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CRC Press eBooks

1 On “Improper” Functions in Quantum Mechanics

L.D. Faddeev, L.A. Khalfin, I.V. Komarov

Published May 21, 2004• 0 citations
Publisher

Abstract

T (x, y) of an integral operator T which has no true kernel. The statement that an improper function T (x, y) is the limit of a sequence Tn(x, y) of usual functions, strictly speaking, makes no sense. The absence of the limit is just the reason why the function T (x, y) is “improper.”
DOI
10.1201/9780203643204-13

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The usual definition of the time evolution operator e−iHt/ℏ=∑n=0∞1n!(−iℏHt)n, where H is the Hamiltonian of the system, as given in almost every book on quantum mechanics, causes problems in some situations. The operators that appear in quantum mechanics are either bounded or unbounded. Unbounded operators are not defined for all the vectors (wave functions) of the Hilbert space of the system; when applied to some states, they give a non-normalizable state. Therefore, if H is an unbounded operator, the definition in terms of the power series expansion does not make sense because it may diverge or result in a non-normalizable wave function. In this article, we explain why this is so and suggest, as an alternative, another definition used by mathematicians.
It is explicitly shown the weak points (being not rigorous) in the reasoning of Dirac, of Landau, of Von Neumann, of Weyl, of Fadeev and of others, deriving many fundamental relations, which later became classical and are being commonly used in Quantum Theory up to now. It is because of their derivations were made when functional analysis was not yet sufficiently developed. Those weak points are of two following reasons:1. Unbounded property of the physical operators is not yet paid by attention.2. The “Dirac delta function” defined by Dirac is not always distinguished clearly with the Dirac distribution defined by Schwartz and with the generalized functions defined by Gel'fand.Particularly, the formulation of the closure relation by using equality of the functionals is not rigorous. Consequently, it is necessary to improve Quantum Theory.