Abstract
We will now turn to the problem that conventional quantum mechanics courses generally begin with that of the quantum system describing a free particle moving in physical space \(\mathbf R^3\). This is something quite different from the classical mechanical description of a free particle, which will be reviewed in chapter 14. A common way of motivating this is to begin with the 1924 suggestion by de Broglie that just as photons may behave like either particles or waves, the same should be true for matter particles.
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Notes
- 1.
This requires restricting attention to a specific class of functions for which the Taylor series converges to the function.
- 2.
For the case of \(\mathcal H\) infinite dimensional, this is known as Stone’s theorem for one-parameter unitary groups, see, for instance, section 10.2 of [41] for details.
- 3.
We are here neglecting questions of whether these integrals are well defined, which require more care in specifying the class of functions involved.
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© 2017 Peter Woit
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Woit, P. (2017). Momentum and the Free Particle. In: Quantum Theory, Groups and Representations. Springer, Cham. https://doi.org/10.1007/978-3-319-64612-1_10
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DOI: https://doi.org/10.1007/978-3-319-64612-1_10
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