Perturbation-Theory Examples

Perturbation theory turns a solvable problem into an approximation scheme for a nearby problem. A worked example is trustworthy only when it identifies the unperturbed problem, the small parameter, the order kept, and the failure mode.

Example Routes

Need	Learn	Formula Check
First-order energy shift	Nondegenerate Perturbation Theory	First-Order Perturbation Theory
Explicit oscillator correction	Anharmonic Oscillator Perturbation	oscillator matrix elements
Degenerate subspace	Degenerate Perturbation Theory	diagonalize perturbation in the degenerate space
Variational estimate	Helium Atom Variational Estimate	Variational Bound
Time-dependent transition	Fermi’s Golden Rule	Fermi Golden Rule
Semiclassical barrier	WKB Barrier Tunneling	WKB Quantization

Minimal Worked Check

For an unperturbed normalized eigenstate $\lvert n^{(0)}\rangle$ of $H_0$ and perturbation $\lambda V$, the first-order shift is

$$E_n^{(1)} =\langle n^{(0)}\rvert V\lvert n^{(0)}\rangle.$$

The expression is not the whole method. It assumes a nondegenerate level or a basis already chosen after diagonalizing $V$ in the degenerate eigenspace.

Common Mistakes

• Applying nondegenerate formulas inside a degenerate multiplet.
• Treating a large correction as a controlled approximation.
• Forgetting selection rules that make a matrix element vanish.
• Mixing the perturbation parameter into $V$ inconsistently.
• Using WKB through a turning point without the needed connection formula.
• Treating a variational upper bound as an error estimate by itself.

Record for Each Example

• $H_0$ and its known spectrum;
• perturbation $V$ and expansion parameter;
• degeneracy status;
• order of approximation;
• symmetry or selection rule;
• comparison with an exactly solvable or limiting case.

References

• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, Wiley, 1977.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.

Exercises

1. Why is the first-order formula above unsafe for a hydrogenic level with fixed principal quantum number $n$?

Solution

The Coulomb spectrum has degeneracies beyond the magnetic quantum number. A perturbation can mix states inside the degenerate subspace, so one must first diagonalize the perturbation restricted to that subspace.