Abstract
This chapter describes the quantum mechanics of a harmonic oscillator, which is of essential importance in treating a phonon, using creation and annihilation operators. A number state, a coherent state, and a squeezed state are introduced. The number state is an eigenstate of the energy of the harmonic oscillator. The coherent state is a minimum uncertainty state. In the squeezed state, fluctuation of one of the conjugated variables is reduced and lowered than that of the vacuum state. Coherent and squeezed states are important in quantum optics and phononics. Time evolution of the mean value of position and variance for the harmonic oscillator is discussed.
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Notes
- 1.
\(\hat{n}^\dag =(\hat{a}^\dag \hat{a})^\dag = (\hat{a})^\dag (\hat{a}^\dag )^\dag = \hat{a}^\dag \hat{a} =\hat{n}\), because \((\hat{A}\hat{B})^\dag = \hat{B}^\dag \hat{A}^\dag \).
- 2.
m is, of course, a nonnegative integer, because it is number of times for the operation.
- 3.
The occupation number state is also called the Fock state for photons in quantum optics.
- 4.
The wave function \(\varphi _(x)\) is obtained by projection of the ket vector \(|\varphi \rangle \) into position space and has the relationship \(\varphi _(x) = \langle x | \varphi \rangle \).
- 5.
The concept of the coherent state is first proposed by Schrödinger [3]. Much work on the coherent state in quantum optics was performed by R. J. Glauber. The coherent state is also called the Glauber state in quantum optics.
- 6.
The Poisson distribution \(P(n) = e^{-\lambda } \lambda ^k / k !\) represents the probability of events occurring k times per unit time, when the events occur randomly \(\lambda \) per unit time.
- 7.
This is because \((\hat{x}+i\hat{p}) \exp (-i \theta /2) = (\hat{x}+i\hat{p})(\cos (\theta /2)-i \sin (\theta /2)) = (\cos (\theta /2) \hat{x}+ \sin (\theta /2)\hat{p})+i (-\sin (\theta /2)\hat{x}+\cos (\theta /2)\hat{p})\).
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I wrote this chapter referencing Refs. [1–5] and following books: Yoshio Kuramoto and Junichi Ezawa, Ryoushi Rikigaku (in Japanese, Quantum Mechanics), Asakura Shyoten (2008); Keiji Igi and Hikari Kawai, Kiso Ryousi Rikigaku (in Japanese, Fundamental Quantum Mechanics), Kodansya (2007); Masahito Ueda Gendai Ryoushi Buturigaku (in Japanese, Modern Quantum Physics), Baifukan (2004); Akira Shimizu, Shin-han Rryoushi Ron no Kiso (in Japanese, New edition Fundamental of Quantum Physics), Science Shya (2004); Kyo Inoue, Kogaku Kei no Tameno Ryoushi Kogaku (in Japanese, Quantum Optics for Engineer), Morikita Shyoten (2015); Masahiro Matsuoka, Ryoushi Kogaku (in Japanese, Quantum Optics), Shokabou (2000)
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Nakamura, K. (2019). Harmonic Oscillator and Coherent and Squeezed States. In: Quantum Phononics. Springer Tracts in Modern Physics, vol 282. Springer, Cham. https://doi.org/10.1007/978-3-030-11924-9_3
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DOI: https://doi.org/10.1007/978-3-030-11924-9_3
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