Central Potentials

A central potential is a three-dimensional potential that depends only on distance from a chosen origin:

$$V(\mathbf r)=V(r), \qquad r=\lvert\mathbf r\rvert.$$

Central potentials are the bridge from general three-dimensional wave mechanics to atoms, radial equations, spherical harmonics, partial waves, and hydrogenic systems. They are important not because every real atom is exactly central, but because rotational symmetry gives a clean organizing structure that more realistic models perturb or refine.

Definition and Examples

For a single particle of mass $m$, a central-potential Hamiltonian has the form

$$\hat H = \frac{\hat{\mathbf p}^{\,2}}{2m} +V(\hat r).$$

Standard examples include:

Potential	Form	Main use
Free particle	$V(r)=0$	radial free waves and partial-wave language
Coulomb attraction	$V(r)=-e^2/(4\pi\epsilon_0r)$	hydrogenic atoms
Isotropic oscillator	$V(r)=\frac12m\omega^2r^2$	radial oscillator and hidden degeneracy
Spherical well	$V(r)=-V_0$ inside a radius, $0$ outside	model bound states and scattering
Screened Coulomb	$V(r)\propto e^{-r/a}/r$	short-range central-force models

The exact functional form controls the radial spectrum. The angular structure is shared by all scalar central potentials.

Rotational Symmetry

A rotation changes direction but preserves distance:

$$r\mapsto r.$$

Therefore a central potential is rotationally invariant. For the Hamiltonian this means

$$[\hat H,\hat L_i]=0, \qquad i=x,y,z,$$

and hence

$$[\hat H,\hat{\mathbf L}^2]=0, \qquad [\hat H,\hat L_z]=0.$$

The choice of $z$ axis is conventional. Because $\hat H$, $\hat{\mathbf L}^2$, and one component such as $\hat L_z$ commute, energy eigenstates can be chosen to carry angular-momentum labels. This does not mean the particle has a classical orbit; it means the angular dependence transforms in a definite representation of rotations.

The symmetry proof of these commutators is developed in Central Potentials and Rotational Symmetry.

Separated Form

For a scalar central potential, the stationary wavefunction may be written as

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

or, for a discrete bound spectrum,

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

The angular labels are

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

The spherical harmonics obey

$$\hat{\mathbf L}^2Y_\ell^m = \hbar^2\ell(\ell+1)Y_\ell^m, \qquad \hat L_zY_\ell^m = \hbar mY_\ell^m.$$

The radial label depends on the problem. In hydrogen it is conventional to use a principal quantum number $n$. In a generic central potential, one may instead use a radial node label $n_r$, an energy label $E$, or a scattering wavenumber.

The separation step itself is derived in Angular and Radial Separation. This page records what that structure means physically.

Radial Dynamics

After angular separation, the radial factor satisfies a one-dimensional equation on the half-line. In reduced radial form,

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

the equation is

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

The effective radial potential is

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

The second term is the centrifugal term. It is angular kinetic energy expressed in the radial equation, not an extra interaction inserted into the Hamiltonian.

This equation looks like a one-dimensional problem, but it is not a full-line problem. The domain is $0\lt r\lt\infty$, regularity at the origin matters, and the three-dimensional normalization is inherited from the measure $r^2dr\,d\Omega$. The canonical treatment is Radial Schrödinger Equation.

Quantum Numbers and What They Mean

The central-potential labels have distinct jobs:

Label	Meaning	What controls it
$\ell$	orbital angular momentum magnitude	angular eigenvalue $\ell(\ell+1)$
$m$	projection along a chosen axis	basis choice inside an $\ell$ multiplet
radial label	radial nodes, bound levels, or continuum energy	the specific potential and boundary behavior

For a generic bound central potential, one expects energies of the form

$$E=E_{n_r\ell}.$$

The energy is independent of $m$, but it usually depends on $\ell$ because the centrifugal term changes the radial equation. Hydrogen is exceptional: the ideal Coulomb bound-state energy depends only on $n=n_r+\ell+1$. That special degeneracy belongs to Degeneracy of the Hydrogen Atom.

