On the Quantum Potential and Pulsating Wave Packet in the Harmonic Oscillator

in Dubois, Daniel (Ed.) COMPUTING ANTICIPATORY SYSTEMS (2008)

A fundamental mathematical formalism related to the Quantum Potential factor, Q, is presented in this paper. The Schrödinger equation can be transformed to two equations depending on a group velocity and ... [more ▼]

A fundamental mathematical formalism related to the Quantum Potential factor, Q, is presented in this paper. The Schrödinger equation can be transformed to two equations depending on a group velocity and a density of presence of the particle. A factor, in these equations, was called “Quantum Potential” by D. Bohm and B. Hiley. In 1999, I demonstrated that this Quantum Potential, Q, can be split in two Quantum Potentials, Q1, and Q2, for which the relation, Q=Q1+Q2, holds. These two Quantum Potentials depend on a fundamental new variable, what I called a phase velocity, u, directly related to the probability density of presence of the wave-particle, given by the modulus of the wave function. This paper gives some further developments for explaining the Quantum Potential for oscillating and pulsating Gaussian wave packets in the Harmonic Oscillator. It is shown that the two Quantum Potentials play a central role in the interpretation of quantum mechanics. A breakthrough in the formalism of the Quantum Mechanics could be provoked by the physical properties of these Quantum Potentials. The probability density of presence of the oscillating and pulsating Gaussian wave packets in the Harmonic Oscillator is directly depending on the ratio Q2/Q1 of the two Quantum Potentials. In the general case, the energy of these Gaussian wave packets is not constant, but is oscillating. The energy is given by the sum of the kinetic energy, T, the potential energy, V, and the two Quantum Potentials: E=T+V+Q1+Q2. For some conditions, given in the paper, the energy can be a constant. The first remarkable result is the fact that the first Quantum Potential, Q1, is related to the ground state energy, E0, of the Quantum Harmonic Oscillator: Q1=[barred aitch]omega/2=E0. The second result is related to the property of the second Quantum Potential, Q2, which plays the role of an anti-potential, Q2=-V(x), where V is the harmonic oscillator potential. This Quantum Potential counter-balances the harmonic oscillator potential, so there is no more harmonic potential in the quantum harmonic oscillator. It remains just a constant potential given by the first Quantum Potential, Q1. The interpretation is as follows: a quantum system can annihilate a classical potential, and so gives rise to a quantum tunnelling, which violates the principles of Classical Mechanics. [less ▲]