Abstract
The harmonic oscillator is one of the most important systems of physics.
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Notes
- 1.
These names arise less from the simple harmonic oscillator as we treat it here, but rather from quantum field theory. There, one uses the ladder operators to describe the creation and annihilation of photons, phonons, and so on. Moreover, these operators are used in many-particle quantum mechanics.
Generalized ladder operators may also be defined in general one-dimensional potentials. This leads to supersymmetric quantum mechanics (see, e.g. Schwabl, p. 351ff; Hecht, p. 130, and other relevant literature).
- 2.
In three dimensions, it is \(\hbar \omega \left( n+ \frac{3}{2}\right) \).
- 3.
This is quite similar to the case of the angular momentum.
- 4.
The oscillator length \(L\) essentially specifies the positions of the classical turning points; see the exercises.
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Pade, J. (2014). The Harmonic Oscillator. In: Quantum Mechanics for Pedestrians 2: Applications and Extensions. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-00813-4_18
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DOI: https://doi.org/10.1007/978-3-319-00813-4_18
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