Understanding the Separation of Solutions in Partial Differential Equations"

[ChrisVer]

I am not sure again, whether this belongs here or in mathematics.
When we have a partial differential equation, in general we can write the solution of $F(r,t)$ as:
$F(r,t)=R(r)T(t)$
In the procedure of separating the differential equation, we find ourselves dividing with $F(r,t)$.
Isn't that actually problematic for points where $R$ or $T$ happen to be zero? I know that they can't be zero everywhere coz the solution would be trivial, but what stops them from being zero at distinct points? Actually nothing...
The only solution to this problem would be the imposing of continuity at the point of interest, setting the left side solution equal to the right side. But dividing with zero is still a problem :(

Thanks.

 

[MisterX]

I am not sure again, whether this belongs here or in mathematics.
When we have a partial differential equation, in general we can write the solution of $F(r,t)$ as:
$F(r,t)=R(r)T(t)$

Actually, you can not do this in general.

Also this is not something particular to quantum mechanics.

 

[ChrisVer]

Actually, you can not do this in general.

Also this is not something particular to quantum mechanics.

when can you do it?

I know, I was skeptic about it in the first place... I just came across it right now in QM for Schrod equation solving, and the question just stroke in my head.
For example, in the Hydrogen atom, there exist $r$ such that $F,R(r)=0$

 

[Meir Achuz]

Your question is more easily answered by a physicist than by a mathematician.
Physicists answer: Although you do write F/(RT), tha this only temporary.
You never actually divide by the R unless it is in the form R(r)/R(r)=1.
I something like R"/R=k, the first thing you do is change it to R"=kR.

 

[Hypersphere]

when can you do it?

It's quite tricky to find a simple and general criterion, but I remember seeing it be done for classes of equations, e.g. diffusion equations. If you are looking for a general answer, it probably will have something to do with symmetries and be rather distanced from how physicists would tend to proceed.

In practice, you do what one of my differential equations professors used to tell us: try the separation of variables method to every equation. If it works, it works and you probably saved a lot of time. If it doesn't work, this becomes clear after two steps of calculation or so, so not much time was lost there.

 

[ChrisVer]

hmm I think it's OK... what is your opinion about the continuity criterion?
I mean, what about if we indeed divide by $F(r,t)$ without caring about the zeros of it? but instead, solve automatically for those points by using the continuity of the $F(r,t)$ function?
Let's say that $R(r_{0})=0$ for some radius $r_{0}$... then we can solve it at $R(r_{0}\pmε)$ and then send $ε \rightarrow 0$ and imposing:
$R(r_{0}+ε)=R(r_{0}-ε)$

I think mathematically that's not incorrect (?)

 

[Hypersphere]

hmm I think it's OK... what is your opinion about the continuity criterion?
I mean, what about if we indeed divide by $F(r,t)$ without caring about the zeros of it? but instead, solve automatically for those points by using the continuity of the $F(r,t)$ function?
Let's say that $R(r_{0})=0$ for some radius $r_{0}$... then we can solve it at $R(r_{0}\pmε)$ and then send $ε \rightarrow 0$ and imposing:
$R(r_{0}+ε)=R(r_{0}-ε)$

I think mathematically that's not incorrect (?)

Generally, one would treat the case of $R=0$ separetely, and solve the differential equation for this case too, which is obviously permitted. Your method is essentially the same, if one matches the solutions using the smoothness property (the derivative of the wave function is continuous too), just as in a potential barrier problem. So yeah, I think it should work fine for the Schrödinger equation.

However, it can't be a general method for all partial differential equations. Non-linear PDEs (for which separation of variables admittedly don't work as often) and linear PDE problems with non-smooth initial conditions can give rise to discontinuous solutions, for example.