Subsections


A.6 Constant spherical potentials

This addendum describes the solutions of the Hamiltonian eigenvalue problem in spherical coordinates if the potential is constant.

These solutions are important for many reasons. For example, you might want to create a simplified model for the hydrogen atom that way. To do so, you could, for example, assume that the potential energy has a constant negative value up to say the Bohr radius and is zero beyond it. That is not really a very good model for the hydrogen atom. However, it works much better for nucleons in atomic nuclei, chapter 14.12.

The solutions in this note are also important for describing experiments in which particles are scattered from some target, {A.30}. And more fundamentally, they give the energy states of definite angular momentum for particles in empty space.


A.6.1 The eigenvalue problem

The Hamiltonian eigenvalue problem is

\begin{displaymath}
- \frac{\hbar^2}{2m} \nabla^2 \psi + V \psi = E \psi
\end{displaymath}

In this note it is assumed that the potential $V$ is a constant in the radial region of interest.

To clean the problem up a bit, take the potential energy term to the other side, and also the coefficient of the Laplacian. That produces

\begin{displaymath}
- \nabla^2 \psi = \frac{p_{\rm {c}}^2}{\hbar^2} \psi
\end{displaymath}

where

\begin{displaymath}
p_{\rm {c}} \equiv \sqrt{2m(E-V)}
\end{displaymath}

The constant $p_{\rm {c}}$ is what classical physics would take to be the linear momentum of a particle with total energy $E$ and potential energy $V$.


A.6.2 The eigenfunctions

Because the potential is spherically symmetric like for the hydrogen atom, the eigenfunctions are of similar form:

\begin{displaymath}
\fbox{$\displaystyle
\psi_{Elm}(r,\theta,\phi) = R_{El}(r) Y_l^m(\theta,\phi)
$} %
\end{displaymath} (A.17)

Here the $Y_l^m$ are again the spherical harmonics. These eigenfunctions have definite square angular momentum $l(l+1)\hbar^2$ where $l$ is an nonnegative integer. They also have definite angular momentum in the chosen $z$-​direction equal to $m\hbar$, where $m$ is an integer that satisfies $\vert m\vert$ $\raisebox{-.3pt}{$\leqslant$}$ $l$.

The radial functions $R_{El}$ in the eigenfunctions $\psi_{Elm}$ are different from those of the hydrogen atom. Depending on whatever is easiest in a given application, they can be written in two ways, {D.16}.

The first way is as

\begin{displaymath}
\fbox{$\displaystyle
R_{El} = c_{\rm{s}} j_l(p_{\rm{c}}r...
...{c}}r/\hbar)
\qquad p_{\rm{c}} \equiv \sqrt{2m(E-V)}
$} %
\end{displaymath} (A.18)

Here $c_{\rm {s}}$ and $c_{\rm {c}}$ are arbitrary constants. The functions $j_l$ and $n_l$ are called the “spherical Bessel functions” of the first and second kinds. The $n_l$ are also called the “Neumann functions” and might instead be indicated by $y_l$ or $\eta_l$.

Expressions for these Bessel functions can be found in advanced mathematical handbooks, [1]:

\begin{displaymath}
\fbox{$\displaystyle
j_l(x)
= (-x)^l \left(\frac{1}{x} \frac{{\rm d}}{{\rm d}x}\right)^l \frac{\sin x}{x}
$} %
\end{displaymath} (A.19)


\begin{displaymath}
\fbox{$\displaystyle
n_l(x)
= - (-x)^l \left(\frac{1}{x} \frac{{\rm d}}{{\rm d}x}\right)^l\frac{\cos x}{x}
$} %
\end{displaymath} (A.20)

The spherical Bessel functions are often convenient in a region of constant potential that includes the origin, because the Bessel functions of the first kind $j_l$ give the solutions that are finite at the origin. (To see that, note that the Taylor series of $\sin{x}$ divided by $x$ is a power series in $x^2$, and that $x{\rm d}{x}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\frac12{\rm d}{x}^2$.) In particular for small $x$:

\begin{displaymath}
\fbox{$\displaystyle
j_l(x)
= \frac{2^l l!}{(2l+1)!}x^l + O(x^{l+2})
\equiv \frac{x^l}{(2l+1)!!} + O(x^{l+2})
$} %
\end{displaymath} (A.21)

Here !! is one of these unfortunate notations. The second ! means that all even factors are dropped from the factorial.

