Infinite Spherical Potential Well

Consider a particle of mass $m$ and energy $E>0$ moving in the following simple central potential:

$$V(r) = \left\{\begin{array}{lcl} 0&\mbox{\hspace{1cm}}&\mbox... ...leq a$}\ [0.5ex] \infty&&\mbox{otherwise} \end{array}\right..$$ (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

$$\psi(r,\theta,\phi) = R_{n,l}(r) Y_{l,m}(\theta,\phi),$$ (648)

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

$$\frac{d^2 R_{n,l}}{dr^2} + \frac{2}{r}\frac{dR_{n,l}}{dr} + \left(k^2 - \frac{l (l+1)}{r^2}\right) R_{n,l} = 0$$ (649)

in the region $0\leq r \leq a$, where

$$k^2 = \frac{2 m E}{\hbar^2}.$$ (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.

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

$$\frac{d^2 R_{n,l}}{dz^2} + \frac{2}{z}\frac{dR_{n,l}}{dz} + \left[1 - \frac{l (l+1)}{z^2}\right] R_{n,l} = 0.$$ (651)

The two independent solutions to this well-known second-order differential equation are called spherical Bessel functions,and can be written

$$j_l(z) = z^l\left(-\frac{1}{z}\frac{d}{dz}\right)^l\left(\frac{\sin z}{z}\right),$$ (652)

$$y_l(z) = -z^l\left(-\frac{1}{z}\frac{d}{dz}\right)^l\left(\frac{\cos z}{z}\right).$$ (653)

Thus, the first few spherical Bessel functions take the form

$$j_0(z) = \frac{\sin z}{z},$$ (654)

$$j_1(z) = \frac{\sin z}{z^2} - \frac{\cos z}{z},$$ (655)

$$y_0(z) = - \frac{\cos z}{z},$$ (656)

$$y_1(z) = - \frac{\cos z}{z^2} - \frac{\sin z}{z}.$$ (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

$$k a = z_{n,l},$$ (658)

for $n=1,2,3,\ldots$. Hence, from (650), the allowed energy levels are

$$E_{n,l} = z_{n,l}^{ 2} \frac{\hbar^2}{2 m a^2}.$$ (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.,

$$\int_0^a j_l(z_{n,l} r/a) j_{l}(z_{n',l} r/a) r^2 dr = 0$$ (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.