How Does Griffith Prove Psi Stays Normalized in Equation [1.25]?

[nymphidius]

I'm having trouble understanding what David Griffith did in equation [1.25]. In this section he's trying to prove that psi stays normalized and I'm following him from [1.21] to [1.25] and where I'm getting stuck is understanding how:

∂/∂t|ψ|^2= i$\hbar$/2m(ψ*∂²/∂x²[ψ]-∂²/∂x²[ψ*]ψ) [1.25]

He makes this equal to yet another confusing equation, but I think if just understood what substitution he made then I'd be able to progress. (This is independent study so I have no university professor to ask).

Also, I'm assuming someone has the book to look at what the confusion is. If someone wants to help but doesn't have the book, I'm willing to write out steps 1.21 to 1.25.

 

[jfy4]

Consider these things equal for the time being
$$i\hbar \frac{\partial}{\partial t}=\hat{H}=-\frac{\hbar^2}{2m}\nabla^2$$
Then apply this to $|\Psi |^2$ and don't forget the product rule.

 

[nymphidius]

Thanks a million man. I understand it now.

 

[nymphidius]

okay so I get that part, but in the same line (within 1.25) he writes:

iℏ/2m(ψ*∂²/∂x²[ψ]-∂²/∂x²[ψ*]ψ) = iℏ/2m(∂/∂x)(ψ*∂/∂x[ψ]-∂/∂x[ψ*]ψ) [1.25]

how did he factor out the partial derivative with respect to x on the RHS? To my knowledge, if you apply the partial derivative on the RHS then I WOULDN'T get the LHS of 1.25.

(to my understanding ψ=ψ(x,t))

 

[jfy4]

try it out again. Watch the negative signs, and make sure to do the product rule when you differentiate, they are equal.