Blurry trajectories

Ever since the beginnings of quantum mechanics, there has been the question around how to get to the classical limit. First was Schrödinger's paper where he presented wavepackets oscillating inside a harmonic well (nowadays known as "coherent states"). It soon turned out that this example was a little too special, as it is the exception to have no spreading of the wave packet over time. Semiclassical methods were soon developed that tried to make use of the classical trajectories, at least in order to describe the propagation of wave packets at high energies, for wave lengths much shorter than the typical length scales of the potential. In this vein of thinking, the Lagrangian was introduced into quantum mechanics by Dirac, and then fully exploited by Feynman to construct the path integral approach.

However, you don't need to go very far away from the ground state in order to illustrate something resembling classical motion. Take, for example, an electron quantum wire – a "waveguide for electrons", infinitely extended in one direction, and bounded in the other directions. The energy eigenstates are freely propagating plane waves in the direction of motion along the wire, multiplied by some transverse eigenfunctions depending on the confining potential in the other directions. Different transverse eigenmodes have different energies, forming several "transport channels" (just like in a microwave guide or an optical fibre for electromagnetic waves).

The general rules of quantum mechanics tell us the following: If you superimpose two states of equal energy but different transverse channels (with different momenta along the wire), you can again form an energy eigenstate. It turns out that this eigenstate resembles the kind of zig-zag-motion, that classically you would expect if you were to launch a particle at an angle into the wire, such that it reflects off the boundaries.

This is best illustrated in pictures. They show the probability density (stationary in time, since this is an energy eigenstate!). The only difference between them is the relative weight of the two superimposed states (with more weight given to the first excited transverse mode in the second picture).

Zig-zag-motion in a quantum wire (1)

Zig zag motion 2

This looks already like a blurry version of the classical motion, doesn't it?

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