Quantum mechanics

Quantum perturbation theory

Perturbation theory estimates shifts in energy levels and eigenstates when a Hamiltonian is close to a solvable problem.

Many realistic Hamiltonians cannot be solved exactly. Write $H=H_0+\lambda V$ , where the spectrum of $H_0$ is known and $V$ is a small correction. Expanding in the dimensionless parameter $\lambda$ gives controlled predictions when levels are nondegenerate and the correction is weak.

E_n=E_n^{(0)}+\lambda\langle n|V|n\rangle+\lambda^2\sum_{m\ne n}\frac{|\langle m|V|n\rangle|^2}{E_n^{(0)}-E_m^{(0)}}+O(\lambda^3)

Definition: First-order correction

For a nondegenerate level $|n\rangle$ , the first-order energy shift is $\Delta E_n^{(1)}=\lambda\langle n|V|n\rangle$ . The eigenstate also changes; its first-order admixture of $|m\rangle$ scales as $\langle m|V|n\rangle/(E_n^{(0)}-E_m^{(0)})$ . A small denominator signals that degeneracy must be treated explicitly or another method used.

Track a two-state spectrum as a coupling perturbation is gradually turned on.

Degeneracy and energy spectra

If several states share the unperturbed energy, first diagonalize $V$ within the degenerate subspace. For two levels separated by $\delta$ and coupled by $g$ , exact eigenvalues split by $2\sqrt{(\delta/2)^2+|g|^2}$ around their mean; at resonance the gap is $2|g|$ .

Example: Example: shift in a box

A constant perturbation $V(x)=V_0$ inside an infinite well has expectation value $\langle n|V|n\rangle=V_0$ . Every level shifts by $\lambda V_0$ at first order, leaving level spacings unchanged at that order.

Solution

Using $\Delta E_n^{(1)}=\lambda\langle n|V_0|n\rangle$ and normalization $\langle n|n\rangle=1$ gives $\Delta E_n^{(1)}=\lambda V_0$ .

At first order, the nondegenerate energy shift is $E_n^{(1)}=\langle n|V|n\rangle$ . At second order, other states contribute $E_n^{(2)}=\sum_{m\ne n}|V_{mn}|^2/(E_n^{(0)}-E_m^{(0)})$ . The denominator explains why a small perturbation can have a large effect near degeneracy; there one first diagonalizes $V$ in the degenerate subspace. The state correction likewise contains first-order mixing. These formulas assume a sufficiently weak perturbation and an isolated state; continuous spectra require different normalization care.

Quick check

For a nondegenerate level, what is the first-order energy shift?

How should a perturbation be treated for a degenerate level?

References

David J. Griffiths, Darrell F. Schroeter (2018). Introduction to Quantum Mechanics