Time-Dependent Perturbations in Quantum Mechanics
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 s
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Time-Dependent Perturbations in Quantum Mechanics
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 [38]. 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. So far all the Hamiltonians which we had studied were time-independent. This property was particularly important for the time-energy Fourier transformation from the time-dependent Schrödinger equation to a time-independent Schrödinger equation. Time-independence of H also ensures conservation of energy, as will be discussed in detail in Chap. 16. Time-dependent perturbation theory, on the other hand, is naturally also concerned with time-dependent Hamiltonians H (t) (although it provides very useful results also for time-independent Hamiltonians, and we will see later that most of its applications in quantum field theory concern sytems with time-independent Hamiltonians). We will therefore formulate all results in this chapter for time-dependent Hamiltonians, and only specify to time-independent cases where it is particularly useful for applications.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_13
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13 Time-Dependent Perturbations in Quantum Mechanics
13.1 Pictures of Quantum Dynamics As a preparation for the discussion of time-dependent perturbation theory (and of field quantization later on), we now enter the discussion of different pictures of quantum dynamics. The picture which we have used so far is the Schrödinger picture of quantum dynamics: The time evolution of a system is encoded in its states |ψS (t) which have to satisfy a Schrödinger equation ihd|ψ ¯ S (t)/dt = H (t)|ψS (t). However, every transformation on states and operators |ψ → U |ψ, A → U · A · U + with a unitary operator U leaves the matrix elements φ|A|ψ and therefore the observables of a system invariant. If U is in particular a time-dependent unitary operator, then this changes the time-evolution of the states and operators without changing the time-evolution of the observables. Application of a time-dependent U (t) corresponds to a change of the picture of quantum dynamics, and two important cases besides the Schrödinger picture are the Heisenberg picture and the i
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