The Schrödinger Equation for Power Potentials

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

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

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 purpose of this note will be to understand when square-integrable wave functions are possible based upon an analysis of the singular points of the equation. This will rely upon a discussion on this website of the s-rank multisymbol of an equation and its implications for solutions. Square-integrable wave functions are those than can be normalized, a necessary feature for interpreting the wave function in terms of probability.

First consider positive integral values of the exponent of the power potential. With the abbreviations

$e = \frac{2 m E}{ℏ^{2}} a = \frac{2 m α}{ℏ^{2}}$

rewrite the Schrödinger equation without denominators of powers of the radial variable:

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

This equation has a regular singular point at the origin and an irregular singular point at infinity, and its s-rank multisymbol is ${1; 2 + \frac{k}{2}}$ . The singularity structure of the equation is unchanged if the angular momentum term is eliminated by setting l = 0 .

The behavior of the radial part of the wave function at the origin is determined by the standard indicial equation

$\begin{array}{c} λ (λ - 1) + (n - 1) λ - l (l + n - 2) = 0 \\ λ = l, - (l + n - 2) \end{array}$

so that a solution regular at the origin can always be found with the first choice for the leading power of the radial variable.

The leading behavior of the radial part of the wave function at infinity is given by an exponential of a power of the radial variable one less than the s-rank of the singular point. With the appropriate substitution, it is simple to determine that the wave function at infinity has the form

$R (r) = exp [\pm \frac{\sqrt{a}}{1 + \frac{k}{2}} r^{1 + k / 2}] f (r)$

For k = 2 one immediately has the Gaussian decay of the standard solution for the simple harmonic oscillator. The coefficient accompanying the radial variable is here proportional to the square root of the coupling constant, as also occurs for the simple harmonic oscillator.

The appearance of a negative sign in the form at infinity indicates that one can always find a wave function that is square integrable. Elimination of the solution with the positive sign is the entire purpose of quantification: only particular values of the energy will correspond to decaying wave functions. Determination of these eigenvalues is in general not simple, but can always be approximated by numerically integrating the differential equation while varying the energy parameter until a decaying function is found within desired limits of accuracy.

It should be noted that the coupling constant α must be positive for a square-integrable result. A negative value would lead to the imaginary exponential of traveling waves that are not normalizable. The positive value of the coupling constant translates to positive total energy for these wave functions.

Now consider positive rational values of the exponent of the power potential. In order to apply the rule for evaluating the s-rank value of the singular point at infinity, the equation must be transformed so that all powers of the independent variable are integral. For k = i / p with these two latter quantities integral, a change of variable $r = u^{p}$ leads to the Schrödinger equation

$u^{2} \frac{d^{2} R}{d u^{2}} + [p (n - 2) + 1] u \frac{d R}{d u} - p^{2} [l (l + n - 2) - e u^{2 p} + a u^{i + 2 p}] R = 0$

The s-rank multisymbol is now ${1; 1 + p + \frac{i}{2}}$ , again unchanged if l = 0 . The behavior of the radial part of the wave function at the origin in terms of the new independent variable is determined by the indicial equation

$\begin{array}{c} λ (λ - 1) + [p (n - 2) + 1] λ - p^{2} l (l + n - 2) = 0 \\ λ = p l, - p (l + n - 2) \end{array}$

so that in terms of the original radial variable it is unchanged. Likewise, the behavior of the radial part of the wave function at infinity has the form

$R (u) = exp [\pm \frac{p \sqrt{a}}{p + \frac{i}{2}} u^{p + i / 2}] f (u)$

which also is unchanged in terms of the original radial variable. Since any real number can be approximated as closely as desired by a rational number, the same form can be extended to continuous real values of the exponent. A square-integrable wave function can thus always be found for a power potential with positive real exponent.

Now consider negative integral values of the exponent of the power potential. With $k = - κ$ , the Schrödinger equation without denominators of powers of the radial variable is

$r^{κ + 2} \frac{d^{2} R}{d r^{2}} + (n - 1) r^{κ + 1} \frac{d R}{d r} - [l (l + n - 2) r^{κ} - e r^{κ + 2} + a r^{2}] R = 0$

The equation now has an irregular singular point at infinity with a constant s-rank of two. That means that the behavior of the radial part of the wave function at infinity is given by a single exponentiated power of the radial variable, or an exponential function. It is straightforward to determine that

$R (r) = exp [\pm \sqrt{- e} r] f (r)$

This exponential appears in the standard solution for the hydrogen atom, and it is clear that it applies to all inverse power potentials. Only systems with negative energy have the possibility of square-integrable wave functions, with positive energy leading to nonnormalizable traveling waves representing particle scattering. Negative energy implies the usual negative coupling constant.

