Boundary Conditions for Radial Wavefunctions

Boundary conditions for radial wavefunctions are the part of a central-potential problem most likely to be hidden by notation. The reduced radial equation resembles a one-dimensional Schrödinger equation, but it lives on the half-line, inherits the three-dimensional measure, and has a special endpoint at the origin.

The general boundary-condition philosophy is in Boundary Conditions. This page records the radial rules needed after deriving the Radial Schrödinger Equation.

The Two Radial Functions

For a separated central-potential state,

$$\psi(r,\theta,\phi) = R_\ell(r)Y_\ell^m(\theta,\phi),$$

the reduced radial wavefunction is

$$u_\ell(r)=rR_\ell(r).$$

These are not interchangeable objects. If the angular function is normalized on the sphere, then

$$\int_{\mathbb R^3} \lvert\psi\rvert^2\,d^3r = \int_0^\infty \lvert R_\ell(r)\rvert^2r^2\,dr = \int_0^\infty \lvert u_\ell(r)\rvert^2\,dr .$$

Thus $R_\ell$ is the radial factor in the three-dimensional wavefunction, while $u_\ell$ is the half-line wavefunction with ordinary $dr$ measure. Many wrong radial solutions come from normalizing one while interpreting the other.

Standard Boundary Conditions

For ordinary bound states in a nonsingular or Coulomb-like central potential, the standard radial conditions are:

• the full wavefunction is regular at the origin;
• the reduced function is square-normalizable on $0\lt r\lt \infty$;
• the bound-state solution decays at infinity;
• matching conditions are imposed at any finite discontinuities or singular shells;
• singular potentials are treated as domain questions, not by casual analogy with regular potentials.

In compact form, a standard bound state satisfies

$$\int_0^\infty \lvert u_\ell(r)\rvert^2\,dr \lt \infty, \qquad u_\ell(r)\to0 \quad \text{as} \quad r\to\infty,$$

together with the appropriate origin condition described below.

Regularity at the Origin

For potentials that are finite at the origin, or less singular than $1/r^2$, the leading near-origin radial equation is controlled by angular momentum:

$$\frac{1}{r^2} \frac{d}{dr} \left( r^2\frac{dR}{dr} \right) - \frac{\ell(\ell+1)}{r^2}R \approx0 .$$

Trying $R(r)\sim r^s$ gives

$$s(s+1)=\ell(\ell+1),$$

so

$$s=\ell \qquad \text{or} \qquad s=-\ell-1 .$$

The regular branch is

$$R_\ell(r)\sim r^\ell, \qquad u_\ell(r)\sim r^{\ell+1}.$$

The singular branch is

$$R_\ell(r)\sim r^{-\ell-1}, \qquad u_\ell(r)\sim r^{-\ell}.$$

In the standard no-contact central-potential problem, the singular branch is rejected because the original three-dimensional wavefunction is not regular at the origin. The practical reduced-function condition is therefore

$$u_\ell(0)=0$$

for the usual regular radial domain.

For $\ell=0$, one must be especially careful. The singular branch behaves as $R_0(r)\sim 1/r$ and $u_0(r)\sim\text{constant}$. It may look harmless in the $u$ equation, but it is not a regular three-dimensional wavefunction for an ordinary nonsingular central potential. Allowing it corresponds to changing the domain, often to include contact physics at the origin.

Normalizability at Infinity

For a bound state, square normalizability requires

$$\int_0^\infty \lvert u_\ell(r)\rvert^2\,dr \lt \infty .$$

When $V(r)\to0$ and $E\lt 0$, the large-$r$ reduced equation usually gives exponential decay:

$$u_\ell(r)\sim e^{-\kappa r}, \qquad \kappa=\frac{\sqrt{-2mE}}{\hbar}.$$

For continuum states with $E\gt 0$, normalizability is replaced by a continuum or flux convention. One then imposes oscillatory asymptotic behavior rather than decay. The boundary condition at infinity is part of the physical question: bound state, standing wave, incoming scattering state, outgoing wave, or phase-shift convention.

Matching at Finite Radii

If $V(r)$ has a finite jump at $r=a\gt 0$, the radial wavefunction and its derivative match across the jump. In reduced form,

$$u(a^-)=u(a^+), \qquad u'(a^-)=u'(a^+).$$

Equivalently, because $a$ is nonzero, $R$ and $R'$ are continuous at the same point. This is the radial version of ordinary matching across a finite step.

For a delta-shell potential,

$$V(r)=\lambda\delta(r-a), \qquad a\gt 0,$$

the reduced wavefunction is continuous, but its derivative jumps:

$$u(a^-)=u(a^+),$$

and

$$u'(a^+)-u'(a^-) = \frac{2m\lambda}{\hbar^2}u(a).$$

This condition follows by integrating the reduced radial equation across a small interval around $a$. It should not be confused with the origin condition.

Singular Potentials

The simple origin rule above assumes the potential is not more singular than the centrifugal term. The Coulomb potential,

$$V(r)=-\frac{\kappa}{r},$$

is singular at the origin, but it is still milder than $1/r^2$. The usual regular behavior remains the correct bound-state condition.

