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# $$\S 16.3$$ Solving Radial Equation

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### Radial Equation for Piece-wise Constant potentials.

The solutions of the radial equation for a constant potential are known in terms of Bessel functions. We shall list these solutions and discuss their properties before taking specific examples such as free particle, square well potential. Let us assume $V(r)=V_0$for some range of values of $r$. Then for this range of values the radial equation takes the form

$$\pp{r}\left( r^2 \pp[R]{r}\right) + \frac{2m}{\hbar^2}\left(E-V_0)- \frac{\ell(\ell+1)}{r^2} \right) R(r)=0 \label{E1}$$

We shall consider the cases $E-V_0 >0$ and $E-V_0< 0$ separately.

\subsubsection*{\underline{CASE I: $E-V_0 > 0$}}

We define $\dfrac{2m(E-V_0)}{\hbar^2}=k^2$ and the two linearly independent solutions of the radial equation are given by $j_\ell(kr)$ and $n_\ell(kr)$ known as spherical Bessel functions.Angular Momentum in  coordinate representation;  Separation of variables in polar coordinates; Solution of radial equation for a constant potential; Free particle solution in polar coordinates ;Square well and hard sphere; General properties of solutions; Isotropic oscillator in three dimension; Hydrogen atom energy levels in Schr\"odinger mechanics; Accidental degeneracy; Energy levels of positronium  and alkali Atoms These are related to the Bessel functions $J_-\nu(kr)$ as follows

\begin{eqnarray} j_\ell(kr)&=&\left(\frac{\pi}{2kr}\right)^{1/2}J_{\ell+\frac{1}{2}}(kr)\\ n_\ell(kr)&=&\left(\frac{\pi}{2kr}\right)^{1/2}(-1)^{\ell+1} J_{-\ell-\frac{1}{2}}(kr) \end{eqnarray}

and the most general solution of the radial equation is a linear combination of the above solutions.

$$R(r) = A j_\ell(kr) + B n_\ell(kr)$$

We need to know the behaviour of the solutions for $r\approx0$ and for $r \to \infty.$

\begin{description}

\item[{ Small $r$} :]

The solution $j_\ell(kr)$ goes to zero but $n_\ell(kr)$ is singular for $r\approx 0$.In fact as $\rho \to 0$, we have

\begin{eqnarray}Angular Momentum in  coordinate representation;  Separation of variables in polar coordinates; Solution of radial equation for a constant potential; Free particle solution in polar coordinates ;Square well and hard sphere; General properties of solutions; Isotropic oscillator in three dimension; Hydrogen atom energy levels in Schr\"odinger mechanics; Accidental degeneracy; Energy levels of positronium  and alkali Atoms j_\ell(\rho) &\rightarrow& \frac{\rho^\ell}{(2\ell+1)!!}\\ n_\ell(\rho) &\rightarrow& (2\ell-1)!! \rho^{-\ell-1} \end{eqnarray}

\item[{Large $r$} :]

For large $\rho$ both $j_\ell$ and $n_\ell$ are oscillatory. As $\rho \to \infty$

\begin{eqnarray} j_\ell(\rho) &\rightarrow& \frac{1}{\rho}\cos(\rho-(\ell+1)\pi/2)\\ n_\ell(\rho) &\rightarrow& \frac{1}{\rho}\sin(\rho-(\ell+1)\pi/2) \end{eqnarray}

Thus  for $E > V_0$ both $j_\ell(\rho)$ and $n_\ell(\rho)$ are acceptable solutions as $\rho \rightarrow \infty$ \end{description}

\subsubsection*{\underline{CASE II : $E-V_0 <0$}}

In this case we define

$$\frac{2m(E-V_0)}{\hbar^2} = -\alpha^2, \qquad \alpha=\mbox{\rm real}$$

In this case two linearly indepndent solutions are $j_\ell(i\alpha r)$ and $n_\ell(i \alpha r)$ and the most general solution is

$$R(r) = A j_\ell(i\alpha r) + B n_\ell(i\alpha r)$$

Again $n_\ell(i\alpha r)$ has unacceptable singular behaviour at $r=0$. To discuss large $r$ behaviour, introduce  Hankel functions of first and second kinds by

\begin{eqnarray} h^{(1)}_\ell(\rho) &=& j_\ell{\rho} + i n_\ell(\rho)\\ h^{(2)}_\ell(\rho) &=& j_\ell{\rho} - i n_\ell(\rho) \end{eqnarray}

