More mass = longer

I hope you don't have ambiguous thoughts with this title. But that's the only title I could think about for this topic. And the topic is an excellent piece of math that will draw the crowds [1].

The Schrödinger equation, stripped of constants, looks something like this:



Now we switch to mass-weighted coordinates:



And we change the wave function to mass-weighted coordinates:



Now we don't need that tedious mass in front of the derivative because:


(Usually I would complain if someone shows a prove like this. But I think it's easier to see this way. And I also did it the "clean" chain rule way.)

Using this we get Schrödingers equation without the mass:



That means: the product of mass^1/2 and length is more natural than just length.
It also means: if an elephant walks for one meter it's the same thing as if a mouse walks 1000m (assuming that the squareroot of an elephant's mass is 1000 as much as the squareroot of a mouse's mass).

The idea is from this article. It turns out that the approach is handy if you want to propagate wave packets over a grid you obtained from linear interpolation. And who doesn't want to do that?

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[1] Just like my diploma thesis never ceases to catch the interest of everyone I talk to.