Perturbation theory (quantum mechanics)

Perturbation theory (in quantum mechanics) is a set of approximation schemes for reducing the mathematical analysis of a complicated quantum system to a simpler mathematical solution. The simpler quantum system is considered as being perturbed by weak physical disturbances, leading to a useful mathematical approximation for the original, more complicated system. Quantum perturbation theory is a generalization of classical perturbation theory.

Quotes

• Logarithmic perturbation theory is an alternative way of solving the perturbation equations in the coordinate representation. It was developed many years ago ... and has lately been widely discussed and applied to many problems in quantum mechanics. ... perturbation theory allows us to transform the nonlinear Riccati equation ... into a set of linear differential equations ...
  • Francisco M. Fernández: Introduction to Perturbation Theory in Quantum Mechanics. CRC Press. 19 September 2000. pp. 17–18. ISBN 978-1-4200-3964-1. 

• The separation of the hamiltonian into an unperturbed part and a perturbation is not unique, but in most problems of interest there is a separation which presents itself in a most natural way. In quantum electrodynamics for example, the unperturbed system consists of the electron-positron field and the photon field without interaction. In the theory of an imperfect gas the unperturbed system will be taken as the ideal gas obtained by neglecting interparticle interactions. In the application of perturbation theory to large quantum systems one encounters problems not met with in the usual perturbation theory of systems with a finite number of degrees of freedom. These problems are related to the following phenomena:

1. Self-energy and cloud effects of individual particles in excited states.
2. The perturbation of the system as a whole.

• • N. M. Hugenholtz: (1957). "Perturbation theory of large quantum systems". Physica 23 (1): 481–532. DOI:10.1016/S0031-8914(57)92950-6. (quote from p. 482)

• The great success of calculations in quantum electrodynamics using the renormalization idea generated a new enthusiasm for quantum electrodynamics. After this change of mood, probably most theorists simply didn’t worry about having to deal with infinite renormalizations. Some theorists thought that these infinities were just a consequence of having expanded in powers of the electric charge of the electron, and that not only observables but even quantities like the “bare” electron charge (the charge appearing in the field equations of quantum electrodynamics) would be found to be finite when we learned how to calculate without perturbation theory. But at least two leading theorists had their doubts about this, and thought that the appearance of infinite renormalizations in perturbation theory was a symptom of a deeper problem, a problem not with perturbation theory but with quantum field theory itself. They were Lev Landau, and Gunnar Källén.
  • Steven Weinberg: (2009). "Living with Infinities". ArXiv:0903.0568. (p. 15)