Abstract
This chapter, the longest in the book, introduces three of the basic assumptions of quantum mechanics and then illustrates them, using mainly the example of the harmonic oscillator. Though some historical remarks are included, neither the historical development nor any other heuristic way towards quantum mechanics is followed. The basic assumptions are formulated, explained, and applied. In Sections II.2, II.4, the basic assumptions are introduced; in Sections II.3, II.5, II.7 the harmonic oscillator is used to illustrate them. Section II.6 contains the derivations of some general consequences and might be omitted in first reading. The discussion for the continuous spectra, important for the description of the scattering and decay phenomena in the second part of the book, is given in Section II.S. Several remarks throughout this chapter emphasize the particular problems connected with generalized eigenvalues and eigenvectors and our unified treatment of continuous and discrete spectra. In Section II.9 we are ready to explain the physical meaning of the quantum-mechanical constant of nature, ℏ.
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References
From G. J. Schulz, Phys. Rev. 135, A988 (1964), with permission.
Multiple scattering of electrons and CO molecules is negligible because the intensity of the electron current e and the density of CO molecules are sufficiently low.
For operators with continuous spectra one has to start with (I.4.4c) instead of (I.4.4d); the spectral theorem looks similar to (4.42) with the sum replaced by an integral, cf. K. Maurin (1967), p. 131.
In fact (4.45a) is the precise statement of commutativity; in order to make (4.44) precise one has to add the requirement that A 2 + B 2 be essentially self-adjoint (its Hilbert space closure is self-adjoint).
For the mathematical problems connected with this definition see K. Maurin (1967), Section V.5.
Cf. Appendix of Section I.6 or N. N. Lebedev, Special Functions and Their Applications, Prentice Hall, Englewood Cliffs, N.J., 1965, Section 10.4.
An elementary discussion of this subject is given in A. Bohm, The Rigged Hilbert Space and Quantum Mechanics, Springer Lecture Notes in Physics, vol. 78, 1978.
For the applications of coherent states consult Sargent Scully-Lamb (1974), Ch. XV.
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© 1986 Springer Science+Business Media New York
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Bohm, A. (1986). Foundations of Quantum Mechanics—The Harmonic Oscillator. In: Quantum Mechanics: Foundations and Applications. Texts and Monographs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-01168-3_2
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DOI: https://doi.org/10.1007/978-3-662-01168-3_2
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