What Bohmian mechanic says about arrival times of 1D vacuum squeezed states

Abstract We calculate the time of arrival probability distribution of a quantum particle using the Bohmian formalism. The pilot-wave is given by the wave function of the one dimensional vacuum squeezed state written in the Schrödinger representation. To obtain this pilot-wave, we used the unitary representation of the symplectic group in the Hilbert space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> . The solution to the Bohmian equation is a closed expression in time thus allowing for a closed expression of the time of arrival distribution. We show the dependence of the time of arrival distribution as a function of the squeezing parameter, the ratio of the detector’s position and the proper length of the oscillator and the squeezing phase parameter.