Stationary Perturbations in Quantum Mechanics
We denote a quantum system with a time-independent Hamiltonian H0 as solvable (or sometimes also as exactly solvable) if we can calculate the energy eigenvalues and eigenstates of H0 analytically. The harmonic oscillator and the hydrogen atom provide two
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Stationary Perturbations in Quantum Mechanics
We denote a quantum system with a time-independent Hamiltonian H0 as solvable (or sometimes also as exactly solvable) if we can calculate the energy eigenvalues and eigenstates of H0 analytically. The harmonic oscillator and the hydrogen atom provide two examples of solvable quantum systems. Exactly solvable systems provide very useful models for quantum behavior in physical systems. The harmonic oscillator describes systems near a stable equilibrium, while the Hamiltonian with a Coulomb potential is an important model system for atomic physics and for every quantum system which is dominated by Coulomb interactions. However, in many cases the Schrödinger equation will not be solvable, and we have to go beyond solvable model systems to calculate quantitative properties. In these cases we have to resort to the calculation of approximate solutions. The methods developed in the present chapter are applicable to perturbations of discrete energy levels by timeindependent perturbations V of the Hamiltonian, H0 → H = H0 + V .
9.1 Time-Independent Perturbation Theory Without Degeneracies We consider a perturbation of a solvable time-independent Hamiltonian H0 by a time-independent term V , and for bookkeeping purposes we extract a coupling constant λ from the perturbation, H = H0 + V → H = H0 + λV .
(9.1)
After the relevant expressions for shifts of states and energy levels have been calculated to the desired order in λ, we usually subsume λ again in V , such that e.g. λφ (0) |V |ψ (0) → φ (0) |V |ψ (0) . © 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_9
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9 Stationary Perturbations in Quantum Mechanics
We know the unperturbed energy levels and eigenstates of the solvable Hamiltonian H0 , H0 |ψj(0) = Ej(0) |ψj(0) .
(9.2) (0)
In the present section we assume that the energy levels Ej are not degenerate, and we want to calculate in particular approximations for the energy level Ei which arises from the unperturbed energy level Ei(0) due to the presence of the perturbation V . We will see below that consistency of the formalism requires that the differences (0) (0) |Ei − Ej | for j = i must have a positive minimal value, i.e. the unperturbed (0)
energy level Ei for which we want to calculate corrections has to be discrete.1 Orthogonality of eigenstates for different energy eigenvalues implies (0)
(0)
ψi |ψj = δij .
(9.3)
In the most common form of time-independent perturbation theory we try to find an approximate solution to the equation H |ψi = Ei |ψi
(9.4)
in terms of power series expansions in the coupling constant λ, |ψi =
λn |ψi(n) , ψi(0) |ψi(n≥1) = 0, Ei =
n≥0
λn Ei(n) .
(9.5)
n≥0
Depending on the properties of V , these series may converge for small values of |λ|, or they may only hold as asymptotic expansions for |λ| → 0. The book by Kato [94] pro
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