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Infinite Spherical Potential Well

Consider a particle of mass $m$ and energy $E>0$ moving in the following simple central potential:
\begin{displaymath}
V(r) = \left\{\begin{array}{lcl}
0&\mbox{\hspace{1cm}}&\mbox...
...leq a$}\ [0.5ex]
\infty&&\mbox{otherwise}
\end{array}\right..
\end{displaymath} (647)

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 Sect. 5.2). Writing the wavefunction in the standard form
\begin{displaymath}
\psi(r,\theta,\phi) = R_{n,l}(r) Y_{l,m}(\theta,\phi),
\end{displaymath} (648)

we deduce (see previous section) that the radial function $R_{n,l}(r)$ satisfies
\begin{displaymath}
\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
\end{displaymath} (649)

in the region $0\leq r \leq a$, where
\begin{displaymath}
k^2 = \frac{2 m E}{\hbar^2}.
\end{displaymath} (650)

Figure 20: The first few spherical Bessel functions. The solid, short-dashed, long-dashed, and dot-dashed curves show $j_0(z)$, $j_1(z)$, $y_0(z)$, and $y_1(z)$, respectively.
\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter09/fig01.eps}}
\end{figure}

Defining the scaled radial variable $z=k r$, the above differential equation can be transformed into the standard form

\begin{displaymath}
\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.
\end{displaymath} (651)

The two independent solutions to this well-known second-order differential equation are called spherical Bessel functions,[*]and can be written
$\displaystyle j_l(z)$ $\textstyle =$ $\displaystyle z^l\left(-\frac{1}{z}\frac{d}{dz}\right)^l\left(\frac{\sin z}{z}\right),$ (652)
$\displaystyle y_l(z)$ $\textstyle =$ $\displaystyle -z^l\left(-\frac{1}{z}\frac{d}{dz}\right)^l\left(\frac{\cos z}{z}\right).$ (653)

Thus, the first few spherical Bessel functions take the form
$\displaystyle j_0(z)$ $\textstyle =$ $\displaystyle \frac{\sin z}{z},$ (654)
$\displaystyle j_1(z)$ $\textstyle =$ $\displaystyle \frac{\sin z}{z^2} - \frac{\cos z}{z},$ (655)
$\displaystyle y_0(z)$ $\textstyle =$ $\displaystyle - \frac{\cos z}{z},$ (656)
$\displaystyle y_1(z)$ $\textstyle =$ $\displaystyle - \frac{\cos z}{z^2} - \frac{\sin z}{z}.$ (657)

These functions are also plotted in Fig. 20. 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
\begin{displaymath}
k a = z_{n,l},
\end{displaymath} (658)

for $n=1,2,3,\ldots$. Hence, from (650), the allowed energy levels are
\begin{displaymath}
E_{n,l} = z_{n,l}^{ 2} \frac{\hbar^2}{2 m a^2}.
\end{displaymath} (659)

The first few values of $z_{n,l}$ are listed in Table 1. It can be seen that $z_{n,l}$ is an increasing function of both $n$ and $l$.


Table 1: The first few zeros of the spherical Bessel function $j_l(z)$.
  $n=1$ $n=2$ $n=3$ $n=4$
$l=0$ 3.142 6.283 9.425 12.566
$l=1$ 4.493 7.725 10.904 14.066
$l=2$ 5.763 9.095 12.323 15.515
$l=3$ 6.988 10.417 13.698 16.924
$l=4$ 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 Eq. (648). As is clear from Sect. 8, 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, etc. 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, etc. 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, etc. 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 (659) 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: i.e.,

\begin{displaymath}
\int_0^a j_l(z_{n,l} r/a) j_{l}(z_{n',l} r/a)  r^2 dr = 0
\end{displaymath} (660)

when $n\neq n'$. Given that the $Y_{l,m}(\theta,\phi)$ are mutually orthogonal (see Sect. 8), this ensures that wavefunctions (648) corresponding to distinct sets of values of the quantum numbers $n$, $l$, and $m$ are mutually orthogonal.


next up previous
Next: Hydrogen Atom Up: Central Potentials Previous: Derivation of Radial Equation
Richard Fitzpatrick 2010-07-20