A spacetime singularity is located at the center of a black hole and surrounded by an event horizon, separating spacetime into two disjunct regions: one of them accessible to an outside observer and one that is not. At the event horizon, a logarithmic phase singularity emerges in the mode functions of a massless scalar field, being characteristic for Hawking radiation emitted by the black hole. We demonstrate that related features are present in the elementary quantum system of an inverted harmonic oscillator. Central to our analysis are the energy eigenfunctions of this system and their phase space representation. At first glance, neither a horizon nor a logarithmic phase dependence are apparent. However, both features are hidden in phase space and revealed by a suitable coordinate transformation. In particular, we show that the Fourier transform of a logarithmic phase leads to an expression that is reminiscent of a specific quantum statistics, governing the reflection and transmission coefficients of the inverted harmonic oscillator.
Fifty years ago Kostin (J Chem Phys 57(9):3589–3591, 1972. https://doi.org/10.1063/1.1678812) proposed a description of damping in quantum mechanics based on a nonlinear Schrödinger equation with the potential being governed by the phase of the wave function. We show for the example of a moving Gaussian wave packet, that the deceleration predicted by this equation is the result of the same non-dissipative, homogeneous but time-dependent force, that also stops a classical particle. Moreover, we demonstrate that the Kostin equation is a special case of the linear Schrödinger equation with three potentials: (i) a linear potential corresponding to this stopping force, (ii) an appropriately time-dependent parabolic potential governed by a specific time dependence of the width of the Gaussian wave packet and (iii) a specific time-dependent off-set. The freedom of the width opens up the possibility of engineering the final state by the time dependence of the quadratic potential. In this way the Kostin equation is a precursor of the modern field of coherent control. Motivated by these insights, we analyze in position and in phase space the deceleration of a Gaussian wave packet due to potentials in the linear Schrödinger equation similar to those in the Kostin equation.
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