Quantum Mechanics and Classical Probability Theory

V. I. Man’ko
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V. I. Man’ko: Lebedev Physical Institute

A chapter in Symmetries in Science IX, 1997, pp 225-242 from Springer

Abstract: Abstract Standard quantum mechanics is based on notion of a complex wave function which satisfies the Schrödinger equation [1]. The attempts to give classical-like interpretations of the wave function were done in [2–4]. It turned out that the new formulation of standard quantum mechanics may be given in terms of classical probabilities for the position [5, 6] based on symplectic tomography scheme [7, 8]. Recently, the energy levels of the harmonic oscillator were discussed in frame of classical formulation of quantum mechanics [9] as well as transition probabilities between the levels. Using marginal distributions for homodyne observable [10, 11] to describe the quantum states is based on some relations of the density matrix to characteristic functions for the observable [12]. The marginal distribution for the position in ensemble of shifted, rotated, and scaled reference frames in classical phase space of the system under study has been introduced [7]. It was shown that this marginal distribution determines the quantum state since the Wigner function is given by a Fourier component of the marginal distribution. The invariant form for the connection of the marginal distribution for the position to the density matrix was found [8] and the approach was extended to the system with several degrees of freedom.

Keywords: Reference Frame; Quantum State; Density Matrix; Harmonic Oscillator; Marginal Distribution (search for similar items in EconPapers)
Date: 1997
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DOI: 10.1007/978-1-4615-5921-4_16

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