Complex Quantum Wave

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© William P. Reinhardt.

The mathematical description of a particle makes use of complex numbers, and can be divided into real and imaginary parts. Experiments can only observe the particle's probability density, which is the absolute value of the wavefunction squared and therefore a real, positive number. However, we can calculate the effects of the complex wavefunction itself. The top image illustrates the ground state wavefunction of a particle trapped in a box. The color scale represents the complex phase of the quantum wavefunction, which is changing periodically in time, and the amplitude of the wave represents the real part of the particle's wavefunction. If we follow the particle's time evolution by drawing a line across the graph parallel to the time axis, we see that this quantum amplitude undulates up and down, behaving like a wave. Note that the undulations correspond to the changes in phase. The lower image shows the probability density for the same particle—which would actually be observed in an experiment. We know that the particle is in a stationary state because the density is the same at all times. However, the stripes that correspond to the complex phase of the wavefunction indicate that the underlying complex wavefunction—which we cannot directly observe—is still waving away. (Unit: 6)