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Canadian Journal of Physics

On the orthogonality condition for the excited states of stationary quantum systems

C. S. Sharma

Published Jan 1, 1968• 14 citations
Publisher

Abstract

A theorem establishing the correct orthogonality condition for the perturbation expansions of the state vectors for the excited states of stationary quantum systems is enunciated. A common misunderstanding on this subject is discussed and corrected. Implications of the theorem to the use of the variation perturbation method for calculating approximate eigenvalues and eigenvectors for excited states is discussed.

DOI
10.1139/p68-006

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A theorem establishing the correct orthogonality condition for the perturbation expansions of the state vectors for the excited states of stationary quantum systems is enunciated. A common misunderstanding on this subject is discussed and corrected. Implications of the theorem to the use of the variation perturbation method for calculating approximate eigenvalues and eigenvectors for excited states is discussed.
It is pointed out that the notation of the theory of functionals and the theorem in it analogous to Taylor's theorem facilitates a compact discussion of perturbation and variation methods in quantum mechanics, both for bound state and scattering problems. With this aid it is shown (i) how stationary expressions required for variation methods may be sought and found, (ii) that the perturbation and variation methods are virtually identical, (iii) that second order perturbations and expressions stationary to the second order may be found similarly, though these are complicated. The form found for the second order perturbation of bound states differs from that commonly used in that it does not explicitly involve the eigenfunctions of states other than that perturbed.
The steepest descent perturbation theory is extended to the calculation of the energy and eigenfunction of the excited state of a quantum system. In case of the orthogonality of the trial function for the excited state to those for lower-energy state or ground state in the same symmetry class is preserved, the variational collapse to lower energy state can be avoided in this proposal. An iterative procedure is given for generating better eigenvalue and eigenfunction of the excited state without requiring an infinite summation over reference states as in conventional perturbation theory. This new perturbation method can be applied to calculate the excitation energy and wave function of excited states for any many-body quantum system to a high degree of accuracy without so much computational effort as in conventional method.
Potential scatterings in L2L^2L2 space: (1) non-orthogonality of stationary states
Orthogonality of eigenstates of different energies and its implications in potential scattering are unlabeled. Scalar products of scattering states of different energies are found to have finite non-orthogonal terms in potentials of finite widths. Their superpositions have time-dependent norms, and are not suitable for isolate states. In these systems, a perturbative method and a variational method are viable methods for finding a rigorous transition probability that describes phenomena completely. In various exceptional potentials, an orthogonality is satisfied.
In this chapter, we present some fundamental issues about approximation methods that are often used when a quantum-mechanical system is perturbed and about the relationship between classical and quantum mechanics. In Sec. 10.1 we introduce the stationary perturbation theory, while Sec. 10.2 is devoted to time-dependent perturbations. In Sec. 10.3 we briefly examine the adiabatic theorem. In Sec. 10.4 we introduce the variation method, an approximation method that is not based on perturbation theory. In Sec. 10.5 we discuss the classical limit of the quantum-mechanical equations, whereas in Sec. 10.6 we deal with the semiclassical approximation, in particular the WKB method. On the basis of the previous approximation methods in Sec. 10.7 we present scattering theory. Finally, in Sec. 10.8 we treat a method that has a wide range of applications: the path-integral method. Stationary perturbation theory Perturbation theory is a rather general approximation method that may be applied when a small additional force (the perturbation) acts on a system (the unperturbed system), whose quantum dynamics is fully known. If the disturbance is small, it modifies both the energy levels and the stationary states. This allows us to make an expansion in power series of a perturbation parameter, which is assumed to be small. Perturbation theory may be applied both to the case where the additional force is time-independent (in which case a stationary treatment suffices – the subject of the present section) as well as to the case where it explicitly depends on time.