Abstract
In this chapter I present a somewhat more formal introduction to the theory that underlies quantum mechanics. Although the discussion is limited mainly to single particles, many of the results apply equally well to many-particle systems. Some of the postulates of the theory depend on the properties of Hermitian operators, operators that play a central role in quantum mechanics. First I state the postulates, then discuss Hermitian operators, and finally explore some results that follow directly from Schrödinger’s equation.
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Notes
- 1.
The fact that \(\left \vert c_{12}\right \vert <1\) follows from the Schwarz inequality, \(\left \vert \left ( \psi _{a_{n}}^{(1)},\psi _{a_{n}}^{(2)}\right ) \right \vert ^{2}\leq \left ( \psi _{a_{n}}^{(1)},\psi _{a_{n}}^{(1)}\right ) \left ( \psi _{a_{n}}^{(2)},\psi _{a_{n}}^{(2)}\right ) =1.\)
- 2.
Note that, in the momentum representation, the eigenfunctions of \(\hat {p}\) are Φ q (p) = δ(p − q) and the eigenfunctions of \(\hat {x}\) are \(\Phi _{x}(p)=e^{-ipx/\hbar }/\sqrt {2\pi \hbar }\).
- 3.
See, for example, M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, U.K., 1997), Chapter 19.
- 4.
Experiments of this type were pioneered in the group of Serge Haroche, who was awarded the Nobel prize in recognition of these and other experiments.
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Berman, P.R. (2018). Postulates and Basic Elements of Quantum Mechanics: Properties of Operators. In: Introductory Quantum Mechanics. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-68598-4_5
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DOI: https://doi.org/10.1007/978-3-319-68598-4_5
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