So I’m trying to visualize what is going on in QM geometrically. More specifically I would like to visualize the time dependent wave equation in 3 dimensions. So let’s start with dimensionality. Normally when I think of a function of some variables, I picture it in a space with the number of dimensions equal to the number of variables plus one for the function. For instance, y(x) is usually mapped in 2 dimensions. So then my first impression of the wave equation is that it is a function of four variables, psi(x,y,z,t), so I should picture it in 5 dimensional space right?(adsbygoogle = window.adsbygoogle || []).push({});

This seemed obvious to me but when I talked to my professor about it he didn’t like the idea but he didn’t really explain himself. So I thought about it some more and I guess his point was that because you can still describe any point in that function with 4 coordinates then it is only 4 dimensional (the definition of dimensionality). I think that does make sense. If we take y(x)=x^2, you could transform the coordinate system so that it is just a straight line with unique points or you could map it as a scalar field on the x axis. So then it really is only 1 dimensional. We then think of the wave function as a scalar field in 4 dimensions so I guess problem solved right?

I’m not so sure. Look back at y(x)=x^2 and add y(x)=x to that plot. Now there is no way to describe those two functions in one dimension because you need more than one coordinate to describe any point. Even if you transform the coordinate system or plot as a scalar field you will have overlap. This system is truly 2 dimensional. So what if we consider two electron guns firing together once every second. If you plot the scalar field of the probabilities of the two electrons then you need 5 coordinates to describe any point in that system (the x y z position, the time, and which particle). Trust me this sounds crazy to me too so I’m thinking I’m just missing something simple and obvious. Anyone care to explain?

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# Can someone give me a geometrical description of the math in QM?

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