Potentials with inverse-square behavior require separate analysis:

$$V(r)\sim -\frac{g}{r^2}.$$

Such a term competes directly with the centrifugal term in the effective radial potential. Depending on the sign and size of $g$, the radial Hamiltonian may require extra boundary data at the origin. Strong attractive singularities can produce the fall-to-the-center pathology if no short-distance physics is supplied.

The safe rule is: when the potential is as singular as $1/r^2$ or worse, do not import the regular-potential boundary condition without checking the operator domain.

Current and Self-Adjointness

Boundary conditions are not only aesthetic. They control probability conservation and the self-adjointness of the radial Hamiltonian. In the $u$ representation, the boundary term obtained by integration by parts has the form

$$\left[ v^*u' - {v'}^*u \right]_{0}^{\infty}.$$

For a closed radial bound-state problem, the domain should make this boundary term vanish for allowed functions $u$ and $v$. Regularity at the origin and decay at infinity accomplish this for the standard bound-state domain.

In current language, the reduced half-line current is

$$J_u = \frac{\hbar}{2mi} \left( u^*u'-u{u'}^* \right).$$

An isolated bound-state problem should not leak probability through $r=0$ or $r=\infty$. Scattering problems instead choose asymptotic current conventions deliberately.

Practical Checklist

When solving a radial problem, check the following before trusting a spectrum or wavefunction:

1. Have you decided whether the physical radial function is $R(r)$ or $u(r)$ in each formula?
2. Is the normalization using $r^2\,dr$ for $R$ or $dr$ for $u$?
3. Is the near-origin solution regular as a three-dimensional wavefunction?
4. Does the large-$r$ behavior match bound-state decay or the intended scattering convention?
5. Are finite jumps, delta shells, or hard cutoffs matched correctly?
6. Is the potential singular enough to require a domain analysis beyond the standard rule?

Common Mistakes

• Treating the reduced equation as a full-line one-dimensional problem.
• Requiring $R(0)=0$ for every regular radial state. For $\ell=0$, regular $R$ can be finite and nonzero at the origin.
• Allowing $u(0)\ne0$ in an ordinary no-contact $s$-wave problem.
• Normalizing $R$ with $dr$ instead of $r^2\,dr$.
• Forgetting that continuum radial states are not square-normalized like bound states.
• Assuming Coulomb singularity and inverse-square singularity have the same domain behavior.
• Imposing derivative continuity across a delta-shell potential.

Where This Is Used

• Radial Schrödinger Equation derives the equation whose endpoint behavior is summarized here.
• Effective Radial Potential uses regularity and normalizability to interpret allowed radial states.
• Coulomb Potential explains why the ordinary Coulomb singularity is milder than an inverse-square singularity.
• Hydrogen Atom uses the standard Coulomb radial boundary conditions.
• Normalization Conventions explains the $R$ and $u$ normalization measures.
• Sturm–Liouville Theory gives the mathematical background for endpoint conditions and eigenvalue problems.
• Hermitian vs Self-Adjoint Operators explains why domains matter for quantum observables.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• G. Teschl, Mathematical Methods in Quantum Mechanics, 2nd ed., American Mathematical Society, 2014.

Exercises

1. For a nonsingular potential near $r=0$, derive the two possible powers of $R(r)\sim r^s$.

Solution

Near the origin, neglect $E$ and a nonsingular $V(r)$ compared with the angular term. The radial equation reduces to

$$\frac{1}{r^2} \frac{d}{dr} \left( r^2\frac{dR}{dr} \right) - \frac{\ell(\ell+1)}{r^2}R \approx0 .$$

With $R\sim r^s$,

$$\frac{1}{r^2} \frac{d}{dr} \left( r^2sr^{s-1} \right) = s(s+1)r^{s-2}.$$

The indicial equation is

$$s(s+1)=\ell(\ell+1).$$

Therefore

$$s=\ell \qquad \text{or} \qquad s=-\ell-1 .$$

The regular branch is $R\sim r^\ell$.

2. Show that the regular branch implies $u_\ell(0)=0$.

Solution

For the regular branch,

$$R_\ell(r)\sim r^\ell.$$

Since $u_\ell=rR_\ell$,

$$u_\ell(r)\sim r^{\ell+1}.$$

For all allowed orbital angular momenta $\ell=0,1,2,\ldots$, this tends to zero as $r\to0$. Thus the regular radial domain has

$$u_\ell(0)=0.$$

3. Derive the derivative jump condition for a delta-shell potential $V(r)=\lambda\delta(r-a)$ with $a\gt 0$.

Solution

Integrate the reduced radial equation across $[a-\epsilon,a+\epsilon]$:

$$\int_{a-\epsilon}^{a+\epsilon} \left[ -\frac{\hbar^2}{2m}u'' + \lambda\delta(r-a)u \right]dr = \int_{a-\epsilon}^{a+\epsilon} \left[ E - \frac{\hbar^2\ell(\ell+1)}{2mr^2} \right]u\,dr .$$

The right side vanishes as $\epsilon\to0$ for finite $u$. The left side becomes

$$-\frac{\hbar^2}{2m} \left[ u'(a^+)-u'(a^-) \right] + \lambda u(a) =0.$$

Therefore

$$u'(a^+)-u'(a^-) = \frac{2m\lambda}{\hbar^2}u(a).$$