Abstract
The chapter is devoted to the solution of the Schrödinger equation for the linear harmonic oscillator, and to a number of important application of the results. The importance of the problem has already been outlined in the sections devoted to the classical treatment: its results can in fact be applied, with little or no modification, to mechanical situations where the positional force acting on the particle can be replaced with a first-order expansion, or to more complicate systems whose degrees of freedom can be separated into a set of Hamiltonian functions of the linear-harmonic-oscillator type. Such systems are not necessarily mechanical: for instance, the energy of the electromagnetic field in vacuo is amenable to such a separation, leading to the formal justification of the concept of photon. Similarly, a system of particles near a mechanical-equilibrium point can be separated in the same manner, providing the justification of the concept of phonon. An interesting property of the Fourier transform of the Schrödinger equation for the linear harmonic oscillator is shown in the complements.
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Notes
- 1.
The term “ladder” is introduced in Sect. 13.3.
- 2.
The eigenfunction’s nature of being either even or odd is related to a general property of the one-dimensional Schrödinger equation, specifically: if the eigenvalue E is simple and the potential energy V is even, the eigenfunction corresponding to E is either even or odd (Prob. 8.4).
- 3.
To complete the description of the photon it is necessary to work out also the quantum expression of its momentum. This is done in Sect. 12.4. The concept of photon was introduced by Einstein in 1905 [45] (the English translation of [45] is in [133]). The quantization procedure shown here is given in [41].
- 4.
The equilibrium distribution of the phonons’ occupation numbers ζ(σ) is the Bose-Einstein statistics (Sect. 15.8.2).
- 5.
Compare with (C.83), where the property is demonstrated for the Gaussian function; the latter, apart from scaling factors, coincides with the eigenfunction of the linear harmonic oscillator belonging to the eigenvalue corresponding to n = 0.
References
P.A.M. Dirac, The quantum theory of the emission and absorption of radiation. Proc. R. Soc. Lond. A 114, 243–265 (1927)
A. Einstein, Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt. Annalen der Physik 17(6), 132–148 (1905) (in German). English translation: D. ter Haar, The Old Quantum Theory, Pergamon Press, pp. 91–107, 1967
I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, New York, 1980)
A. Messiah, Mécanique Quantique (Dunod, Paris, 1969) (in French). English edition: Quantum Mechanics (Dover, New York, 1999)
D. ter Haar, On a heuristic point of view about the creation and conversion of light, in The Old Quantum Theory, pp. 91–107 (Pergamon Press, London, 1967)
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Rudan, M. (2018). Cases Related to the Linear Harmonic Oscillator. In: Physics of Semiconductor Devices. Springer, Cham. https://doi.org/10.1007/978-3-319-63154-7_12
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DOI: https://doi.org/10.1007/978-3-319-63154-7_12
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