The quantization of energy characteristic of quantum systems is too often presented as if it only occurs for certain mathematically simple systems. One can of course quantize any nicely behaved system: one just cannot always write down a simple expression for the quantized energy levels and numerical integration becomes necessary.

This presentation will demonstrate graphically what quantization means in general for spherically symmetric potentials in an arbitrary number of spatial dimensions greater than one. With this condition the angular behavior of the wave function can be separated from the radial behavior. The *n*-dimensional Schrödinger equation for the radial part of the wave function is

$[\frac{{d}^{2}}{d{r}^{2}}+\frac{n-1}{r}\frac{d}{dr}-\frac{l(l+n-2)}{{r}^{2}}+2[E-V(r\left)\right]\phantom{\rule{.2em}{0ex}}]R\left(r\right)=0$

where $m=\hslash =1$ and other symbols here have their expected quantum mechanical meanings. A derivation of this form can be found in this presentation.

Letting $R\sim {r}^{k}$ in the first three terms leads to the indicial equation

$\begin{array}{c}k(k-1)+(n-1)k-l(l+n-2)=0\\ k=l\phantom{\rule{.5em}{0ex}},\phantom{\rule{.5em}{0ex}}-(l+n-2)\end{array}$

so that a regular solution approaches the origin like a power of the radial variable. Partly in order to regularize the behavior for zero and nonzero angular momentum, the substitution

$R\left(r\right)=\frac{u\left(r\right)}{r}$

is often made, leading to the differential equation

$[\frac{{d}^{2}}{d{r}^{2}}+\frac{n-3}{r}\frac{d}{dr}-\frac{n-3+l(l+n-2)}{{r}^{2}}+2[E-V(r\left)\right]\phantom{\rule{.2em}{0ex}}]u\left(r\right)=0$

This radial function has the benefit of a uniform initial condition $u\left(0\right)=0$ for all angular momenta. The second initial condition can be taken as ${u}^{\prime}\left(0\right)=1$ since the actual value merely alters the normalization of the wave function but not the qualitative behavior. This form of the equation also lacks a first derivative term for $n=3$ , which allows application of Numerov’s method in its integration.

Unfortunately this form of the differential equation also has a rather nasty singularity at the origin that requires special attention to avoid numerical inaccuracy. For this reason it is preferable to make instead the substitution

$R\left(r\right)={r}^{l}f\left(r\right)$

which removes the angular momentum term entirely:

$[\frac{{d}^{2}}{d{r}^{2}}+\frac{n-1+2l}{r}\frac{d}{dr}+2[E-V(r\left)\right]\phantom{\rule{.2em}{0ex}}]f\left(r\right)=0$

The initial condition on this radial function is now $f\left(0\right)=1$ for all angular momenta, where again the normalization of the wave function is irrelevant for the qualitative behavior to be visualized. For potentials that are regular at the origin, the second initial condition can be taken as ${f}^{\prime}\left(0\right)=0$ to obviate the singularity of the first-derivative term. The numerical integration is then straightforward to perform for potentials nonsingular at the origin.

As a first application, consider one of the canonically simple cases solved in every quantum mechanics textbook: the simple harmonic oscillator with unit coupling constant in three dimensions. The radial equation is in this case

$[\frac{{d}^{2}}{d{r}^{2}}+\frac{2(l+1)}{r}\frac{d}{dr}+2E-{r}^{2}\phantom{\rule{.2em}{0ex}}]f\left(r\right)=0$

It can be shown that the quantized energy levels have the extremely simple behavior

$E=2q+l+\frac{3}{2}$

for some integer *q*. In order to keep total probability finite, an energy eigenfunction must decrease identically to zero as the radial variable increases. A numerical integration with finite precision will not show this exact behavior, but what one can expect is the so-called ‘wagging of the tail’ as the wave function changes sign for large radii. This is the tell-tale indication of the existence of an energy eigenvalue and can be seen quite easily in this interactive graphic:

As the value of the energy is manipulated, the wave function flips from one side of the vertical axis to the other when passing through an eigenvalue. Incremental changes around the eigenvalue allow one to confirm that the behavior of the graphic is consistent with the exact eigenvalue expression. The numerical integration is fast enough that one can observe the dynamic development of radial nodes of the wave function by holding down a cursor key in the input box for this slider.

An even better way to see the overall behavior across energy values is to integrate the differential equation for consecutive values and connect them into a wave function surface as a function of energy and radial variable for the system. The step size in energy is deliberately large so as not to overload the browser excessively. The result looks like this:

The vertical portions of the surface mark where the energy passes through an eigenvalue at equally spaced intervals. The first vertical portion moves out the energy axis as the angular momentum is increased, consistent again with the exact expression for eigenvalues.

