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Perturbations and Quantum Relaxation

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Abstract

We investigate whether small perturbations can cause relaxation to quantum equilibrium over very long timescales. We consider in particular a two-dimensional harmonic oscillator, which can serve as a model of a field mode on expanding space. We assume an initial wave function with small perturbations to the ground state. We present evidence that the trajectories are highly confined so as to preclude relaxation to equilibrium even over very long timescales. Cosmological implications are briefly discussed.

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Notes

  1. Note that the properties of a general nonequilibrium ensemble cannot be described by \(\psi \) alone.

  2. An exception is an early paper by Bohm [35], which considered an ensemble of two-level molecules subject to random external collisions and argued that the molecules would relax to equilibrium.

  3. Specifically, the signature amounts to a primordial power deficit at long wavelengths with a specific (inverse-tangent) dependence on wavenumber k [21, 23]. A large-scale power deficit has in fact been reported in the Planck data [39, 40], though the extent to which it matches our prediction is still being evaluated [41].

  4. The initial phases were \(\theta _{00} = 0.5442\), \(\theta _{01} = 2.3099\), \(\theta _{10} = 5.6703\), \(\theta _{11} = 4.5333\).

  5. For the illustrative figures in this subsection, the initial phases were as follows: \(\theta _{00} = 4.8157\), \(\theta _{01} = 1.486\), \(\theta _{10} = 2.6226\), \(\theta _{11} = 3.8416\).

  6. For the trajectories in Fig. 12, the initial phases were \(\theta _{00} = 4.2065\), \(\theta _{01} = 0.1803\), \(\theta _{02} = 2.0226\), \(\theta _{10} = 5.5521\), \(\theta _{11} = 3.3361\), \(\theta _{20} = 2.6561\).

  7. For a square centered at the origin, the coordinates of the 13 points would be as follows: \((-\,0.02,0.02)\), \((-\,0.02,0.0)\), \((-\,0.02,-\,0.02)\), \((0.0,-\,0.02)\), \((0.02,-\,0.02)\), (0.02, 0.0), (0.02, 0.02), (0.0, 0.02), \((-\,0.01,0.01)\), \((-\,0.01,-\,0.01)\), \((0.01,-\,0.01)\), (0.01, 0.01), (0.0, 0.0).

  8. For the trajectories displayed in Fig. 13, the initial phases used were \(\theta _{00} = 1.2434\), \(\theta _{01} = 4.411\), \(\theta _{02} = 4.3749\), \(\theta _{10} = 4.2427\), \(\theta _{11} = 1.5574\), \(\theta _{20} = 5.7796\).

  9. For the trajectories displayed in Fig. 15, the initial phases used were \(\theta _{00} = 4.0857\), \(\theta _{01} = 0.2194\), \(\theta _{02} = 4.6059\), \(\theta _{10} = 1.2201\), \(\theta _{11} = 0.439\), \(\theta _{20} = 4.0563\).

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  1. Kinard Laboratory, Department of Physics and Astronomy, Clemson University, Clemson, SC, 29634-0978, USA

    Adithya Kandhadai & Antony Valentini

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  1. Adithya Kandhadai
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  2. Antony Valentini
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Correspondence to Antony Valentini.

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Kandhadai, A., Valentini, A. Perturbations and Quantum Relaxation. Found Phys 49, 1–23 (2019). https://doi.org/10.1007/s10701-018-0227-3

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