Maxwell speed distribution and analogue Hawking–Unruh temperature in an ontological model of a Harmonic oscillator ground state

Within an ontological (hidden variable) model of quantum fluctuation, one can discuss the actual   properties of a system regardless (independent) of measurement. Here we apply an ontological model proposed earlier to investigate a Harmonic oscillator in the quantum mechanical ground state. We first show that the actual speed of the oscillator fluctuates randomly following the Maxwell–Boltzmann distribution. On the other hand, the actual energy obeys a broad Gamma distribution with an average 3ħω/23ħω/2, where ωω is the classical angular frequency, so that one may conclude that the outcome of a single energy measurement reveals the average of the actual energy. The distribution of actual speed (energy) thus formally resembles the distribution of speed (energy) of an ideal gas in thermal equilibrium of temperature Tg=ħω/2Tg=ħω/2. We shall then argue that TgTg can be written in a form analogous to the Hawking temperature for a Schwarzschild black hole in which the average distance of the oscillator from the origin plays the analogous role of the radius of the black hole event horizon. It can also be written in a form analogous to the Unruh temperature experienced by a body moving with a uniform acceleration. In the analogy, the oscillator suffers an effective acceleration which balances the attractive force of the trapping Harmonic potential, thus keeps its average position away from the origin.