Quantum billiards are an important class of systems showing a large variety of dynamical behavior ranging from regular motion through quasiperiodic behavior to strongly chaotic behavior. Suppose a single quantum particle, an atom, is in a superposition of two energy eigenstates, in the absence of measurement. This could be achieved by exciting the atom with coherent laser pulses. For this system, a transformation from regular to quasiperiodic motion is shown in the Bohm trajectories for this "unobserved" system. If, before emitting a photon, a position or energy measurement is made, the atom will remain in the superposition state. A quantum trajectory in an infinite spherical potential well of radius  could be described by the de Broglie–Bohm approach [1, 2], using spherical Bessel functions. Due to the large oscillations of the superposed wavefunction in configuration space, the trajectory could become very unstable and it could leave the billiard boundary for certain time intervals. To ensure that the trajectories lie within the billiard boundary, the initial point is restricted to the region  . In the graphic, you can see the wave density (if enabled); a possible orbit of a quantum particle, where the trajectory (blue) depends on the initial starting point (  ,  ,  ); the initial starting point of the trajectory (shown as a small black sphere) and the position (shown as a small blue sphere).
An unnormalized wavefunction  of the Schrödinger equation with mass  is separable in spherical polar coordinates, such that  , where  is a spherical harmonic and  a spherical Bessel function [3]. The boundary condition that  at  is fulfilled when  is the  zero of the spherical Bessel function  . The quantized energy levels are then given by  , where  is the central potential energy with  . In this Demonstration, the total wavefunction  is defined by a superposition of two stationary eigenstates: with  . For simplicity, set the mass  and  equal to 1 (atomic units [4]). In this case, the wavefunction for the quantum particle in an infinite spherical well in spherical polar coordinates [1] reads  . The velocity field  with the position  can be calculated from the current or from the gradient of the total phase function  from the wavefunction  in the eikonal form (often called polar form)  :  . From the total phase function  the gradient  , with the partial derivative  and so on, is given by:  ,  . For the initial point  the velocity becomes indeterminate, and for  the velocity field  becomes autonomous, with  , or in Cartesian coordinates,  , and the atom orbits the  axis along a circle of constant radius and with a constant angular speed depending on the azimuthal quantum number  (here,  , which is a multiple of  or  . For greater accuracy, increase PlotPoints, AccuracyGoal, PrecisionGoal and MaxSteps. Due to the computational limitations, here only two superposed states are investigated. For this case, there is no chaotic behavior in the quantum motion [5, 6]. [1] Bohmian-Mechanics.net. (Oct 18, 2019) [3] O. F. de Alcantara Bonfim, J. Florencio and F. C. Sá Barreto, "Chaotic Bohm’s Trajectories in a Quantum Circular Billiard," Physics Letters A, 277(3) , 2000 pp. 129–134. doi:10.1016/S0375-9601(00)00705-2. [5] R. H. Parmenter and R. W. Valentine, "Deterministic Chaos and the Causal Interpretation of Quantum Mechanics," Physics Letters A, 201(1), 1995 pp. 1–8. doi:10.1016/0375-9601(95)00190-E. [6] R. H. Parmenter and R. W. Valentine, "Erratum: Deterministic Chaos and the Causal Interpretation of Quantum Mechanics ( Physics Letters A 210 (1995) 1)," Physics Letters A, 213(5–6), 1996 p. 319. doi:10.1016/0375-9601(96)00096-5.
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