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Visual Examples: Bound States of the Harmonic Oscillator

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The Bohmian trajectories for a particle in a 1-dimensional parabolic (harmonic oscillator) potential with wave function psi = sqrt(3)/2 |0> + 1/2 |1>, where |n> stands for the n-th energy eigenstate. So this is a linear combination of the two lowest eigenstates. Since the two frequencies of these eigenstates are 1/2 and 3/2, the relative frequency is 1, thus the wave function is periodic in time. In one dimension, the trajectories inherit periodicity from the wave function. One period is shown in the image.

The numerics was done with Maple V.

The |psi|2 distribution for the same psi as before, sqrt(3)/2 |0> + 1/2 |1>. You can see it is periodic in time.

"The head of the bat". These are the trajectories for a slightly more complex wave function, psi' = 1/sqrt(3) (|0> + |1> + |2>). They still have the same period length.

Now we take a look at the 2-dimensional harmonic oscillator. The easiest to compute are the trajectories for product states, here psi(x,y,t) = phi(x,t) phi(y,t) with phi = sqrt(3)/2 |0> + 1/2 |1> (a function also depicted above).

The 100 points shown with their Bohmian motion in the animation are roughly |psi|2 distributed, but the discretization grid is still visible. For points near the diagonal, the motion has some similarity to a classical pendulum with zero angular momentum, swinging from one direction to the opposite. But since the velocity on the diagonal is tangent to the diagonal, the motion cannot cross the diagonal, and so points away from the diagonal go back and forth on some C- or even L-shaped trajectory, a rather non-classical sort of motion.

Different wave function, different motion. Here, psi(x,y,t) = phi(x,t) phi(y,t + period/4) with the same phi as before, sqrt(3)/2 |0> + 1/2 |1>. Again, 100 points are roughly |psi|2 distributed and act out the Bohmian motion in the animation.

This time, the motion resembles rather the circular motion of a classical 2-d oscillator. However, the trajectories are neither circles nor centered at the minimum of the potential (which lies at the center of the image), in contrast to the classical solution. For a typical wave function, the trajectories will not even be closed, let alone have the same period. In our example, this is a consequence of psi's being a product state.

By Roderich Tumulka.

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