Quantum theory of motion of a time-dependent harmonic oscillator in the pilot-wave theory

The de Broglie-Bohm quantum trajectories are found in analytically closed forms for the eigenstates and the coherent state of the Lewis-Riesenfeld (LR) invariant of a time-dependent harmonic oscillator. It is also shown that an eigenstate (a coherent state) of an invariant can be interpreted as squeezed states obtained by squeezing an eigenstate (a coherent state) of another invariant. This provides ways for a whole description of squeezed states. ∗Electronic address: [email protected] †Electronic address: [email protected] 1 The quantum theory of motion introduced by de-Broglie and Bohm provides another viewpoints on the world of quantum mechanics [1,2]. There are many examples which explain the structure of the atom, the interference and the tunneling in the quantum motional scheme [3]. The quantum trajectories have been numerically solved in a time-dependent scattering from square barriers and square potential wells [4,5]. Recently, the damped harmonic oscillator is analyzed according to de Broglie-Bohm theory [6]. However explicitly time-dependent systems have not yet been dealt with. Since the exact wave function for an explicitly time-dependent harmonic oscillator is recently found [7], we expect that one may calculate the quantum trajectories for an eigenstate, a coherent state, and a squeezed state of the system. In this Letter, we find the de Broglie-Bohm quantum trajectories for a time-dependent harmonic oscillator. The LR invariants [8] are constructed using the creation and the annihilation operators which are expressed in terms of a classical solution. It is argued that since any linear combination of two independent classical solutions is also a solution, there are many invariants and the invariant can be specified uniquely (except for a constant factor) if we fix the classical solution. It is shown that the invariants with different solutions are obtained by squeezing each other with a squeezing operator. Here the squeezing operator describes the Bogoliubov transformation between two sets of creation and annihilation operators. For an invariant, we consider the eigenstates and coherent states as a guiding wave in the de Broglie-Bohm theory and find the corresponding quantum trajectories. Because these states are squeezed states for another invariant, we have a whole description of squeezed states. The solutions are applied to a time-independent oscillator and a damped oscillator. In the damped oscillator we find the new type of quantum trajectories which have not been found in Ref. [6]. Here the quantum motions are oscillating while damping. In the causal interpretation of quantum mechanics, the wave is mathematically described by Ψ(x, t), a solution to Schrödinger’s wave equation