The time-independent perturbation theory in quantum mechanics is formulated using projection operator techniques. The determination of the perturbed eigenvalue can be decoupled from that of the perturbed eigenstate. Both the Brillouin–Wigner theory and the Rayleigh–Schrödinger theory come out straightforwardly. Both degenerate and nondegenerate cases can be treated in a unified way for arbitrarily high order perturbations.

## REFERENCES

1.

E. Schrödinger,

*Collected Papers on Wave Mechanics*(Blackie, London and Glasgow, 1928), pp. 64–76.2.

R. L. Liboff,

*Introductory Quantum Mechanics*(Addison–Wesley, Reading, MA, 1998), 3rd ed., pp. 701–730.3.

E. P. Wigner,

*The Collected Works of Eugene Paul Wigner*(Springer-Verlag, Berlin, 1997), Part A, Vol. IV, pp. 131–136.4.

K. Hannabuss,

*Introduction to Quantum Theory*(Clarendon, Oxford, 1997), pp. 221–226.5.

P.

Löwdin

, “Studies in perturbation theory. IV. Solution of eigenvalue problem by projection operator formalism

,” J. Math. Phys.

3

(5

), 969

–982

(1962

).6.

C. E.

Soliverez

, “An effective Hamiltonian and time-independent perturbation theory

,” J. Phys. C

2

(11

), 2161

–2174

(1969

).7.

Both authors have taught the

*Quantum Mechanics*course recently and became interested in the problem after having successfully used the projection operator method in research;see D. Yao and J. Shi, “Method of projection operator for the study of angle-averaged distribution function of beam particles in hadron storage rings,” Phys. Rev. Special Topics-Accelerators and Beams

**1**, 084001, 1–11 (1998).8.

A. Dalgarno, “Stationary perturbation theory,” in

*Quantum Theory*, edited by D. R. Bates (Academic, New York, 1961), Vol. I, pp. 171–209.9.

P.

Löwdin

, “Partitioning technique, perturbation theory, and rational approximation

,” Int. J. Quantum Chem.

21

(1

), 69

–92

(1982

).10.

J. M. Ziman,

*Elements of Advanced Quantum Theory*(Cambridge U.P., Cambridge, 1969), pp. 53–56.11.

W.

Silvert

, “Comparison of Rayleigh–Schrödinger and Brillouin–Wigner theories

,” Am. J. Phys.

40

(4

), 557

–561

(1972

).12.

M. L. Goldberger and K. M. Watson,

*Collision Theory*(Wiley, New York, 1964), Chap. 8, pp. 424–509.13.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg,

*Atom-Photon Interactions*(Wiley, New York, 1992), Chap. III, pp. 165–255.14.

A. Bohm,

*Quantum Mechanics*(Springer-Verlag, New York, 1993), 3rd ed., pp. 242–252.15.

L.

Mower

, “Decay theory of closely coupled unstable states

,” Phys. Rev.

142

(4

), 799

–816

(1966

).16.

C. Cohen-Tannoudji, “

Optical Pumping and Interaction of Atoms with the Electromagnetic Field,” in Cargèse Lectures in Physics, Vol. 2, edited by M. Lévy (Gordon and Breach, New York, 1968), pp. 347–393.

This content is only available via PDF.

© 2000 American Association of Physics Teachers.

2000

American Association of Physics Teachers

AAPT members receive access to the

*American Journal of Physics*and*The Physics Teacher*as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.