The harmonic oscillator

Charles G. Wohl

I. H. Umirzakov

As known all physical properties of solids are described well by the system of quantum linear harmonic oscillators. It is shown in the present paper that the system consisting of classical linear harmonic oscillators having temperature dependent masses or (and) frequencies has the same partition function as the system consisting of quantum linear harmonic oscillators having temperature independent masses and frequencies while the means of the square displacements of the positions of the oscillators from their mean positions for the system consisting of classical linear harmonic oscillators having: the temperature dependent masses; temperature dependent frequencies; and temperature dependent masses and frequencies differ from each other and from that of the system consisting of quantum linear harmonic oscillators, and hence, the system consisting of classical linear harmonic oscillators describes well the thermodynamic properties of the system consisting of quantum linear harmonic oscillators and solids.

Chung‐In Um, Kyu-Hwang Yeon, Thomas F. George +1 more

The minimum uncertainty and other relations are evaluated in the framework of the coherent states of the damped harmonic oscillator. It is shown that the coherent states of the damped harmonic oscillator are the squeezed coherent states of the simple harmonic oscillator. The unitary operator is also constructed, and this connects coherent states with damped harmonic and simple harmonic oscillators.

S. M. Blinder

Can Gokler

We show that the Schrödinger equation for the quantum harmonic oscillator can be derived as an approximation to the Newtonian mechanics of a classical harmonic oscillator subject to a random force for time intervals $O( m / \hbar)$, when $\hbar / m \ll 1$. Conversely, every solution to the Schrödinger equation, including all the superposition states, arises this way. In other words, the quantum harmonic oscillator is approximately nothing but the classical harmonic oscillator, with the same mass and frequency as the quantum harmonic oscillator, subject to a random force. We generalize the result to multiple non-interacting oscillators. We show that the Schrödinger equation for $n$ non-interacting quantum harmonic oscillators with masses $m_1, ..., m_n$ and frequencies $ω_1, ..., ω_n$ can be derived as an approximation to the Newtonian mechanics of $n$ non-interacting classical harmonic oscillators, with the same set of masses and frequencies as the quantum oscillators, subject to random forces. This is valid for time intervals $O( \tilde{m} / \hbar)$, where $\tilde{m}$ is the mass of the minimum mass oscillator, when $\hbar/\tilde{m} \ll 1$. Conversely, every solution, including all the entangled states, to the Schrödinger equation arises this way. In other words, $n$ non-interacting quantum harmonic oscillators are approximately nothing but $n$ non-interacting classical harmonic oscillators, with the same set of masses and frequencies as the quantum oscillators, subject to random forces. This provides a local Newtonian model of entanglement of non-interacting quantum oscillators. The correlations required by entangled states are embedded in the phase space probability density of the classical oscillators.

Jochen Pade