Re:Is the Schrödinger Equation True?
by ur-whale on Tuesday January 12, 2021 @05:54AM
(#25745116)
>Contrary to this article's claim, the equation describes the behaviour of Helium atoms perfectly well;
Yes, I was disappointed to read this as well. I suspect what he meant is that we can't derive closed form solutions for the Schrödinger equation in the case of the Helium atom.
But the equation can perfectly be integrated numerically ... in these days of rather powerful computers, closed-form solutions to diffeqs aren't as important as they used to. As a matter of fact, for most problems in QM, I suspect there's just no other way.
One step further, when reading QM books, there seem to be a strong tendency to try and assign "physical meaning" to purely mathematical techniques used to find closed form solution to diffeqs (eg separation of variables). I am not convinced that it is a fruitful endeavor.
Another reason he may be saying this could also be because with the Schrödinger equation for the Helium atom, assuming it isn't separable along space variables, the solution ψ is a map from the configuration space, (i.e. ℝ6) to ℂ, something that's hard and annoying to visualize, whereas for the Hydrogen atom, you can draw pretty orbital probability density pictures in ℝ3, something everyone can "understand".
Yes, I was disappointed to read this as well. I suspect what he meant is that we can't derive closed form solutions for the Schrödinger equation in the case of the Helium atom.
But the equation can perfectly be integrated numerically ... in these days of rather powerful computers, closed-form solutions to diffeqs aren't as important as they used to. As a matter of fact, for most problems in QM, I suspect there's just no other way.
One step further, when reading QM books, there seem to be a strong tendency to try and assign "physical meaning" to purely mathematical techniques used to find closed form solution to diffeqs (eg separation of variables). I am not convinced that it is a fruitful endeavor.
Another reason he may be saying this could also be because with the Schrödinger equation for the Helium atom, assuming it isn't separable along space variables, the solution ψ is a map from the configuration space, (i.e. ℝ6) to ℂ, something that's hard and annoying to visualize, whereas for the Hydrogen atom, you can draw pretty orbital probability density pictures in ℝ3, something everyone can "understand".