Angular and Radial Separation

Angular and radial separation is the step that turns a three-dimensional central-potential problem into angular eigenfunctions on the sphere plus a radial equation on the half-line. It is the bridge between spherical coordinates and the full Radial Schrödinger Equation.

The central assumption is rotational symmetry:

$$V(\mathbf r)=V(r).$$

When the domain and boundary conditions are rotationally invariant as well, the angular basis is universal. The details of the potential enter the radial equation.

Helpful background. Schrödinger Equation in Three Dimensions to use the three-dimensional kinetic operator.

Central-potential Schrödinger equation

For a particle of mass $m$ in a time-independent central potential, the stationary equation is

$$\left[ -\frac{\hbar^2}{2m}\nabla^2 +V(r) \right]\psi(\mathbf r) =E\psi(\mathbf r).$$

In spherical coordinates, the Laplacian splits into radial and angular pieces:

$$\nabla^2 = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2\frac{\partial}{\partial r} \right) + \frac{1}{r^2}\Delta_{S^2},$$

where

$$\Delta_{S^2} = \frac{1}{\sin\theta} \frac{\partial}{\partial\theta} \left( \sin\theta\frac{\partial}{\partial\theta} \right) + \frac{1}{\sin^2\theta} \frac{\partial^2}{\partial\phi^2}$$

is the Laplacian on the unit sphere. The three-dimensional equation becomes

$$\left[ -\frac{\hbar^2}{2m} \left( \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2\frac{\partial}{\partial r} \right) + \frac{1}{r^2}\Delta_{S^2} \right) +V(r) \right]\psi =E\psi.$$

This is the point where spherical symmetry matters: $V(r)$ contains no angular derivatives and no angular coordinate dependence.

Product Ansatz

Try a separated form

$$\psi(r,\theta,\phi)=R(r)Y(\theta,\phi).$$

Away from nodes, substitute into the equation, divide by $R(r)Y(\theta,\phi)$, and multiply by $2mr^2/\hbar^2$. The terms can be arranged as

$$\frac{1}{R} \frac{d}{dr} \left( r^2\frac{dR}{dr} \right) + \frac{2mr^2}{\hbar^2} \left( E-V(r) \right) = -\frac{1}{Y}\Delta_{S^2}Y.$$

The left side depends only on $r$. The right side depends only on $\theta$ and $\phi$. Therefore both sides must equal the same separation constant:

$$-\frac{1}{Y}\Delta_{S^2}Y=\lambda.$$

Equivalently,

$$\Delta_{S^2}Y=-\lambda Y.$$

The allowed values of $\lambda$ are not chosen by the radial potential. They come from the eigenvalue problem on the unit sphere, together with regularity and single-valuedness of ordinary scalar wavefunctions.

Angular Equation

The angular eigenfunctions are the spherical harmonics:

$$Y(\theta,\phi)=Y_\ell^m(\theta,\phi),$$

with

$$\lambda=\ell(\ell+1), \qquad \ell=0,1,2,\ldots, \qquad m=-\ell,-\ell+1,\ldots,\ell.$$

They obey

$$\Delta_{S^2}Y_\ell^m = -\ell(\ell+1)Y_\ell^m.$$

With the usual orbital angular-momentum operator,

$$\hat L^2=-\hbar^2\Delta_{S^2},$$

this is the same as

$$\hat L^2Y_\ell^m = \hbar^2\ell(\ell+1)Y_\ell^m.$$

The magnetic label $m$ distinguishes the $2\ell+1$ angular functions in a fixed $\ell$ sector. It does not enter the radial equation for a purely central potential, because the Hamiltonian has no preferred axis.

Radial Equation

Putting $\lambda=\ell(\ell+1)$ into the radial side gives

$$\frac{1}{R_\ell} \frac{d}{dr} \left( r^2\frac{dR_\ell}{dr} \right) + \frac{2mr^2}{\hbar^2} \left( E-V(r) \right) = \ell(\ell+1).$$

Multiplying back into Schrödinger form,

$$-\frac{\hbar^2}{2m} \left[ \frac{1}{r^2} \frac{d}{dr} \left( r^2\frac{dR_\ell}{dr} \right) - \frac{\ell(\ell+1)}{r^2}R_\ell \right] +V(r)R_\ell =ER_\ell.$$

This is the radial equation in $R$ form. The subscript $\ell$ reminds us that each angular-momentum sector has a different radial equation. For a bound state with additional discrete labels, one often writes

$$\psi_{n\ell m}(r,\theta,\phi) = R_{n\ell}(r)Y_\ell^m(\theta,\phi).$$

For continuum states, the radial label is usually an energy or wavenumber rather than a principal quantum number.