Then, as  $\rho \rightarrow \infty$, we have

$$h^{(1)}_\ell(\rho) \rightarrow -\frac{1}{\rho}\exp(-\rho)$$

and $h^{(2)}_ell(\rho)$ blows up and becomes infinite as $\rho \to \infty.$ and is unacceptable. It may be remarked that the solutions $j_\ell(\rho)$ and $n_\ell(\rho)$ are linear combinations of $\cos \rho ,\sin\rho$ multiplied by powers of $\rho$. Similarly, $h^{(1,2)}_\ell(\rho)$ are exponentials multiplied by terms containing powers of $\rho$.

\begin{center}

{\bf Table: Forms of acceptable solutions of radial equation} \\[3mm]

\begin{tabular}{llll}

\hline && &\\

& Near $r=0$  & For large $r$ & $r$ in $(a,b)$\\

\hline &&&\\[3mm]

$E-V_0 >0$ & $j_\ell(kr)$ & $A j_\ell(kr)+ B n_\ell(kr)$  &

$A j_\ell(kr)+ B n_\ell(kr)$ \\[3mm]

$E-V_0 <0$ & $j_\ell(ikr)$ & $j_\ell(ikr) + i n_\ell(ikr)$ &

$A j_\ell(ikr)+ B n_\ell(ikr)$ \\

&& $\equiv h_\ell^{(1)}(ikr)$ & $\equiv h_\ell^{(1)}(ikr)$ \\  [3mm]

\hline

\end{tabular}

\end{center}

 {cccc} {\bf ---------------} {\bf ---------------} {\bf ---------------} \underline{$n^\prime=3$~~~~~~~} \underline{$n^\prime=3$~~~~~~~} $n^\prime=3$ {\bf ---------------} \underline{$n^\prime=2$~~~~~~~} \underline{$n^\prime=2$~~~~~~~} $n^\prime=2$ {\bf---------------} \underline{$n^\prime=1$~~~~~~~} \underline{$n^\prime=1$~~~~~~~} $n^\prime=1$ {\bf---------------} \underline{$n^\prime=0$~~~~~~~} \underline{$n^\prime=0$~~~~~~~} $n^\prime=0$ {\bf---------------} $l=0$ $l=1$ $l=2$ nondegenerate $m=-1,0,1$ $m=-2,-1,0,1,2$ 3 fold degenerate 5 fold degenerate

Some Spherical Bessel functions

We shall now tabulate first few spherical Bessel functions.

\begin{eqnarray} j_0(\rho) &=& \frac{\sin\rho}{\rho}\\ j_1(\rho) &=& \frac{\sin\rho}{\rho^2} -\frac{\cos\rho}{\rho}\\ j_2(\rho) &=& \left( \frac{3}{\rho^3}-\frac{1}{\rho}\right)\sin\rho - \frac{3}{\rho^2} \cos\rho \\ n_0(\rho) &=& -\frac{\cos\rho}{\rho}\\ n_1{\rho} &=& -\frac{\cos\rho}{\rho^2}-\frac{\sin\rho}{\rho}\\ n_2{\rho} &=& -\left(\frac{3}{\rho^3}-\frac{1}{\rho} \right)\cos\rho - \frac{3}{\rho}\sin\rho\\ h_0^{(1)}(i\rho) &=& -\frac{1}{\rho}\exp(-\rho)\\ h_1^{(1)}(i\rho) &=& i \left( \frac{1}{\rho} +\frac{1}{\rho^2} \right)\\ h_2^{(2)}(i\rho) &=& \left(\frac{1}{\rho} + \frac{3}{\rho} +\frac{3}{\rho^3} \right)\exp(-\rho) \end{eqnarray}

