# Quantum Motion in an Infinite Spherical Well

Quantum Motion in an Infinite Spherical Well

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.

R=1

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 .

-0.36⩽,=⩽0.36

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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).

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