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
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Notes
- 1.
P.A.M. Dirac, Proc. Roy. Soc. London A 112, 661 (1926).
- 2.
F.J. Dyson, Phys. Rev. 75, 1736 (1949).
- 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 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 | k⟩ at infinity, similar to the construction of incoming approximate plane wave states in Section \e13.5, see also the discussion of the photoeffect in [2].
- 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.
J.R. Oppenheimer, Z. Phys. 55, 725 (1929).
- 8.
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).
- 9.
Alternatively, we could have used box normalization for the incoming plane waves, \(\langle x\vert k\rangle =\exp (\mathrm{i}k \cdot x)/\sqrt{V}\) both in dw k → k′ and in j ( ⇒ j = ℏk ∕ (mV ) = v ∕ V ), or we could have rescaled both dw k → k′ and j with the conversion factor 8π3 ∕ V to make both quantities separately dimensionally correct, [dw k → k′ ] = s − 1, [j] = cm − 2s − 1. All three methods yield the same dimensionally correct result for the scattering cross section, of course.
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Dick, R. (2012). Time-dependent Perturbations in Quantum Mechanics. In: Advanced Quantum Mechanics. Graduate Texts in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8077-9_13
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