# Non-Heisenberg states of the harmonic oscillator

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## Abstract

*The effects of the vacuum electromagnetic fluctuations and the radiation reaction fields on the time development of a simple microscopic system are identified using a new mathematical method. This is done by studying a charged mechanical oscillator (frequency Ω*_{0})*within the realm of stochastic electrodynamics, where the vacuum plays the role of an energy reservoir. According to our approach, which may be regarded as a simple mathematical exercise, we show how the oscillator Liouville equation is transformed into a Schrödinger-like stochastic equation with a free parameter h′ with dimensions of action. The role of the physical Planck's constant h is introduced only through the zero-point vacuum electromagnetic fields. The perturbative and the exact solutions of the stochastic Schrödinger-like equation are presented for h′>0. The exact solutions for which h′<h are called sub-Heisenberg states. These nonperturbative solutions appear in the form of Gaussian, non-Heisenberg states for which the initial classical uncertainty relation takes the form 〈(δx*^{2})*〉〈(δp)*^{2}*〉=(h′/2)*^{2},*which includes the limit of zero indeterminacy (h → 0). We show how the radiation reaction and the vacuum fields govern the evolution of these non-Heisenberg states in phase space, guaranteeing their decay to the stationary state with average energy hΩ*_{0}*/2 and 〈(δx)*^{2}*〉〈(δp)*^{2}*〉=h*^{2}*/4 at zero temperature. Environmental and thermal effects-are briefly discussed and the connection with similar works within the realm of quantum electrodynamics is also presented. We suggest some other applications of the classical non-Heisenberg states introduced in this paper and we also indicate experiments which might give concrete evidence of these states*.

## Keywords

Average Energy Quantum Electrodynamic Liouville Equation Stochastic Equation Mechanical Oscillator## Preview

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