On the time-dependent Rindler Hamiltonian single particle eigenstates in momentum space

Abstract

We have developed a formalism in the non-relativistic scenario to obtain the time evolution of the eigenstates of Rindler Hamiltonian in momentum space. Hence, the particle wave function in spacetime coordinates is obtained using Fourier transform of the momentum space wave function. We have discussed the difficulties with characteristic curves, and re-cast the time evolution equations in the form of two-dimensional Laplace equation. The solutions are obtained both in polar coordinates as well as in the Cartesian form. It has been observed that in the Cartesian coordinate, the probability density is zero both at [Formula: see text] (the initial time) and at [Formula: see text] (the final time) for a given [Formula: see text]-coordinate. The reason behind such peculiar behavior of the eigenstate is because it satisfies (1 + 1)-dimensional Laplace equation. This is of course the mathematical explanation, whereas physically we may interpret that it is because of the Unruh effect.