Magnetic Degeneracy

For fixed $\ell$, the allowed values of $m$ give

$$2\ell+1$$

states. In a purely central potential these states have the same energy:

$$E_{\ell m}=E_\ell \qquad \text{for fixed radial label and fixed }\ell.$$

This degeneracy is called magnetic degeneracy because $m$ is the magnetic quantum number. The name comes from the fact that external magnetic fields can split these levels by selecting a preferred axis. Without such an axis, different orientations inside the same angular-momentum multiplet are physically equivalent.

The phrase “magnetic degeneracy” does not mean a magnetic field is present. It means the degeneracy is associated with the $m$ label.

Bound States and Scattering States

Central potentials support different spectral situations:

• attractive potentials may have square-normalizable bound states;
• short-range potentials may also have continuum scattering states;
• repulsive potentials may have no bound states but still scatter waves;
• the Coulomb potential has both bound states below zero energy and continuum states above zero energy.

For bound states, $R(r)$ or $u(r)$ is normalized. For scattering states, one uses continuum normalization and asymptotic incoming and outgoing behavior. Partial-wave scattering expands a scattering state into central-potential angular-momentum sectors; the detailed scattering formalism belongs to the approximation and scattering volume.

Common Mistakes

• Thinking that central means Coulomb. Coulomb is one important central potential, not the definition.
• Assuming every central potential has hydrogen-like $\ell$ degeneracy.
• Forgetting that $m$ degeneracy follows from rotational symmetry, while $\ell$ dependence is usually radial dynamics.
• Treating $u(r)$ as if it were an ordinary full-line wavefunction.
• Ignoring the origin as a boundary point of the radial half-line.
• Confusing the arbitrary chosen $z$ axis with a physical direction when no external field is present.
• Treating orbitals or angular wavefunctions as classical particle trajectories.

Where This Is Used

• Angular and Radial Separation derives the separated central-potential form.
• Central Potentials and Rotational Symmetry explains the angular-momentum symmetry argument behind the labels.
• Radial Schrödinger Equation gives the radial equation, normalization, and boundary conventions in detail.
• Effective Radial Potential interprets the centrifugal barrier, radial turning points, and qualitative spectra.
• Propagators in Multiple Dimensions gives the corresponding partial-wave decomposition of the time-domain kernel.
• Boundary Conditions for Radial Wavefunctions collects the regularity, normalization, and singular-potential cautions for radial problems.
• Coulomb Potential sets the charge, reduced-mass, Bohr-radius, and Rydberg-scale conventions for hydrogenic systems.
• Hydrogen Atom solves the Coulomb central potential.
• Degeneracy of the Hydrogen Atom explains why Coulomb has more degeneracy than a generic central potential.
• Degeneracy in Separable Systems compares $m$ degeneracy with other repeated-energy mechanisms.
• Spherical Harmonics gives the angular functions used in all scalar central potentials.

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.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.

Exercises

1. Explain why $V(\mathbf r)=V(r)$ is rotationally invariant.

Solution

A rotation changes $\mathbf r$ to $R\mathbf r$, where $R$ is an orthogonal rotation matrix. Rotations preserve length:

$$\lVert R\mathbf r\rVert=\lVert\mathbf r\rVert.$$

Therefore

$$V(R\mathbf r) = V(\lVert R\mathbf r\rVert) = V(\lVert\mathbf r\rVert) = V(\mathbf r).$$

Thus the potential is unchanged by rotations.

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

Solution

The radial equation receives angular information through the eigenvalue of $\hat{\mathbf L}^2$:

$$\hbar^2\ell(\ell+1).$$

The magnetic quantum number $m$ labels eigenvalues of $\hat L_z$, which distinguish orientations inside the same $\ell$ multiplet. A central potential has no preferred axis, so the radial dynamics cannot depend on this orientation label.

3. A central potential has bound-state energies $E_{n_r\ell}$. Which degeneracy is guaranteed by rotational symmetry, and which is not?

Solution

For each fixed $n_r$ and $\ell$, the states with

$$m=-\ell,-\ell+1,\ldots,\ell$$

are degenerate, giving $2\ell+1$ states. This is guaranteed by rotational symmetry.

Degeneracy between different $\ell$ values is not guaranteed. Changing $\ell$ changes the centrifugal term in the radial equation. Hydrogen has extra $\ell$ degeneracy because the Coulomb problem has special hidden structure, not because all central potentials do.