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 central potential occurs when the force field has a distinguished centre and the potential energy depends only on the distance from that centre. It does not depend on direction. This is the situation for an ideal Coulomb field, gravitational field, or any spherically symmetric potential.

A centrally symmetric field is described by

\[V=V(r).\]

Such systems are important because rotational symmetry allows the Schrodinger equation to be separated into radial and angular parts. The angular part is handled by spherical harmonics, while the radial part carries the details of the particular potential.

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.

This form already shows why angular momentum is important in a central potential. The angular dependence enters through $L^2$, whose eigenfunctions are known.

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.

Effective potential

The radial equation for $u(r)$ has the same form as a one-dimensional Schrodinger equation with

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

The second term is positive for $l\ne0$ and becomes large near $r=0$. It represents the centrifugal barrier. For $l=0$, this barrier is absent, and the radial wavefunction may have non-zero value at the origin.

Answer form for separation

To derive the radial equation in an answer, the clean sequence is:

1. write the Hamiltonian in spherical coordinates;
2. state that $[H,L^2]=[H,L_z]=0$ for $V(r)$;
3. choose $\psi=R(r)Y_l^m(\theta,\phi)$;
4. use $L^2Y_l^m=\hbar^2l(l+1)Y_l^m$;
5. define $u=rR$ to remove the first derivative term.

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?