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# 8.2: Infinite Spherical Potential Well

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

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

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 $$l$$.

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
[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)
