The continuous spectra of quantum operators

Foundations of Physics 12 (6):583-606 (1982)

The linear vector space for the quantum description of a physical system is formulated as the intersection of the domains of Hermiticity of the observables characterizing the system. It is shown that on a continuous interval of its spectrum every Hermitian operator on a Hilbert space of one degree of freedom is a generalized coordinate with a conjugate generalized momentum. Every continuous spectral interval of a Hermitian operator is the limit of a discrete spectrum in the same interval. This result is applied to description of the state of a system by a statistical operator p( $\hat q$ ) following measurement of the probabilities of the eigenvalues of an observable $\hat q$ . Each continuous interval in the spectrum of $\hat q$ is replaced by a discrete spectrum in the same interval with a parameterK; the limitK → ∞ in which the spectrum becomes continuous is not attainable physically. In the linear vector space of a quantum mechanical system every continuous interval in the spectra of the observables is similarly replaced by a discrete spectrum with a finite parameter: it is a space of discrete bases and a set of such parameters. The case of degenerate spectrum is also discussed
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DOI 10.1007/BF00731930
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