My goal with this site is to make QM as intuitive as possible by exploring the extent to which visualizations can be used to clarify the subject. One of my primary tools for doing this are what might be called “trajectory plots” of the wavefunction. There are 2 steps to generating these plots: First, create a scatter-plot of the initial probability distribution, then evolve each point along the “streamlines of probability flow”. At each instant in time, the density of the points in a region will correspond to the probability density of the wavefunction (to within numerical error). This is arguably the most compact visual representation of the wavefunction possible. The state’s density and phase are clearly visible, and it allows for the possibility of displaying multiparticle systems.

Attempts at using probability streamlines to interpret the Schrodinger Equation go back to the early days of Quantum Theory. Madelung investigated the idea in 1926, in what he called the Hydrodynamic form of Quantum Theory. Later, David Bohm went down a similar path with his “Pilot Wave” theory.

One of the advantages of their approach is that they unpack Schrodinger’s Equation in a way that is much easier to digest than the standard form. The typical introductory approach is to simply state the equation, out of the blue, then work numerous examples. Eventually, the student will get a feel for how the solutions behave. A much more intuitive presentation, in my opinion, begins with the Hamilton Jacobi Equation from Classical Mechanics, then compares Schrodinger’s Equation to it. They are almost identical in form.

As far as interpreting QM goes, we’re going to explore the idea that “the trajectories are what is really there” and see where it leads us. This approach is surprisingly effective, despite it simplicity. It is an excellent first approximation towards a complete theory, and we will be examining where the idea breaks down.

So without further ado…

Chapter 1

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An essay on free content by Sam Harris