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Journal of Low Temperature Physics

The Kostin Equation, the Deceleration of a Quantum Particle and Coherent Control

Harald Losert, Freyja Ullinger, Matthias Zimmermann, Maxim A. Efremov, Ernst Maria Rasel, Wolfgang P. Schleich

Published Sep 19, 2022• 2 citations

Abstract

Abstract 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.

DOI: 10.1007/s10909-022-02857-y

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Abstract 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.

The structure of time-dependent Gaussian solutions for the Kostin equation in dissipative quantum mechanics is analyzed. Expanding the generic external potential near the center of mass of the wave packet, one conclude that: the center of mass follows the dynamics of a classical particle under the external potential and a damping proportional to the velocity; the width of the wave packet satisfy a non-conservative Pinney equation. An appropriate perturbation theory is developed for the free particle case, solving the long standing problem of finding analytic expressions for square integrable solutions of the free Kostin equation. The associated Wigner function is also studied.

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Abstract The second chapter introduces the notion of wave functions and their equation of motion proposed by Erwin Schrödinger. The interpretation and normalization of the wave function are discussed, leading to the continuity equation. The chapter also introduces basic mathematical operations that can be carried out for wave functions and operators, describing e.g., how to calculate scalar products and commutators, as well as expectation values and variances of observables. With these tools at hand, Heisenberg’s uncertainty principle is re-examined, focusing on the example of a Gaussian wave packet, which is studied both in position space and momentum space. We also discuss the time evolution of general states and of the exemplary wave packet. Finally, we will see how classical physics is recovered as a limiting case of the quantum-mechanical description

Abstract An effective operational approach to quantum mechanics is to focus on the evolution of wave packets, for which the wave function can be seen in the semi-classical regime as representing a classical motion dressed with extra degrees of freedom describing the shape of the wave packet and its fluctuations. These quantum dressing are independent degrees of freedom, mathematically encoded in the higher moments of the wave function. We review how to extract the effective dynamics for Gaussian wave packets evolving according to the Schrödinger equation with time-dependent potential in a 1 + 1-dimensional spacetime, and derive the equations of motion for the quadratic uncertainty. We then show how to integrate the evolution of all the higher moments for a general wave function in a time-dependent harmonic potential.

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