The Bessel functions of the second kind are singular at the origin and normally do not appear if the origin is part of the considered region.

Also, the spherical Bessel functions are real for real $x$. However, in a region where the potential $V$ is larger than the energy $E$ of the particles, the argument of the Bessel functions in (A.18) will be imaginary.

The other way to write the radial functions is as

\begin{displaymath}
\fbox{$\displaystyle
R_{El} = c_{\rm{f}} h_l^{(1)}(p_{\r...
...{c}}r/\hbar)
\qquad p_{\rm{c}} \equiv \sqrt{2m(E-V)}
$} %
\end{displaymath}

where $h_l^{(1)}$ and $h_l^{(2)}$ are called the “spherical Hankel functions.”

The spherical Hankel functions can again be found in advanced mathematical handbooks, [1]:

\begin{displaymath}
\fbox{$\displaystyle
h_l^{(1)}(x) = - {\rm i}(-x)^l\left...
...ght)^l
\frac{e^{{\rm i}x}}{x} = j_l(x)+{\rm i}n_l(x)
$} %
\end{displaymath} (A.22)


\begin{displaymath}
\fbox{$\displaystyle
h_l^{(2)}(x) = {\rm i}(-x)^l \left(...
...ht)^l
\frac{e^{-{\rm i}x}}{x} = j_l(x)-{\rm i}n_l(x)
$} %
\end{displaymath} (A.23)

The given expressions in terms of the spherical Bessel functions are readily inverted to give the Bessel functions in terms of the Hankel functions,
\begin{displaymath}
j_l(x) = \frac{h_l^{(1)}(x)+h_l^{(2)}(x)}{2}
\qquad
n_l(x) = \frac{h_l^{(1)}(x)-h_l^{(2)}(x)}{2{\rm i}} %
\end{displaymath} (A.24)

For large $x$ the spherical Hankel functions can be approximated as

\begin{displaymath}
\fbox{$\displaystyle
h_l^{(1)}(x) \sim (-i)^{l+1} \frac{...
...ad
h_l^{(2)}(x) \sim i^{l+1} \frac{e^{-{\rm i}x}}{x}
$} %
\end{displaymath} (A.25)

This asymptotic behavior tends to make the Hankel functions more convenient far from the origin. Exponentials are mathematically simpler and more fundamental than the sines and cosines in the asymptotic behavior of the Bessel functions.


A.6.3 About free space solutions

The most important case for which the energy eigenfunctions of the previous subsection apply is for particles in empty space. They describe energy states with definite square and $z$ angular momentum. However, sometimes particles in empty space are better described by states of definite linear momentum. And in some cases, like in scattering problems, you need both types of solution. Then you also need to convert between them.

The energy states in empty space with definite square and $z$ angular momentum are

\begin{displaymath}
\fbox{$\displaystyle
\psi_{Elm} = j_l(p_{\rm{c}}r/\hbar) Y_l^m(\theta,\phi)
\qquad p_{\rm{c}} \equiv \sqrt{2mE}
$} %
\end{displaymath} (A.26)

These states have square angular momentum $l(l+1)\hbar^2$ and angular momentum in the chosen $z$-​direction $m\hbar$. They are nonsingular at the origin.

A state that has definite linear momentum $p_{\rm {c}}$ purely in the $z$-​direction has an energy eigenfunction

\begin{displaymath}
\fbox{$\displaystyle
\psi_{{\vec k}} = e^{{\rm i}p_{\rm{c}} z/\hbar}
\qquad p_{\rm{c}} \equiv \sqrt{2mE}
$} %
\end{displaymath} (A.27)

This eigenfunction is not normalized, and cannot be normalized. However, neither can the eigenfunction $\psi_{Elm}$ above be. It is the curse of eigenfunctions in infinite empty space. An introduction to the adjustments that must be made to deal with this is given in chapter 7.9.

It is sometimes necessary to write a linear momentum eigenfunction of the form (A.27) in terms of angular momentum ones of the form (A.26). Rayleigh worked out the correct combination a very long time ago, {D.16}:

\begin{displaymath}
\fbox{$\displaystyle
e^{{\rm i}p_{\rm{c}} z/\hbar}
= \...
... \qquad
c_{{\rm{w}},l} = {\rm i}^l \sqrt{4\pi(2l+1)}
$} %
\end{displaymath} (A.28)

Note that only eigenfunctions with $m$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 are needed.