times. However, the \(y_l(z)\) functions are badly behaved (i.e., they are not square integrable) at \(z=0\), whereas the \(j_l(z)\) functions are well behaved everywhere. It follows from our boundary condition at \(r=0\) that the \(y_l(z)\) are unphysical, and that the radial wavefunction \(R_{n,l}(r)\) is thus proportional to \(j_l(k\,r)\) only. In order to satisfy the boundary condition at \(r=a\) [i.e., \(R_{n,l}(a)=0\)], the value of \(k\) must be chosen such that \(z=k\,a\) corresponds to one of the zeros of \(j_l(z)\). Let us denote the \(n\)th zero of \(j_l(z)\) as \(z_{n,l}\). It follows that
\[k\,a = z_{n,l},\] for \(n=1,2,3,\ldots\). Hence, from Equation ([e9.29]), the allowed energy levels are \[\label{e9.39} E_{n,l} = z_{n,l}^{\,2}\,\frac{\hbar^{\,2}}{2\,m\,a^{\,2}}.\] The first few values of \(z_{n,l}\) are listed in Table [tsph]. It can be seen that \(z_{n,l}\) is an increasing function of both \(n\) and \(l\).
\(n=1\) | \(n=2\) | \(n=3\) | \(n=4\) | |
---|---|---|---|---|
\(l=0\) | 3.142 | 6.283 | 9.425 | 12.566 |
[0.5ex] \(l=1\) | 4.493 | 7.725 | 10.904 | 14.066 |
[0.5ex] \(l=2\) | 5.763 | 9.095 | 12.323 | 15.515 |
[0.5ex] \(l=3\) | 6.988 | 10.417 | 13.698 | 16.924 |
[0.5ex] \(l=4\) | 8.183 | 11.705 | 15.040 | 18.301 |
We are now in a position to interpret the three quantum numbers— \(n\), \(l\), and \(m\)—which determine the form of the wavefunction specified in Equation ([e9.27]). As is clear from Chapter [sorb], the azimuthal quantum number \(m\) determines the number of nodes in the wavefunction as the azimuthal angle \(\phi\) varies between 0 and \(2\pi\). Thus, \(m=0\) corresponds to no nodes, \(m=1\) to a single node, \(m=2\) to two nodes, et cetera. Likewise, the polar quantum number \(l\) determines the number of nodes in the wavefunction as the polar angle \(\theta\) varies between 0 and \(\pi\). Again, \(l=0\) corresponds to no nodes, \(l=1\) to a single node, et cetera. Finally, the radial quantum number \(n\) determines the number of nodes in the wavefunction as the radial variable \(r\) varies between 0 and \(a\) (not counting any nodes at \(r=0\) or \(r=a\)). Thus, \(n=1\) corresponds to no nodes, \(n=2\) to a single node, \(n=3\) to two nodes, et cetera. Note that, for the case of an infinite potential well, the only restrictions on the values that the various quantum numbers can take are that \(n\) must be a positive integer, \(l\) must be a non-negative integer, and \(m\) must be an integer lying between \(-l\) and \(l\). Note, further, that the allowed energy levels ([e9.39]) only depend on the values of the quantum numbers \(n\) and \(l\). Finally, it is easily demonstrated that the spherical Bessel functions are mutually orthogonal: that is, \[\int_0^a j_l(z_{n,l}\,r/a)\,j_{l}(z_{n',l}\,r/a) \,r^{\,2}\,dr = 0\] when \(n\neq n'\) . Given that the \(Y_{l,m}(\theta,\phi)\) are mutually orthogonal (see Chapter [sorb]), this ensures that wavefunctions ([e9.27]) corresponding to distinct sets of values of the quantum numbers \(n\), \(l\), and \(m\) are mutually orthogonal.
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
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