Coulomb Potential

The Coulomb potential is the inverse-radius interaction between two point charges. In nonrelativistic quantum mechanics it is the central potential behind hydrogenic atoms, Coulomb scattering, Rydberg scaling, and many useful approximations in atomic physics.

This page sets up the Coulomb problem before the exact Hydrogen Atom solution. It fixes sign conventions, reduced mass, natural length and energy scales, and the distinction between attractive bound-state physics and long-range continuum scattering.

Charge Convention

Use $e\gt 0$ for the elementary positive charge. The charge of an electron is $-e$. For two point charges $q_1$ and $q_2$, the Coulomb potential energy in SI-compatible units is

$$V(r) = \frac{q_1q_2}{4\pi\epsilon_0r}.$$

The sign is physical:

• if $q_1q_2\gt 0$, the interaction is repulsive;
• if $q_1q_2\lt 0$, the interaction is attractive.

For an electron bound to a point nucleus of charge $+Ze$, with $Z=1,2,\ldots$, the relative-coordinate potential is

$$V(r) = - \frac{Ze^2}{4\pi\epsilon_0r}.$$

It is useful to define

$$\kappa_Z = \frac{Ze^2}{4\pi\epsilon_0},$$

so the attractive Coulomb potential is

$$V(r)=-\frac{\kappa_Z}{r}.$$

The sign convention matters. Many errors in atomic calculations are really sign errors: a repulsive Coulomb potential has no hydrogenic bound states.

Two-Body Reduction

For two particles with masses $m_1,m_2$ and relative separation

$$r=\lvert\mathbf r_1-\mathbf r_2\rvert,$$

the nonrelativistic Hamiltonian separates into center-of-mass and relative parts. The relative Hamiltonian is

$$\hat H_{\mathrm{rel}} = \frac{\hat{\mathbf p}^{\,2}}{2\mu} + \frac{q_1q_2}{4\pi\epsilon_0r},$$

where the reduced mass is

$$\mu = \frac{m_1m_2}{m_1+m_2}.$$

For an electron and a nucleus of mass $M_N$,

$$\mu = \frac{m_eM_N}{m_e+M_N}.$$

The infinite-nuclear-mass approximation replaces $\mu$ by $m_e$. That approximation is often close, but reduced mass is the correct parameter for precision energies and isotope dependence.

Natural Length Scale

For the attractive charge-$Z$ Coulomb problem, the natural length is the reduced-mass Bohr scale

$$a_Z = \frac{4\pi\epsilon_0\hbar^2}{\mu Ze^2} = \frac{\hbar^2}{\mu\kappa_Z}.$$

For $Z=1$, this is the reduced-mass Bohr radius. If one uses $m_e$ instead of $\mu$, it becomes the usual infinite-proton-mass Bohr radius.

The scaling is simple:

$$a_Z\propto\frac{1}{\mu Z}.$$

Larger nuclear charge pulls the wavefunction inward. Larger reduced mass also shrinks the length scale. This is why muonic atoms are much smaller than ordinary electronic atoms, and why isotope shifts can appear even before relativistic or radiative corrections.

Natural Energy Scale

The corresponding Coulomb energy scale is

$$E_Z = \frac{\hbar^2}{2\mu a_Z^2} = \frac{\mu Z^2e^4}{2(4\pi\epsilon_0)^2\hbar^2} = \frac{\mu\kappa_Z^2}{2\hbar^2}.$$

For $Z=1$ and electron mass in the infinite-nuclear-mass approximation, this is the Rydberg energy:

$$E_Z=Z^2\mathrm{Ry} \qquad \text{up to reduced-mass replacement}.$$

The exact nonrelativistic attractive Coulomb bound-state energies are

$$E_n = - \frac{E_Z}{n^2}, \qquad n=1,2,\ldots .$$

The full derivation and wavefunctions belong to the Hydrogen Atom page. Here the important lesson is the scale:

$$\lvert E_n\rvert\propto \mu Z^2.$$

Dimensionless Form

The scales $a_Z$ and $E_Z$ are chosen so that the Schrödinger equation has a clean dimensionless form. Let

$$\rho=\frac{r}{a_Z}, \qquad \epsilon=\frac{E}{E_Z}.$$

For the attractive Coulomb Hamiltonian,

$$\left[ -\frac{\hbar^2}{2\mu}\nabla^2 - \frac{\kappa_Z}{r} \right]\psi = E\psi,$$

the dimensionless equation becomes

$$\left[ -\nabla_\rho^2 - \frac{2}{\rho} \right]\psi = \epsilon\psi.$$

In a fixed angular-momentum sector, the reduced radial equation becomes

$$\left[ -\frac{d^2}{d\rho^2} + \frac{\ell(\ell+1)}{\rho^2} - \frac{2}{\rho} \right]u_\ell = \epsilon u_\ell.$$

This form makes clear why every attractive point-Coulomb problem has the same dimensionless shape. The physical differences are restored by multiplying lengths by $a_Z$ and energies by $E_Z$.

Bound and Continuum States

The attractive Coulomb potential has two spectral regimes:

• $E\lt 0$: an infinite ladder of normalizable bound states;
• $E\gt 0$: continuum states describing ionization and scattering.

