Conversation started Apr 13, 2021 at 2:50.
Andrew Micallef
Andrew Micallef
Apr 13, 2021 02:50
Quick question (will be back in a couple of hours for more details): if I have an equation of a second derivative of a function, can I integrate the other side of the equation to make the expression one of a first order derivative?
I want an expression for the first derivative with respect to x in the schrodinger equation
Karim Mansour
Karim Mansour
@TedShifrin Hi
leslie townes
leslie townes
andrew i am not an expert on the schrodinger equation but very frequently you can indeed integrate both sides of "f''(x) = blah" and get "f'(x) = new blah." the issue is that new blah might not be any easier to analyze than the original formula for f''(x). it's not guaranteed to be a slam dunk unless the RHS is in some way 'simple' or cleverly dealt with.
StudySmarterNotHarder
StudySmarterNotHarder
@AndrewMicallef that sounds like a good question for physics SE or MSE
I'm not a Schrodinger cat yet, so I couldn't tell you the answer
I did just purchase Griffiths though, paid $75
Still on the first chapter
It's astounding to me that particles obey these wave equations, what an amazing theory QM is
leslie townes
leslie townes
you see an analog of this in high school algebra, particularly when students move just one copy of something involving a variable x to the one side and then tries to solve for x by applying an inverse function to the other side. if the other side still has x's in it, that might not be much easier to analyze.
Ted Shifrin
Ted Shifrin
Apr 13, 2021 03:05
@AndrewMicallef with regular derivatives, sure, but with partial derivatives you have to remember that the constant of integration is a function of the remaining variables.
Hi, Karim.
leslie townes
leslie townes
and that. often assisted by the not-always-helpful tendency in physics to omit things like bounds of integration or what depends on what.
StudySmarterNotHarder
StudySmarterNotHarder
@AndrewMicallef it seems to me that if you integrated w.r.t. either $x$ or $t$ (say the 1-D Schrodinger) you'd be left with an integral of $\Psi$ itself, not just an equation with $\Psi$ and its derivatives.
Karim Mansour
Karim Mansour
@TedShifrin I am thinking now of working out the surjectivity of that map by looking at surface theory. In particular Beauville's book on classification of surfaces. I think the surjectivity can be achieved by by dividing it into cases.
will discuss with my supervisor tomorrow. I don't think there exist one technique to prove it for all surfaces without delving it into cases.
Hi @leslietownes
Kodaira was such a boss.
leslie townes
leslie townes
good evening
StudySmarterNotHarder
StudySmarterNotHarder
@AndrewMicallef is $\dfrac{\partial \Psi}{\partial t} = \dfrac{i \bar{h}}{2m} \dfrac{\partial^2 \Psi}{\partial x^2} - \dfrac{i}{\bar{h}} V \Psi$ the equation you're dealing with?
See integrating with respect to $x$ say, and you'd be left with an integral $\int \Psi \rm{dx}$ term
Karim Mansour
Karim Mansour
Apr 13, 2021 03:11
Yeah Kodaira so cool I will read his books definitely at some point.
StudySmarterNotHarder
StudySmarterNotHarder
If it doesn't work in the $1$-D case, it probably (QM is about probability anyway!) won't work in higher dims
@AndrewMicallef did that help at all?
Ted Shifrin
Ted Shifrin
Apr 13, 2021 03:26
@StudySmarterNotHarder @Andrew Not to mention you have no idea what to do with the left-hand side. You can interchange integral and derivative and then you'll have $\frac{\partial}{\partial t} \int \Psi\,dx$.
BTW, it's $\hbar$, not $\bar h$ :P
StudySmarterNotHarder
StudySmarterNotHarder
Nice, $\hbar$ I was also writing incorrectly on paper
fixed
Right now going through the proof that a wave solution normalized at $t=0$ stays normalized as $t$ evolves. Learning some stuff :)
I haven't messed with calculus since high school essentially, and am new to $\Bbb{C}$ analysis, but the author knew this
Surprisingly the table of integrals at the back of Griffiths is only 9 items! Usually these tables have a hundred or so entries
 
Conversation ended Apr 13, 2021 at 3:30.