Central Potential Hamiltonian

Hamiltonian

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

$$H = \frac{\hat{\mathbf p}^{\,2}}{2m} + V(r), \qquad r=\lVert\mathbf r\rVert.$$

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

$$H_\ell = -\frac{\hbar^2}{2m} \frac{d^2}{dr^2} + V(r) + \frac{\hbar^2\ell(\ell+1)}{2mr^2}.$$

Assumptions

• The potential depends only on radius.
• The origin and radial domain are specified.
• Boundary conditions at $r=0$ and infinity are part of the problem.
• Reduced mass should be used for a two-body relative-coordinate problem.

Common Mistakes

• Thinking “central” means “Coulomb” rather than any $V(r)$.
• Assuming hydrogen’s extra $\ell$ degeneracy for every central potential.
• Forgetting the centrifugal term in the reduced radial equation.
• Treating singular potentials without checking radial boundary conditions.

Canonical Links

• Central Potentials
• Radial Schrödinger Equation
• Effective Radial Potential
• Angular Momentum Operator

References

• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.