Central Potentials

Everything we have studied so far has been one-dimensional. Real atoms live in three dimensions, and the most important class of three-dimensional problem in quantum mechanics is one where the potential energy depends only on the distance from a fixed centre — not on direction. Such a potential is called a central potential.

What makes a potential "central"?

In Cartesian coordinates the position of a particle is $(x, y, z)$ . If the potential energy $V$ satisfies

$V(\mathbf{r}) = V(r), \qquad r = \sqrt{x^2 + y^2 + z^2},$

then $V$ depends only on the radial distance $r$ and is isotropic (the same in every direction). The Coulomb potential of the hydrogen atom,

$V(r) = -\frac{e^2}{4\pi\epsilon_0 r},$

is the prime example: it depends only on how far the electron is from the proton.

Why central potentials are special: angular momentum is conserved

In classical mechanics a central force $\mathbf{F} = F(r)\hat{r}$ (directed toward or away from the origin) exerts zero torque about the origin, so the angular momentum $\mathbf{L} = \mathbf{r} \times \mathbf{p}$ is constant in time. The particle's orbit is therefore confined to a fixed plane.

Quantum mechanics inherits this structure. The angular-momentum operators

$L_x,\quad L_y,\quad L_z, \qquad L^2 = L_x^2 + L_y^2 + L_z^2$

all commute with a Hamiltonian of the form $H = p^2/(2m) + V(r)$ . Commuting with $H$ means they are conserved observables: a state with definite angular momentum retains that value for all time. This symmetry is the central reason why quantum numbers labelling angular momentum — namely $\ell$ and $m$ — will appear when we solve for energy eigenstates.

Effective one-dimensional problem

The spherical symmetry reduces the three-dimensional problem. Working in spherical coordinates $(r, \theta, \phi)$ , one can show that the kinetic energy operator splits into a radial part and an angular part. The angular part is precisely $L^2 / (2mr^2)$ . Because $L^2$ has known eigenvalues $\hbar^2 \ell(\ell+1)$ (for non-negative integer $\ell$ ), the Schrödinger equation for each angular-momentum sector reduces to a one-dimensional radial equation:

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

where $u(r) = r\,R(r)$ and $R(r)$ is the radial wavefunction. The second term in the bracket is called the centrifugal barrier (or centrifugal potential). It acts like a repulsive contribution that pushes the particle away from the origin and grows larger for higher $\ell$ .

The full picture

Combining the radial and angular pieces, a stationary state of a central potential can be written as

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

where $Y_\ell^m$ are the spherical harmonics — the angular eigenfunctions common to all central potentials — and $R_{n\ell}$ is the radial wavefunction specific to the choice of $V(r)$ . Three quantum numbers appear naturally:

| Symbol | Name | Origin | |--------|------|---------| | $n$ | principal | radial boundary conditions | | $\ell$ | orbital angular-momentum | $L^2$ eigenvalue: $\hbar^2\ell(\ell+1)$ | | $m$ | magnetic | $L_z$ eigenvalue: $m\hbar$ |

The constraints $n \geq 1$ , $0 \leq \ell \leq n-1$ , and $-\ell \leq m \leq \ell$ emerge from requiring normalisable solutions and will be derived in later lessons when we study the hydrogen atom explicitly.

Summary

A central potential $V(r)$ is spherically symmetric, making angular momentum a conserved quantity. This symmetry reduces the three-dimensional Schrödinger equation to a radial ordinary differential equation supplemented by a centrifugal barrier. Solutions factor into spherical harmonics $Y_\ell^m(\theta,\phi)$ and a radial function $R_{n\ell}(r)$ , and are labelled by the three quantum numbers $n$ , $\ell$ , and $m$ . The hydrogen atom — the centrepiece of the next several lessons — is the most important realization of this structure.