Foundations of Physics

, Volume 25, Issue 11, pp 1599–1620 | Cite as

Non-Heisenberg states of the harmonic oscillator

  • K. Dechoum
  • H. M. FranÇa


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 Ω0)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 〈(δx2)〉〈(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 hΩ0/2 and 〈(δx)2〉〈(δp)2〉=h2/4 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.


Average Energy Quantum Electrodynamic Liouville Equation Stochastic Equation Mechanical Oscillator 
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Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • K. Dechoum
    • 1
  • H. M. FranÇa
    • 1
  1. 1.Instituto de FisicaUniversidade de SÃo PauloSÃo Paulo, SPBrazil

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