[NOTES/QM-16008] Spherically Symmetric Square Well

The potential for a spherical well can be written as \begin{equation} V(r) = \begin{cases} -V_0 ,& 0 < r < a, \qquad V_0>0,\\ 0, & r >a \end{cases} \label{eq01}. \end{equation} The problem is separable in spherical polar coordinates and form of the full wave function is \begin{equation} \psi(r,\theta, \phi) = R(r) Y_{\ell m}(\theta, \phi) \label{eq02}. \end{equation} We need to consider solutions of the radial equation only. No solution can be found for $E< -V_0$, therefore we consider $E>-V_0$. We shall consider two cases of

• $-V_0 < E < 0$. This case corresponds to bound states,
• $E> 0$. In this case the there is no bound state. This case is of interest for scattering from the potential.

 Bound states

The bound states correspond to $-V_0<E< 0$. The radial equation in regions $r<a$ and $r>a$ assumes the forms \begin{eqnarray} \frac{1}{r^2}\dd{r} r^2 \dd[R(r)]{r} +\Big(q^2 -\frac{\ell(\ell+1)}{r^2} \Big) R(r) &=&0,\qquad r> a ,\label{eq03}\\ \frac{1}{r^2}\dd{r} r^2 \dd[R(r)]{r} +\Big( -\alpha^2+ \frac{\ell(\ell+1)}{r^2}\Big) R(r) &=&0, \qquad 0< r < a .\label{eq04} \end{eqnarray} where \begin{equation} q^2=\frac{2m(E+V_0)}{\hbar^2}, \quad \alpha^2=\frac{2m|E|}{\hbar^2}. \label{eq10A} \end{equation} The most general solution of this equation is given in terms of spherical Bessel functions $j_\ell, n_\ell$ and is given by \begin{equation} R(r) =\begin{cases} A j_\ell(qr) + Bn_\ell(qr), & 0<r<a\\ Ch^{(1)}(\alpha r) + D h^{(2)}(\alpha r) & r > a \end{cases} \label{eq05}.\end{equation}Recall that near $r=0$, $n_\ell(r) \sim r^{-\ell-1}$ and blows up as $r\to 0$. Therefore we must set $B=0$ if the solution is to remain finite at $r=0$. Also as $r\to\infty$ the Hankel function $h^{(2)}(\alpha r)$ increases exponentially, hence we must set $D=0$. Thus we get \begin{equation} R_\ell(r) = \begin{cases} A j_\ell(qr), & 0 a \end{cases}.\label{eq06} \end{equation} Next we must demand that the radial wave function $R(r)$ and its derivative must be continuous at $r=0$. These continuity requirements become give the following restrictions of the coefficients $A,C$. \begin{eqnarray} A j_\ell(qa) &=& C h^{(1)}_\ell(\alpha a). \label{eq07}\\ A \dd[j_\ell(qr)]{r}\Big|_{r=a} &=& C\dd[h^{(1)}_\ell(\alpha r)]{r}\Big|_{r=a}. \label{eq08} \end{eqnarray} Noting that $A,C$ cannot be zero and eliminating $A$ and $C$ we get condition on the bound state energy to be \begin{equation} \frac{1}{j_\ell(qr)} \dd[j_\ell(qr)]{r}\Big|_{r=a} = \frac{1}{h_\ell(qr)}\dd[h^{(1)}_\ell(\alpha r)]{r}\Big|_{r=a}. \label{eq09}. \end{equation} The above equation can be solved numerically to obtain allowed values energies. In this case of square well only a finite number of states exist for a given $\ell$ below a maximum value. In general there will be no bound state for $\ell$ greater that a certain values. The states of definite energy depend on quantum number $n\ell$ and the energy does not depend on magnetic quantum number $m$. Therefore for a given azimuthal quantum number $\ell$ we have $(2\ell+1)$ wave functions $N_{n\ell} R_{n\ell}(\rho)Y_{\ell m}(\theta,\phi)$ and the energy levels $E_{n\ell}$ are $(2\ell+1)$ fold degenerate. The energy increases with $\ell$ and also with increasing $n$. Thus energy level diagram would appear as follows.

Continuous energy solutions

The energy levels for $E>0$ are continuous. We shall write the corresponding solutions which are important for discussion of scattering from a square well. When $E>0$ we define \begin{equation} q^2=\frac{2m(E+V_0)}{\hbar^2}, \quad k^2=\frac{2mE}{\hbar^2}. \label{eq10} \end{equation} A most general form of the solution of the radial equation is given by \begin{equation} R_\ell(r) = \begin{cases} A j_\ell(qr) + Bn_\ell(qr), & r<a\\ C j_\ell(kr) + Bn_\ell(kr), & r>a \end{cases}. \end{equation}In order that the radial wave function be finite at $r=0$, we must set $B=0$. This gives \begin{equation} R_\ell(r) = \begin{cases} A j_\ell(qr), & r<a\\ c="" j_\ell(kr)="" +="" bn_\ell(kr),="" &="" r="">a \end{cases}. \end{equation} Next we demand continuity of the radial wave function and its derivative at $r=a$ and get \begin{eqnarray} A j_\ell(qa) &=& C j_\ell(ka) + Bn_\ell(ka)\\ A \dd{r}j_\ell(qr)\Big|_{r=a} &=& C \dd{r}j_\ell(kr)\Big|_{r=a} + B\dd{r} n_\ell(kr)\Big|_{r=a} \end{eqnarray} These two equations constrain the three constants $A,B,C$ and determine their ratios, the overall normalization constant remains, as expected, undetermined. For a given energy $E$ there is solution for each $\ell=0,1,2,...$ and $m= -\ell,...,\ell$ giving rise to infinite degeneracy for $E>0$. These continuous energy solutions will be required for physical applications to scattering problems.