Pairs of Power Potentials in the Schrödinger Equation

It is well known that there is a simple transformation of variable that converts the Schrödinger equation for a simple harmonic oscillator into that for a hydrogen atom and vice versa. It is perhaps less well known that a similar transformation exists linking pairs of power potentials with continuous real exponent. Presentation of this transformation is simple yet instructive.

The n-dimensional Schrödinger equation for the radial part of a wave function in a spherically symmetric power potential with coupling constant a is

$[\frac{d^{2}}{d r^{2}} + \frac{(n - 1)}{r} \frac{d}{d r} - \frac{l (l + n - 2)}{r^{2}} + \frac{2 m}{ℏ^{2}} [E - a r^{k}]] R (r) = 0$

Other symbols here have their expected quantum mechanical meanings. A derivation of this form can be found in this presentation. The exponent k of the power potential is allowed to be an arbitrary real number, both positive and negative.

The constant coefficients on the second and third terms of this equation are meant to have physical significance. The constant on the second term indicates the dimensionality of the space, and the constant on the third term is a corresponding eigenvalue of a Gegenbauer polynomial in that space. The transformation between pairs of potentials will alter these constants, so that they will not have their expected connection to physical space afterwards.

Under the transformation $u = r^{q}$ the Schrödinger equation becomes

$[\frac{d^{2}}{d u^{2}} + \frac{(n + q - 2)}{q u} \frac{d}{d u} - \frac{l (l + n - 2)}{q^{2} u^{2}} + \frac{2 m}{ℏ^{2} q^{2}} u^{2 (1 - q) / q} [E - a u^{k / q}]] R (u) = 0$

What is remarkable here is that the second and third terms of the equation have the same functional dependence with the new independent variable, albeit with slightly altered constants as just noted. The value of the exponent in the transformation can now be chosen to remove the radial dependence of the original potential:

$\frac{2 (1 - q) + k}{q} = 0 \to q = \frac{k + 2}{2}$

Under the transformation $u = r^{(k + 2) / 2}$ the Schrödinger equation for the radial part of the wave function can be written

$[\frac{d^{2}}{d u^{2}} + \frac{ν}{u} \frac{d}{d u} - \frac{λ^{2}}{u^{2}} + \frac{2 m}{ℏ^{2}} [ε - α u^{κ}]] R (u) = 0$

where the coefficients are now functions of the original exponent of the power potential,

$\begin{array}{c} ν = \frac{(2 n + k - 2)}{(k + 2)} λ^{2} = \frac{4 l (l + n - 2)}{(k + 2)^{2}} \\ ε = - \frac{4 a}{(k + 2)^{2}} α = - \frac{4 E}{(k + 2)^{2}} \end{array}$

as is the new exponent of the power potential:

$κ = - \frac{2 k}{k + 2}$

The transformation essentially interchanges energy and coupling constant. The change of sign under this interchange is physically correct, taking an inverse attractive potential with states of total negative energy to an attractive positive-exponent potential of total positive energy, and vice versa. The other coefficients lose their connection to the physical description of the n-dimensional space, so that the transformation should be viewed as more of a mathematical than physical relationship.

Letting the original exponent k = −1 for the hydrogen atom produces a simple harmonic oscillator in the transformed equation, and letting k = 2 for a simple harmonic oscillator leads to the hydrogen atom in the transformed equation. This well-known transformation pair is clearly a single instance of the general set of pairs in a continuous real exponent.

The relation between the pairs of exponents is one branch of a hyperbola with an asymptote at k = −2, and this correlates with a previous presentation on square-integrable wave functions. Such wave functions for a power potential with positive exponent can be put in a one-to-one correspondence with wave functions for inverse power potentials not less negative than an inverse squared potential. The inverse squared potential then corresponds to a positive exponent of infinity. The other branch of the hyperbola links pairs of nonnormalizable wave functions for power potentials with negative exponents.

The transformation between the pairs of potentials is a simple replacement of variable, implying that the wave functions for the pairs will have similar behavior apart from the independent variable. This is born out in the well-known case: the radial wave function for the hydrogen atom consists of a decreasing exponential multiplying a Laguerre polynomial and a power of the radial variable. The wave function for the simple harmonic oscillator consists of a decreasing Gaussian, or an exponential with squared argument, multiplying a Laguerre polynomial with squared argument and a power of the radial variable. The only apparent difference is that the two wave functions have the same overall power of the radial variable, and this is due to a difference in angular quantum numbers between the two cases.

Other wave function pairs will have much more complicated radial behavior, in general not expressible in terms of known functions. They will both, however, have the same general functional behavior, taking into account the arguments related by the transformation.

There is a classical analog to this correspondence of wave function pairs, expressible in terms of the classical radial action. This quantity is defined by the integral

$J_{r} = \oint d r p_{r} = \oint d r \sqrt{2 m (E - a r^{k}) - \frac{L^{2}}{r^{2}}}$

over one cycle of the classical system. Under the transformation $u = r^{(k + 2) / 2}$ of the variable of integration, the radial action is

$J_{r} = \oint d u \sqrt{2 m [\frac{4 E}{(k + 2)^{2}} u^{- 2 k / (k + 2)} - \frac{4 a}{(k + 2)^{2}}] - \frac{4 L^{2}}{(k + 2)^{2} u^{2}}}$

One can see immediately the same interchange of energy and coupling constant with the same coefficients. The angular momentum term displays the same behavior as for the quantum mechanical case, apart from the absence of quantum numbers for the classical system.

Uploaded 2013.08.16 — Updated 2023.07.15 analyticphysics.com