Abstract
The development of time-dependent perturbation theory was initiated by Paul Dirac’s early work on the semi-classical description of atoms interacting with electromagnetic fields. Dirac, Wheeler, Heisenberg, Feynman and Dyson developed it into a powerful set of techniques for studying interactions and time evolution in quantum mechanical systems which cannot be solved exactly. It is used for the quantitative description of phenomena as diverse as proton-proton scattering, photo-ionization of materials, scattering of electrons off lattice defects in a conductor, scattering of neutrons off nuclei, electric susceptibilities of materials, neutron absorption cross sections in a nuclear reactor etc. The list is infinitely long. Time-dependent perturbation theory is an extremely important tool for calculating properties of any physical system.
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Notes
- 1.
P.A.M. Dirac, Proc. Roy. Soc. London A 112, 661 (1926).
- 2.
- 3.
The transformation law for operators from the Schrödinger picture into the interaction picture implies H D (t) ≡ V D (t). The notation V D (t) is therefore also often used for H D (t).
- 4.
If the perturbation V (t) contains directional information (e.g. polarization of an incoming photon or the direction of an electric field), then we might also like to calculate probabilities for the direction of dissociation of the hydrogen atom. This direction would be given by the \(\boldsymbol{k}\) vector of relative motion between the electron and the proton after separation. For the calculation of directional information we would have to combine the spherical Coulomb waves | k, ℓ, m〉 into states which approximate plane wave states \(\vert \boldsymbol{k}\rangle\) at infinity, similar to the construction of incoming approximate plane wave states in Section 13.5, see also the discussion of the photoeffect in [3].
- 5.
Recall that the notation tacitly implies dependence of the operators V and W on x and p (just like we usually write H instead of H(x, p) for a Hamilton operator).
- 6.
G. Wentzel, Z. Phys. 43, 524 (1927).
- 7.
See the discussion after equation (13.38) for an explanation why we can deal with monochromatic perturbations as abridged non-hermitian operators.
- 8.
J.R. Oppenheimer, Z. Phys. 55, 725 (1929).
- 9.
W. Wessel, Annalen Phys. 397, 611 (1930); E.C.G. Stückelberg, P.M. Morse, Phys. Rev. 36, 16 (1930); M. Stobbe, Annalen Phys. 399, 661 (1930).
- 10.
Alternatively, we could have used box normalization for the incoming plane waves, \(\langle \boldsymbol{x}\vert \boldsymbol{k}\rangle =\exp (\mathrm{i}\boldsymbol{k} \cdot \boldsymbol{ x})/\sqrt{V}\) both in \(dw_{\boldsymbol{k}\rightarrow \boldsymbol{k}'}\) and in \(\boldsymbol{j}\) ( ⇒ \(\boldsymbol{j} = \hslash \boldsymbol{k}/(mV ) =\boldsymbol{ v}/V\)), or we could have rescaled both \(dw_{\boldsymbol{k}\rightarrow \boldsymbol{k}'}\) and \(\boldsymbol{j}\) with the conversion factor 8π 3∕V to make both quantities separately dimensionally correct, \([dw_{\boldsymbol{k}\rightarrow \boldsymbol{k}'}] =\mathrm{ s}^{-1}\), \([\boldsymbol{j}] =\mathrm{ cm}^{-2}\mathrm{s}^{-1}\). All three methods yield the same result for the scattering cross section, of course.
- 11.
M.V. Berry, Proc. Roy. Soc. London A 392, 45 (1984).
Bibliography
H.A. Bethe, E.E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Springer, Berlin, 1957)
J. Orear, A.H. Rosenfeld, R.A. Schluter, Nuclear Physics: A Course Given by Enrico Fermi at the University of Chicago (University of Chicago Press, Chicago, 1950)
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Dick, R. (2016). Time-dependent Perturbations in Quantum Mechanics. In: Advanced Quantum Mechanics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-25675-7_13
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