Abstract
In this chapter, we will explore how to describe the position and momentum of a particle in quantum mechanics, and how we set up its Schrödinger equation. To this end, we need to work out how to describe the momentum and energy for a particle. We will conclude with a description of a bizarre phenomenon called quantum tunnelling, which is now widely used in advanced microscopes, and with a qualitative overview of the quantum mechanical foundations of chemistry.
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Notes
- 1.
There are many ways in which we can encounter infinity, and many different types of infinity. The symbol \(\infty \) is a shorthand for how we ended up at infinity; you cannot use this symbol in regular arithmetic without paying attention to the context of how the infinity came about. This is why we have to introduce limiting procedures that provide this context.
- 2.
L. de Broglie, Recherches sur la théorie des quanta, PhD thesis, Paris 1924; Annales de Physique 3 22, 1925.
- 3.
Also, photons in free space can have varying momentum, but their velocity is always c, which doesn’t tell us very much.
- 4.
W. Heisenberg, Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, Zeitschrift für Physik 33 879, 1925.
- 5.
E. Schrödinger, An Undulatory Theory of the Mechanics of Atoms and Molecules, Phys. Rev. 28 1049, 1926.
- 6.
The procedure of replacing classical quantities with operators is called quantisation. Once you know which operators the classical quantities correspond to you can quantise nearly every problem in physics.
- 7.
If you have already taken a course on special relativity, you may recognise \(px-Et\) as the product between the position and momentum four-vectors, even though our discussion has been completely non-relativistic. The quantity \(px-Et\) is the accumulated phase of a wave, which is a Lorentz invariant scalar.
- 8.
G. Binnig and H. Rohrer, Scanning tunneling microscopy, IBM Journal of Research and Development 30 355, 1986.
- 9.
W. Pauli, Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren, Zeitschrift für Physik 31 765, 1925.
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Kok, P. (2023). Motion of Particles. In: A First Introduction to Quantum Physics. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-031-16165-0_8
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