We apply the Walborn [Walborn S P, Taketani B G, Salles A, Toscano F and de Matos Filho R L 2009 Entropic Entanglement Criteria for Continuous Variables Phys. Rev. Lett. 103 160505] Shannon Entropic Entanglement Criterion (SEEC), to simple harmonic oscillators, a system typically discussed in undergraduate quantum mechanics courses. In particular, we investigate the entanglement of a system of coupled harmonic oscillators. A simple form of entanglement criterion in terms of their interaction is found. It is shown that a pair of interacting ground state coupled oscillators are more likely to be entangled for weaker coupling strengths while coupled oscillators in excited states entangle at higher coupling strengths. Interacting oscillators are more likely to be entangled as their interaction increases.
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/ℏ), when
ℏ/m≪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
m1,...,mn and frequencies
ω1,...,ωn 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(m~/ℏ), where
m~ is the mass of the minimum mass oscillator, when
ℏ/m~≪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.