Central Potentials and Hydrogenic Systems

A central potential depends only on distance from one point. Rotational symmetry lets us separate the shape of a wavefunction on a sphere from its radial dependence. The Coulomb potential then provides a complete solvable model of a one-electron atom, including bound orbitals and a continuum above the ionization threshold.

Radial motion and the Coulomb length

For relative motion with reduced mass $\mu$, write

$$\psi(r,\theta,\phi)=R_{n\ell}(r)Y_{\ell m}(\theta,\phi), \qquad u_{n\ell}(r)=rR_{n\ell}(r).$$

The angular function is normalized on the sphere. The radial equation and normalization become

$$\left[-\frac{\hbar^2}{2\mu}\frac{d^2}{dr^2} +V(r)+\frac{\hbar^2\ell(\ell+1)}{2\mu r^2}\right]u(r)=Eu(r), \qquad \int_0^\infty |u(r)|^2\,dr=1.$$

The centrifugal term raises the effective potential when $\ell>0$. This equation is posed on the half-line, with the origin behavior appropriate to the three-dimensional problem. It is not an unrestricted one-dimensional well.

For a point nucleus of charge $+Ze$ and an electron of charge $-e$, define

$$V(r)=-\frac{\kappa}{r},\qquad \kappa=\frac{Ze^2}{4\pi\epsilon_0},\qquad a_Z=\frac{\hbar^2}{\mu\kappa}.$$

The nonrelativistic bound energies are

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

Increasing $Z$ at fixed $\mu$ shrinks orbital lengths as $1/Z$ and increases binding-energy magnitudes as $Z^2$. Changing the nucleus also changes the reduced mass, so accurate isotope comparisons keep both effects distinct.

Rotational symmetry makes the $2\ell+1$ magnetic states degenerate for a given radial state and $\ell$. The Coulomb potential has the additional property that different allowed $\ell$ values at fixed $n$ have the same energy, producing $n^2$ spatial states. This larger degeneracy is not a property of every central force.

The model omits spin, relativistic corrections, nuclear size, and radiative effects. It explains the leading structure of hydrogenic spectra; precision spectroscopy requires the additional physics developed in Atomic, Molecular, and Optical Physics.

Reading sequence

Page	What you will learn
Central Potentials	Rotational invariance, separated states, quantum labels, and generic spectra
Radial Schrödinger Equation	$R$ and $u$ equations, normalization, regularity, and free radial motion
Effective Radial Potential	Centrifugal barrier, turning points, nodes, and qualitative radial spectra
Boundary Conditions for Radial Wavefunctions	Origin and infinity domains, finite-radius matching, and singular-potential cautions
Coulomb Potential	Charge signs, reduced mass, Bohr and Rydberg scales, and model limitations
Hydrogen Atom	Exact nonrelativistic bound spectrum, quantum numbers, and wavefunctions
Hydrogenic Ions	$Z$ and reduced-mass scaling for one-electron ions
Radial Wavefunctions	Laguerre structure, low-lying formulas, radial probabilities, nodes, and moments
Atomic Orbitals	One-electron orbital meaning, real and complex bases, and visualization
Degeneracy of the Hydrogen Atom	$n^2$ count, rotational versus Coulomb degeneracy, and splitting
Continuum States of the Coulomb Problem: Overview	Ionization threshold, Coulomb functions, delta normalization, and long-range phase

A useful first route

Begin with central potentials, the radial equation, and radial boundary conditions. Then work through hydrogen and its radial functions. Orbital pictures become much easier to interpret once you can distinguish the pointwise density $|\psi|^2$ from the probability contained in a spherical shell.

Exercises

For the ground state, $R_{10}(r)=2a_Z^{-3/2}e^{-r/a_Z}$. Find where the radial probability density is largest. Is this also where the spatial density is largest?

Solution

The shell probability density is $P(r)=4r^2a_Z^{-3}e^{-2r/a_Z}$. Its logarithmic derivative for $r>0$ is $2/r-2/a_Z$, so its maximum is at $r=a_Z$. The spatial density is proportional to $e^{-2r/a_Z}$ and is largest at the origin. There is no contradiction: the volume of a thin shell tends to zero at the origin.

Ignoring spin, count the spatial states with $n=3$ in the ideal Coulomb model.

Solution

The allowed values are $\ell=0,1,2$, contributing $1$, $3$, and $5$ magnetic states. Their total is $9=3^2$. A generic rotationally invariant perturbation can split the different $\ell$ sectors while preserving each sector’s magnetic degeneracy.

References

• H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer, 1957.
• 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.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• E. Merzbacher, Quantum Mechanics, 3rd ed., Wiley, 1998.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.