For an arbitrary number of spatial dimensions, the radial differential equation is

$[\frac{{d}^{2}}{d{r}^{2}}+\frac{n-1+2l}{r}\frac{d}{dr}+2E-{r}^{2}\phantom{\rule{.2em}{0ex}}]f\left(r\right)=0$

and it can be shown that the exact expression of for eigenvalues is modified to

$E=2q+l+\frac{n}{2}$

again for some integer *q*. The required modification of the numerical integration is trivial, with only a single coefficient changed. One can even allow all input values to become continuous and the resulting interactive behavior remains consistent with the exact expression:

The separation of angular behavior that leads to the radial equation is of course predicated on integral dimensions and angular momenta, so allowing continuous dimensions and angular momenta is speculative. It is intriguing to see, however, that radial quantization survives intact with the loosening of the meaning of input parameters.

One can repeat the integration for consecutive values of energy to display a wave function surface for the system as an analytic function of continuous dimension and angular momentum,

which again displays the vertical portions indicative of energy eigenvalues. While this continues to be a soluble modification of the simple harmonic oscillator, the essence of quantization begins to appear.

Modify the system further by allowing variation in the potential of the system. Consider a power potential with an exponent of arbitrary positive value:

$V\left(r\right)=\frac{1}{2}{r}^{k}\phantom{\rule{1em}{0ex}},\phantom{\rule{1em}{0ex}}k>0$

The radial differential equation is now

$[\frac{{d}^{2}}{d{r}^{2}}+\frac{n-1+2l}{r}\frac{d}{dr}+2E-{r}^{k}\phantom{\rule{.2em}{0ex}}]f\left(r\right)=0$

and is no longer exactly soluble in general. While it is possible to derive an expression for the energy eigenvalues as a function of exponent using WKB theory, the result would be rather cumbersome and detract from the graphic presentation.

The required modification of the numerical integration is again trivial, but the result is not. Individual radial wave functions look like this,

and the wave function surface for consecutive energy values is

Again one can readily see the ‘wagging of the tail’ passing through eigenvalues, as well as the corresponding vertical portions on the wave function surface. This is the essence of quantization: the clear existence of energy eigenvalues for general conditions beyond simple soluble systems.

As a final modification, consider letting the exponent in the potential take on negative values greater than minus two. The coupling constant and energy must now both be taken as negative numbers for bound states,

$V\left(r\right)=-\frac{1}{2}{r}^{-\left|k\right|}\phantom{\rule{1em}{0ex}},\phantom{\rule{1em}{0ex}}-2<k<0\phantom{\rule{1em}{0ex}};\phantom{\rule{1em}{0ex}}E<0$

so that the radial differential equation becomes

$[\frac{{d}^{2}}{d{r}^{2}}+\frac{n-1+2l}{r}\frac{d}{dr}-2\left|E\right|+{r}^{-\left|k\right|}\phantom{\rule{.2em}{0ex}}]f\left(r\right)=0$

Applying a transformation connecting pairs of power potentials, change the independent variable to

$\rho ={r}^{(k+2)/2}$

The differential equation then has the form

$[\frac{{d}^{2}}{d{\rho}^{2}}+\frac{\alpha}{\rho}\frac{d}{d\rho}+\beta -\gamma {\rho}^{\kappa}]f\left(\rho \right)=0$

with the coefficients and exponent

$\begin{array}{l}\alpha =\frac{2n-2+4l+k}{k+2}\\ \beta =\frac{4}{(k+2{)}^{2}}\\ \gamma =\frac{8\left|E\right|}{(k+2{)}^{2}}\end{array}\phantom{\rule{4em}{0ex}}\kappa =-\frac{2k}{k+2}=\frac{2\left|k\right|}{2-\left|k\right|}>0$

This equation now has a potential that is explicitly finite at the origin, so the same initial conditions as used previously can be applied for the transformed radial variable. For $k=-1$ and the given coupling constant, it can be shown that the exact eigenvalues are

$E=-\frac{1}{8[q+l+\frac{n-1}{2}{]}^{2}}$

which can be used as a check on the validity of the method. Since most of the eigenvalues occur close to zero energy, a logarithmic scaling of the input is employed and the scan for eigenvalues should proceed in the reverse direction compared to previous graphics. Individual radial wave functions now look like this:

For consistency with previous graphics, the wave function surface will be presented without rescaling of the energy axis. This means that only the first couple vertical portions indicating eigenvalues will be visible, but the intent of the graphic should be clear:

The entire procedure can of course be repeated for other spherically symmetric potentials. For those regular at the origin the integration presents no problems. For those singular at the origin and for which no coordinate transformation is available to remove the singularity, the integration can begin displaced slightly from the origin without major disruption of the qualitative behavior.

For a general radial Schrödinger equation, the asymptotic value of the wave function will transition from positive to negative or vice versa as the energy value is successively altered. The points where it flips through zero determine the energy eigenvalues that allow normalization of the wave function. The is the mathematical essence of quantization.

*Uploaded 2018.12.07 — Updated 2023.07.15*
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