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Making sense of the divergent series for reconstructing a Hamiltonian from its eigenstates and eigenvalues

Bender, Carl M., Brody, Dorje C. and Parry, Matthew F. (2019) Making sense of the divergent series for reconstructing a Hamiltonian from its eigenstates and eigenvalues American Journal of Physics.

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Abstract

In quantum mechanics the eigenstates of the Hamiltonian form a complete basis. However, physicists conventionally express completeness as a formal sum over the eigenstates, and this sum is typically a divergent series if the Hilbert space is infinite dimensional. Furthermore, while the Hamiltonian can be reconstructed formally as a sum over its eigenvalues and eigenstates, this series is typically even more divergent. For the simple cases of the square-well and the harmonic-oscillator potentials this paper explains how to use the elementary procedure of Euler summation to sum these divergent series and thereby to make sense of the formal statement of the completeness of the formal sum that represents the reconstruction of the Hamiltonian.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
NameEmailORCID
Bender, Carl M.
Brody, Dorje C.d.brody@surrey.ac.uk
Parry, Matthew F.
Date : 2019
Copyright Disclaimer : © 2019 American Association of Physics Teachers. The following article has been submitted to/accepted by American Journal of Physics. After it is published, it will be found at https://aapt.scitation.org/journal/ajp
Related URLs :
Depositing User : Clive Harris
Date Deposited : 17 Oct 2019 14:34
Last Modified : 17 Oct 2019 14:34
URI: http://epubs.surrey.ac.uk/id/eprint/852950

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