## Tutorial: Perturbation Approximations to Asymptotic Quantum Transition Rates

**Authors:** Steven Kenneth Kauffmann

The orders of the perturbation approximation to the Schroedinger-picture time evolution operator in powers of the presumed-small perturbation part of the Hamiltonian operator are developed by the iteration of an identity; the possibly confusing switch to the "interaction picture" isn't needed. Those time-evolution operator approximations are sandwiched between two orthogonal, normalized eigenstates of the "unperturbed" part of the Hamiltonian operator to produce the transition amplitude approximations, whose calculation is reduced to quadrature when every occurrence of the perturbation part of the Hamiltonian operator in them is expanded in the "unperturbed" basis. That expansion also reveals the time-dependent parts of those approximations to be multiply nested integrals which in the long-time limit approach simple products of non-singular inverses (principle value plus delta function) of differences of "unperturbed" energy eigenvalues. Closely related to the long-time limits of transition amplitudes are the long-time averages of transition rates. When the "unperturbed" basis is that of free-particle states, sums over relevant final states of transition rates from an initial state, divided by the initial state's particle flux, produce cross sections. Here the perturbation approximations to generic quantum transition rates are parlayed to the corresponding approximations to differential cross sections for nonrelativistic-particle potential scattering.

**Comments:** 8 Pages.

**Download:** **PDF**

### Submission history

[v1] 2019-08-27 07:23:31

**Unique-IP document downloads:** 13 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary.
In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution.
Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

**Add your own feedback and questions here:**

*You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.*

*
*