Reduced Radial Equation

Define the reduced radial wavefunction

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

For $r\gt 0$, the radial equation becomes

$$-\frac{\hbar^2}{2m} \frac{d^2u_\ell}{dr^2} + \left[ V(r) + \frac{\hbar^2\ell(\ell+1)}{2mr^2} \right]u_\ell = Eu_\ell.$$

This has the form of a one-dimensional stationary equation on the half-line, with effective potential

$$V_{\ell,\mathrm{eff}}(r) = V(r) + \frac{\hbar^2\ell(\ell+1)}{2mr^2}.$$

The second term is the centrifugal term. It is not an extra force inserted into the model; it is the angular kinetic energy expressed in the radial equation.

The half-line matters. The point $r=0$ is a boundary, and physical regularity at the origin supplies information that a full-line one-dimensional equation would not contain. The detailed boundary conventions are treated on the Radial Schrödinger Equation page.

Normalization

If the angular functions are normalized by

$$\int_{S^2} \lvert Y_\ell^m(\theta,\phi)\rvert^2\,d\Omega =1,$$

then the full bound-state normalization reduces to

$$\int_0^\infty \lvert R_{n\ell}(r)\rvert^2r^2\,dr =1.$$

In terms of $u_{n\ell}=rR_{n\ell}$, this becomes

$$\int_0^\infty \lvert u_{n\ell}(r)\rvert^2\,dr =1.$$

Thus $R$ is the radial factor in the three-dimensional wavefunction, while $u$ is the half-line wavefunction with ordinary $dr$ measure. Confusing these two functions changes both normalization and near-origin behavior.

Physical Interpretation

Central-potential separation organizes the Hilbert space into angular-momentum sectors. The same potential $V(r)$ appears in every sector, but each sector sees a different centrifugal barrier:

$$\frac{\hbar^2\ell(\ell+1)}{2mr^2}.$$

For $\ell=0$, there is no centrifugal term. For larger $\ell$, the term suppresses probability near the origin and raises the short-distance effective potential. The label $m$ counts orientations within a fixed $\ell$ multiplet; rotational symmetry keeps those orientations degenerate unless an external field, boundary condition, or interaction selects an axis.

This separation also explains why spherical harmonics occur far beyond the hydrogen atom. They are the angular basis for any scalar central-potential problem, including free radial motion, finite spherical wells, partial-wave scattering, and the rigid rotor.

Common Mistakes

• Treating $\lambda$ as an arbitrary real parameter after regularity on the sphere has quantized it.
• Forgetting that the radial equation depends on $\ell$ but not on $m$ for a rotationally invariant Hamiltonian.
• Normalizing $R(r)$ with $dr$ instead of $r^2\,dr$.
• Treating $u(r)$ as a full-line wavefunction rather than a half-line wavefunction.
• Calling the centrifugal term a new physical potential instead of angular kinetic energy in radial form.
• Assuming that spherical harmonics are special to the Coulomb problem.

Exercises

1. Starting from the product ansatz $\psi=RY$, show why the separation constant can be chosen as an angular eigenvalue.

Solution

After substitution and division by $RY$, the equation can be written as

$$\frac{1}{R} \frac{d}{dr} \left( r^2\frac{dR}{dr} \right) + \frac{2mr^2}{\hbar^2} \left( E-V(r) \right) = -\frac{1}{Y}\Delta_{S^2}Y.$$

The left side depends only on $r$, while the right side depends only on angles. If the equality holds for all $r$, $\theta$, and $\phi$, both sides must equal a constant. Choosing

$$-\frac{1}{Y}\Delta_{S^2}Y=\lambda$$

makes the angular equation

$$\Delta_{S^2}Y=-\lambda Y.$$

Regular single-valued scalar functions on the sphere give $\lambda=\ell(\ell+1)$.

2. Convert the $R$ equation to the reduced $u=rR$ equation.

Solution

Use $R=u/r$. Then

$$\frac{dR}{dr} = \frac{u'}{r} - \frac{u}{r^2},$$

so

$$r^2\frac{dR}{dr} = ru'-u.$$

Differentiating gives

$$\frac{d}{dr} \left( r^2\frac{dR}{dr} \right) = ru''.$$

Substitute this into the $R$ equation and multiply by $r$. The result is

$$-\frac{\hbar^2}{2m} \frac{d^2u}{dr^2} + \left[ V(r) + \frac{\hbar^2\ell(\ell+1)}{2mr^2} \right]u = Eu.$$

3. Why does the quantum number $m$ not appear in the radial equation for a central potential?

Solution

For a central potential, the Hamiltonian is rotationally invariant and contains no preferred $z$ axis. The angular equation gives the eigenvalue of $\hat L^2$, namely $\hbar^2\ell(\ell+1)$, and this is the only angular information entering the radial equation. Different values of $m$ label different orientations within the same $\ell$ sector, so they have the same radial equation unless rotational symmetry is broken.

References

• G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2012.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• 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.