The bound energies accumulate at $E=0$ from below. Large-$n$ states are weakly bound and spatially extended. The continuum begins at the ionization threshold and is treated as its own overview in Continuum States of the Coulomb Problem.

The repulsive Coulomb potential has no square-normalizable bound states in the two-body point-charge problem. It still has continuum scattering states, but the long-range tail changes the asymptotic form of scattering wavefunctions. This is why Coulomb Scattering requires special conventions beyond ordinary short-range scattering.

Effective Radial Potential

For the attractive Coulomb case, the fixed-$\ell$ effective radial potential is

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

For $\ell=0$, there is no centrifugal barrier. For $\ell\gt 0$, the $1/r^2$ term dominates close to the origin and suppresses small-radius probability. The competition between attraction and the centrifugal barrier gives a qualitative radial scale before the exact polynomial solution is known.

The Effective Radial Potential page develops the turning-point and centrifugal-barrier picture. The Boundary Conditions for Radial Wavefunctions page explains why the ordinary Coulomb singularity is still compatible with regular radial boundary conditions.

What the Coulomb Model Leaves Out

The point-Coulomb Hamiltonian is a controlled starting point, not a complete atomic theory. It omits:

• spin-orbit coupling and relativistic kinetic-energy corrections;
• the Darwin term and full Dirac-Coulomb structure;
• Lamb-shift radiative corrections;
• hyperfine interactions with nuclear spin;
• finite nuclear size and nuclear polarizability;
• many-electron screening and correlation;
• external electric and magnetic fields.

Those effects are smaller than the gross Coulomb binding scale in ordinary hydrogen, but they are essential for precision spectroscopy and for real many-electron atoms. They should be added as corrections to a clearly stated base model, not silently mixed into the Coulomb potential itself.

Common Mistakes

• Forgetting that $e\gt 0$ while the electron charge is $-e$.
• Using the electron mass when the reduced mass is required.
• Calling $a_Z$ the Bohr radius without stating whether reduced mass and nuclear charge have been included.
• Treating the unscreened Coulomb continuum as if it had short-range plane-wave asymptotics.
• Assuming a repulsive Coulomb potential can have hydrogenic bound states.
• Confusing the Rydberg energy with the Hartree energy; the Hartree is twice the Rydberg in the infinite-proton-mass hydrogen convention.
• Folding fine structure or Lamb-shift physics into the basic nonrelativistic Coulomb Hamiltonian.

Where This Is Used

• Hydrogen Atom solves the attractive $Z=1$ point-Coulomb bound-state problem.
• Hydrogenic Ions applies the same Coulomb solution to one-electron ions with nuclear charge $Z$.
• Continuum States of the Coulomb Problem explains the positive-energy sector, threshold, and generalized normalization.
• Degeneracy of the Hydrogen Atom explains the special Coulomb degeneracy beyond ordinary rotational degeneracy.
• Effective Radial Potential interprets the Coulomb centrifugal barrier in fixed-$\ell$ sectors.
• Coulomb Scattering treats the long-range positive-energy problem.
• Units and Constants fixes the charge and $\hbar$ conventions used here.
• Dimensionless Variables and Scaling explains why natural scales simplify model equations.

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.
• H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer, 1957.

Exercises

1. Check that the charge-$Z$ Bohr scale has dimensions of length.

Solution

The Coulomb coupling

$$\kappa_Z=\frac{Ze^2}{4\pi\epsilon_0}$$

has units of energy times length. The quantity

$$a_Z=\frac{\hbar^2}{\mu\kappa_Z}$$

therefore has units

$$\frac{(\text{action})^2}{(\text{mass})(\text{energy})(\text{length})}.$$

Since energy has units mass times length squared over time squared, this reduces to length. Equivalently, $a_Z$ is the length that makes $\hbar^2/(2\mu a_Z^2)$ and $\kappa_Z/a_Z$ comparable.

2. Derive the dimensionless attractive Coulomb equation using $r=a_Z\rho$.

Solution

With $r=a_Z\rho$,

$$\nabla^2=\frac{1}{a_Z^2}\nabla_\rho^2.$$

The Schrödinger equation is

$$\left[ -\frac{\hbar^2}{2\mu a_Z^2}\nabla_\rho^2 - \frac{\kappa_Z}{a_Z\rho} \right]\psi = E\psi.$$

Using

$$E_Z=\frac{\hbar^2}{2\mu a_Z^2}, \qquad \frac{\kappa_Z}{a_Z}=2E_Z,$$

and dividing by $E_Z$ gives

$$\left[ -\nabla_\rho^2 - \frac{2}{\rho} \right]\psi = \epsilon\psi, \qquad \epsilon=\frac{E}{E_Z}.$$

3. If $Z$ is doubled while the reduced mass is held fixed, how do the natural length and energy scales change?

Solution

The charge-$Z$ length scale is

$$a_Z\propto\frac{1}{Z}.$$

Doubling $Z$ halves the length scale. The energy scale is

$$E_Z\propto Z^2.$$

Doubling $Z$ therefore multiplies the binding-energy scale by $4$. The dimensionless spectrum is unchanged, but the conversion back to physical units changes.