Note that the constant here is proportional to the square root of energy for inverse power potentials, as opposed to the square root of the coupling constant for positive values of the exponent of the power potential. This feature is related to a well-known mapping that turns the hydrogen atom into a simple harmonic oscillator while swapping energy and coupling constant.

The value of the s-rank of the singular point at the origin is determined by the smallest power of the radial variable there, and so depends upon the value of the exponent of the power potential. For κ ≤ 2 the multisymbol is ${1; 2}$ , but when κ > 2 the multisymbol is ${\frac{κ}{2}; 2}$ . The former multisymbol is that for the confluent hypergeometric equation, which is why Laguerre polynomials, as a case of the confluent hypergeometric function, appear in the standard solution for the hydrogen atom. These singularity structures are likewise unchanged if the angular momentum term is eliminated by setting l = 0 .

The behavior of the radial part of the wave function at the origin for κ < 2 is determined by the same indicial equation as for the potential with positive exponent,

$\begin{array}{c} λ (λ - 1) + (n - 1) λ - l (l + n - 2) = 0 \\ λ = l, - (l + n - 2) \end{array}$

so that again a solution regular at the origin can be found with the first choice for the leading power of the radial variable. A square-integrable wave function can then be determined by appropriate selection of energy values that lead to decaying exponential radial behavior at infinity.

For κ = 2 the Schrödinger equation is

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

The energy can be removed by rescaling the radial variable with a factor of the square root thereof, leaving nothing to quantize. This is mathematically identical to what happens for κ = 0,

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

where the combination of the energy and coupling constant can be removed by rescaling the radial variable with a square root thereof. This case makes physical sense: if there is no potential then there can be no quantization, apart from the mathematically expedient trick of imagining an infinite square box filling the cosmos. What is interesting is that an inverse square potential behaves as if there is no potential at all, and one cannot find square-integrable wave functions for either of these two cases.

For κ > 2 the singular point in the Schrödinger equation at the origin becomes irregular. Solutions to differential equations with finite singular points always contain inverse powers of the independent variable at the singular point. In this case one cannot construct a solution regular at the origin, and there can be no square-integrable wave functions for inverse power potentials with κ > 2.

Now consider negative rational values of the exponent of the power potential, again transforming the equation so that all powers of the independent variable are integral. For κ = i / p with these two latter quantities integral, a change of variable $r = u^{p}$ leads to the Schrödinger equation

$u^{i + 2} \frac{d^{2} R}{d u^{2}} + [p (n - 2) + 1] u^{i + 1} \frac{d R}{d u} - p^{2} [l (l + n - 2) u^{i} - e u^{i + 2 p} + a u^{2 p}] R = 0$

For i < 2p the s-rank multisymbol of the transformed equation is ${1; 1 + p}$ . The behavior of the radial part of the wave function at the origin in terms of the new independent variable is determined by the indicial equation

$\begin{array}{c} λ (λ - 1) + [p (n - 2) + 1] λ - p^{2} l (l + n - 2) = 0 \\ λ = p l, - p (l + n - 2) \end{array}$

so that in terms of the original radial variable it is unchanged. Likewise, the behavior of the radial part of the wave function at infinity has the form

$R (u) = exp [\pm \sqrt{- e} u^{p}] f (u)$

which also is unchanged in terms of the original radial variable. Since any real number can be approximated as closely as desired by a rational number, the same form can be extended to continuous real values of the exponent. Since i < 2p corresponds to κ < 2, a square-integrable wave function can thus always be found for a power potential with negative real exponent less negative than an inverse squared power.

For i > 2p the s-rank multisymbol of the transformed equation is ${1 + \frac{i - 2 p}{2}; 1 + p}$ . Unsurprisingly, the situation in this case has not changed either. The irregular singular point at infinity still leads to an exponential in the original radial variable, and the irregular singular point at the origin precludes any solution regular there. There can be no square-integrable wave function for an inverse power potential with negative real exponent more negative than an inverse squared power.

As with a positive exponent, the singularity structure of the transformed equation for a negative exponent does not change in either case if l = 0 . The presence or absence of the angular momentum term in the Schrödinger equation does not have any bearing upon the possibility of square-integrable wave functions.

In summary, square-integrable wave functions are possible for any positive real exponent, apart from zero, and any negative real exponent greater than minus two. This follows from the singularity structure of the Schrödinger equation alone and not from explicit solutions.

Uploaded 2013.08.06 — Updated 2023.07.15 analyticphysics.com