Visual Representation of Separation of Varables

[sonnichs]

TL;DR

Are there any good written or video tutorials out there that shows graphically how separation of variables works

Are there any good visualization tutorials, written or video, that show graphically how separation of variables works?
I particularly have the time-independent Schrödinger Equation in mind. There are hundreds of demonstrations out there which essentially distill to copies of one another. However I am trying to visualize in my mind how this process looks graphically - for example plotting t on one axis and x on the other for f(x,t).
I have seen other good visual representations of mathematical methods but none for this one.
thanks-fritz

 

[jedishrfu]

I searched around but couldn't a geometric way of explaining separation of variables.

It seems it's really a niche technique that works for a small set of partial differential equations. You make an assumption that the solution is a product of functions of each independent variable and see where it leads you.

It leaves open the issue of when it does and doesn't work and whether it generates a unique solution or any solution at all. However, there are certain partial differential equations that are common to physics and engineering like the wave equation where separation of variables does work and that makes it a worthwhile technique to learn.

What would be enlightening is if you or someone at PF contacted Grant Sanderson of the 3brown1blue youtube channel to see if he knows a visual way of presenting separation of variables. He has many videos already where he presents beautiful visualizations for calculus, linear algebra and many famous math problems.

 

[sonnichs]

Thank You Jedishrfu !
I too am a fan of 3brown1blue and have sent a message as you suggested. I got my degrees 50 years ago and I think years later Grant, unlike so many others, has gone beyond the common use of computers and made use of them to visualize math.
My knowledge of math is largely related to Quantum mechanics and as you say maybe separation of variables is not common.

I have long had similar thoughts about exact differentials which appear a lot in thermodynamics. Graphing a "smooth" surface, a point on that surface can be approached continuously from many different directions. But I think when the differential is not exact, there is a discontinuity depending upon the direction of approach. I can "think" this but don't quite know how to graph it.

I will report here if I get word from Grant.
thanks Fritz

 

[jedishrfu]

You might be interested in Geometric Algebra and Calculus too. Its a repackaging of vector analysis ideas among other topics and can be used in many fields of ohysics including quantum mechanics.

Its also more intuitive and works across all dimensions not just 3.

They define the geometric product and how vectors are multiplied that makes more sense than having a dot product and cross product. At least the was my feeling.

 

[jedishrfu]

Prof David Hestenes is a proponent of GA and GC for physicists.

https://davidhestenes.net/geocalc/html/Evolution.html#References

And QM

https://davidhestenes.net/geocalc/html/Impl_QM.html#CAIntQM