# Time dependence in quantum theory

1. Mar 23, 2015

### TrickyDicky

I'm trying to elucidate certain concepts about time dependence and perturbation theory in quantum mechanics and QFT.
I get the impression that most of the important results that in principle can be considered to have a time dependence can actually be calculated in terms of time-independent perturbation methods. I'm thinking for example about the case of scattering both in QM and QFT particle physics, where most of the important calculations like scattering cross-sections, decay rates, energies of bound states, etc can be made by matching the asymptotic states of the time-independent Hamiltonian in the context of the S-matrix.
Or the use of the Dyson series in the context of the Feynman diagrams in QED perturbative renormalization.
And Fermi's golden rule wich is a time-dependent perturbation derivation that can also be used in the context of time-independent problems.

It seems the strict time-dependent perturbation is only needed to obtain the quantum transition probabilities under a external perturbation.

Is the above impression correct, or is it wrong and if so can someone give examples of applications or physical problems where the time-dependent perturbation is used in the everyday practice either within the formal obtention of transition probabilities or beyond?

2. Mar 23, 2015

### vanhees71

Yes, it's correct. By definition, scattering theory is about transition probabilities between asymptotically free in and out states. Although the time-dependent approach with wave packets is the most intuitive and physically correct way to derive the definition of the corresponding S-matrix elements, you can also entirely work in terms of time-independent perturbation theory. See, e.g., S. Weinberg, The Quantum Theory of Fields, Vol. 1, where this is very well explained in the context of relativistic QFT.