06 Jun 2026

Motion in a Centrally Symmetric Field

Quantum motion in a central potential, separation of variables, radial equation, and angular momentum quantum numbers.

msc semester-i quantum-mechanics central-potential spherical-harmonics

A centrally symmetric field is described by a potential that depends only on the distance from the origin:

\[V=V(r).\]

Such systems are important because rotational symmetry allows the Schrodinger equation to be separated into radial and angular parts.

Hamiltonian

For a particle of mass $m$ in a central potential,

\[H=-\frac{\hbar^2}{2m}\nabla^2+V(r).\]

In spherical coordinates,

\[\nabla^2= \frac{1}{r^2}\frac{\partial}{\partial r} \left(r^2\frac{\partial}{\partial r}\right) - \frac{L^2}{\hbar^2r^2}.\]

Therefore the Hamiltonian can be written as

\[H= -\frac{\hbar^2}{2m} \frac{1}{r^2}\frac{\partial}{\partial r} \left(r^2\frac{\partial}{\partial r}\right) +\frac{L^2}{2mr^2} +V(r).\]

The term

\[\frac{L^2}{2mr^2}\]

acts like an angular kinetic energy term.

Separation of variables

Since $V(r)$ is spherically symmetric,

\[[H,L^2]=0,\qquad [H,L_z]=0.\]

The wavefunction can be chosen as a simultaneous eigenfunction:

\[\psi(r,\theta,\phi)=R_{nl}(r)Y_l^m(\theta,\phi).\]

The angular part is determined by spherical harmonics:

\[L^2Y_l^m=\hbar^2l(l+1)Y_l^m,\] \[L_zY_l^m=\hbar mY_l^m.\]

Radial equation

Substitution into the Schrodinger equation gives the radial equation

\[-\frac{\hbar^2}{2m} \frac{1}{r^2}\frac{d}{dr} \left(r^2\frac{dR}{dr}\right) +\frac{\hbar^2l(l+1)}{2mr^2}R +V(r)R =ER.\]

It is often simplified by defining

\[u(r)=rR(r).\]

Then the radial equation becomes

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

The effective potential is

\[\boxed{ V_{\mathrm{eff}}(r) = V(r)+\frac{\hbar^2l(l+1)}{2mr^2}. }\]

The second term is called the centrifugal term.

Quantum numbers

For central potentials, stationary states are labeled by

\[n,\quad l,\quad m.\]

Here:

• $n$ labels the radial energy level,
• $l$ labels orbital angular momentum,
• $m$ labels the $z$-component of angular momentum.

The degeneracy in $m$ follows from spherical symmetry. For a fixed $l$, the number of $m$ states is

\[2l+1.\]

Boundary conditions

The radial wavefunction must be finite at the origin and normalizable at infinity:

\[R(r)\ \text{finite at }r=0,\] \[\int_0^\infty |R(r)|^2r^2\,dr<\infty.\]

Equivalently,

\[\int_0^\infty |u(r)|^2\,dr<\infty.\]

These boundary conditions make the energy spectrum discrete for bound states.

Main points

• Central potentials depend only on $r$.
• The Hamiltonian commutes with $L^2$ and $L_z$.
• The wavefunction separates as $R(r)Y_l^m(\theta,\phi)$.
• The radial equation contains an effective centrifugal potential.
• The degeneracy in $m$ reflects rotational symmetry.

Practice questions

1. Show that $[H,L_z]=0$ for $V=V(r)$.
2. Derive the radial equation for a central potential.
3. Explain the origin of the centrifugal term.
4. Why does the wavefunction separate into radial and angular parts?
5. For $l=2$, how many possible $m$ values exist?