Quantum Mechanics and Classical Probability Theory

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.