Abstract
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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Leaf, B. The continuous spectra of quantum operators. Found Phys 12, 583–606 (1982). https://doi.org/10.1007/BF00731930
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DOI: https://doi.org/10.1007/BF00731930