### Free particle solution in polar coordinates

The radial equation for a free particle, $V(r)=0$, \underline{for all} $r$ is

$$\frac{1}{r^2} \pp{r} \left(r^2\pp[R]{r}\right) + \left( k^2 - \frac{\ell(\ell+1)}{r^2}\right)R =0,$$

where $k^2 =\dfrac{2mE}{\hbar^2}$. The solution of the radial equation has the most general form

$$R(r) = A j_\ell(kr) + B n_\ell(kr)$$

but we must set $B=0$ because $n_\ell(kr) \to \infty$ as $r \to 0$. Hence we get

$$R_\ell(r) = A j_\ell(kr)$$

and the full free particle wave function is

$$\Psi(r,\theta,\phi) = N j_\ell(kr) Y_{\ell m}(\theta,\phi)$$

For a given value of energy $E$, $\ell$ can take all values $0,1,2,\ldots.$ and $m$ has $2(\ell+1)$ values from $-\ell$ to $\ell$. Therefore, for every energy value $E>0$ there are infinte number of solutions. If we take linear combinations of solutions with fixed energy $E$ we get most general form of the solution for a given energy as

$$\Phi(\vec{r}) = \sum C_{\ell m} j_\ell(kr) Y_{\ell m}(\theta,\phi)$$

In cartesian coordinates the free particle solutions for energy $E$ are plane waves $$\exp(i\vec{k}\cdot\vec{r})$$ Thus it is possible to write each of these two type of solutions as a linear combination of the other type. In particular we have

$$\exp(\vec{k}\cdot\vec{r}) = \sum C_{\ell m} j_\ell(kr) Y_{\ell m}(\theta,\phi)$$

In a particular case of this equation, when  $k$  is along the $z$ axis and $\vec{k}\cdot\vec{r}=kz$, we have the expansion of plane waves

$$\exp(ikz) = \sum_0^\infty (2\ell+1)i^\ell j_\ell(kr) P_\ell(\cos\theta)$$

Note that only $m=0$ terms contribute in the above equation.

### Hard Sphere

The potential for a rigid spherical box can be written as

$$V(r) = \begin{cases} 0 ,& 0 < r < a\\ \infty, & r >a \end{cases} \label{EQ01}.$$

The problem is separable in spherical polar coordinates and form of the full wave function is

$$\psi(r,\theta, \phi) = R(r) Y_{\ell m}(\theta, \phi) \label{EQ02}.$$

We need to consider solutions of the radial equation only. No solution can be found for $E< 0$, therefore we consider $E>0$. For $0r>a$ the potential is infinite and hence the radial wave function must be zero, Next we consider $r< a$, where the potential is zero. The radial  equation assumes the form

$$-\frac{1}{r^2}\dd{r} r^2 \dd[R(r)]{r} + \frac{\ell(\ell+1)\hbar^2}{2mr^2} R(r) - E R(r) =0. \label{EQ03}$$

The most general solution of this equation  is given in terms of spherical Bessel functions $j_\ell, n_\ell$ and we write it as

$$R_{E\ell(r)} = A j_\ell(kr) + Bn_\ell(kr), \qquad k^2 = \frac{2mE}{\hbar^2} \label{EQ04}.$$

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 $r=0$. Thus we get

$$R_\ell(r) = \begin{cases} A j_\ell(kr), & 0<r < a\\ 0 & r > a \end{cases}.\label{EQ05}$$

Next we must demand that the radial wave function $R(r)$ must be continuous at $r=0$ Remember that there is no corresponding requirement on the derivative for this case of infinite jump in the potential at $r=a$  The continuity requirement of $R_{E\ell}(r)$ becomes

$$j_\ell(ka) = 0. \label{EQ06}$$

The solutions of the above equation determine allowed values of $k$ and hence allowed bound state energies.

Energy levels and degeneracy

To get all the solutions, one proceeds as follows. First set $\ell=0$ and locate the roots  of $j_0(ka)=0$. We call the roots as $\rho_{0n}, n=0,1,2,\ldots$ and the corresponding energies are given by $E=\frac{\hbar^2\rho_{0n}^2}{2ma^2}$.. Here $n$ denotes the number of nodes of the radial wave function for $\ell=0$. Next, we set $\ell=1$ and find the roots of $j_1(kr)=0$, calling these roots as $\rho_{1n}, n=0,1,2,\ldots$ the $\ell=1$ energy levels are given by $E=\frac{\hbar^2\rho_{1n}^2}{2ma^2}$. This process is to be repeated for all values of angular momentum $\ell$ and the number of bound states for each $\ell$ turns out to be infinite. The states of definite energy  depend on quantum numbers $n\ell m$ 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 $N_{n\ell} R_{n\ell}(r/\rho_{n\ell})Y_{\ell m}(\theta,\phi), (m=-\ell, -\ell+1,\cdots, \ell)$ and the energy levels $E_{n\ell}$ are $(2\ell+1)$ fold degenerate. The energy increases with $\ell$ and also with increasing $n$. Thus schematic energy level diagram would appear as follows.

\vskip0.1cm

\begin{tabular}{cccc}
&&&{\bf ---------------}\\
&  & {\bf ---------------}\\
& {\bf ---------------}\\
&&& \underline{$n^\prime=3$~~~~~~~}\\
&& \underline{$n^\prime=3$~~~~~~~}\\
$n^\prime=3$ & {\bf ---------------}\\
&&& \underline{$n^\prime=2$~~~~~~~}\\
&& \underline{$n^\prime=2$~~~~~~~}\\
$n^\prime=2$ & {\bf---------------}\\
&&& \underline{$n^\prime=1$~~~~~~~}\\
&& \underline{$n^\prime=1$~~~~~~~}\\
$n^\prime=1$ & {\bf---------------}\\
&&& \underline{$n^\prime=0$~~~~~~~}\\
&& \underline{$n^\prime=0$~~~~~~~}\\
$n^\prime=0$ & {\bf---------------}\\
& $l=0$        & $l=1$ & $l=2$\\
&nondegenerate & $m=-1,0,1$        & $m=-2,-1,0,1,2$\\
&              & 3 fold degenerate & 5 fold degenerate\\
\end{tabular}

### Spherically Symmetric Square Well

The potential for a spherical well can be written as

$$V(r) = \begin{cases} -V_0 ,& 0 < r < a, \qquad V_0>0,\\ 0, & r >a \end{cases} \label{EQ01}.$$

The problem is separable in spherical polar coordinates and form of the full wave function is

$$\psi(r,\theta, \phi) = R(r) Y_{\ell m}(\theta, \phi) \label{EQ02}.$$

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

\begin{AlphaList1} \item $-V_0 < E < 0$. This case corresponds to bound states, \item $E> 0$. In this case the there is no bound state. This case is of interest for      scattering from the potential. \end{AlphaList1}

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> 0 ,\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  r> 0 .\label{EQ04} \end{eqnarray}

where

$$q^2=\frac{2m(E+V_0)}{\hbar^2}, \quad \alpha^2=\frac{2m|E|}{\hbar^2}. \label{EQ10A}$$

The most general solution of this equation  is given in terms of spherical Bessel functions $j_\ell, n_\ell$ and is given by

$$R(r) = \begin{cases} A j_\ell(qr) + Bn_\ell(qr), & r<a\\ C h^{(1)}(\alpha r) + D h^{(2)}(\alpha r) & r > a \end{cases} \label{EQ05}.$$

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 $D=$. Thus we get

$$R_\ell(r) = \begin{cases} A j_\ell(qr), & 0<r < a\\ C h^{(1)}_\ell(\alpha r) & r > a \end{cases}.\label{EQ06}$$

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

$$\frac{1}{j_\ell(qr)} \dd[j_\ell(qr)]{r}\Big|_{r=a} = \frac{1}{h_\ell(qr)}\dd[h^{(1)}_\ell(\alpha r)]{r}\Big|_{r=a}. \label{EQ09}.$$

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

$$q^2=\frac{2m(E+V_0)}{\hbar^2}, \quad k^2=\frac{2mE}{\hbar^2}. \label{EQ10}$$

A most general form of the solution of the radial equation is given by

$$R_\ell(r) = \begin{cases} A j_\ell(qr) + Bn_\ell(qr), & r<a\\ C j_\ell(kr) + Bn_\ell(kr), & r<a \end{cases}.$$

In order that the radial wave function be finite at $r=0$, we must set $B=0$. This gives

$$R_\ell(r) = \begin{cases} A j_\ell(qr), & r<a\\ C j_\ell(kr) + Bn_\ell(kr), & r<a \end{cases}.$$

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.

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