# Where is the particle most likely to be found?

by Levi Tate
Tags: particle
 Mentor P: 10,685 You've got the square of the wave function which is (ignoring the normalization constant) ##\sin^2 (n\pi x / a)##. Think of it as ##sin^2 \theta## where ##\theta = n\pi x / a##. What's the maximum possible value of ##\sin^2 \theta##, and what values of ##\theta## give you that maximum? This is something you should be able to read off a graph of ##\sin^2 \theta##. "Science Advisor" is a title that the Mentors (moderators) here give to posters that we think give reliably good answers and advice. "Homework Helper" is similar, but it's for people who hang out mainly in the homework forums.
 P: 122 Cool thanks for the info mate. See I just did this problem here, I didn't ignore the 2/a, I just divided it out. The problem says for the first excited state, so n=2, so I ended up with θ=Arcsin(2[pi]x/a), (n=2 for the first energy state) I dunno I just am not getting this problem, because arcsin(sin^2x)≠x I don't think. I am just going to ask my teacher this one. I don't think he solves ODEs and uses a computer, so I have a better question, it's pretty straightforward. Thanks for the help.
 P: 122 And definitely, the graph is right next to the problem showing the maxima. Another thing is I just don't get that if a is a constant where the infinite square well ends how I know that, ah! I am letting this one go. It's not even a question that my teacher asked or anything, either is the other one really. Which is irrelevant, but this isn't my homework, this is just me wondering about physics, or really in this case math.
 Mentor P: 10,685 I think you're going at it sort of backwards. Try the approach I gave in my previous post, which is actually the first of two steps. First step: figure out which values of ##\theta## give you the maximum value of ##\sin^2 \theta##. This is where a graph of ##\sin^2 \theta## versus ##\theta## (not ##\psi## versus x!) comes in handy. Second step: use ##\theta = n \pi x / a## to find the corresponding values of x.
HW Helper
Thanks
P: 7,873
 Quote by Levi Tate How can it be that a cosine function and a sine function will have the same maxima and minima here in this physics problem?
That isn't what I said. In hindsight, my post may have been confusing because I was taking an unnecessarily general approach. Suppose f = f(x), and you want the extrema of f2(x). Standard calculus says to find extrema of a function you look for where the derivative of the function is zero, so that will be d f2(x) /dx = 0. But of course that reduces to f(x) f'(x) = 0, which will be true wherever either f(x) = 0 or f'(x) = 0. And since f2(x) ≥ 0, you know straight away that f(x) = 0 will give the minima (and only the minima), while f'(x) = 0 will give all the maxima (but maybe repeats some of the minima).
As jtbell points out, you don't actually need any calculus in the present case. You just need to know what arguments maximise the sine function.

Wrt "Science Advisor", please remember it only indicates a general reliability, not infallibility. I don't think anyone has made Science Pope yet.
 P: 122 Your user name, everytime i read it, i read 'harpsex'. I'm like, he has a bit of a funny name, oh.. ! Then i realize i don't know how to read. We actually took the Schrodinger (time independent) to three dimensions today and while it was extremely beautiful, I do not know what we were doing. I'm trying to catch up with Optics but hopefully you fine fellows might be able to provide some assistance, to be honest, in 3d at some point I didn't even know what we were looking for, or doing, or trying to solve anymore.
P: 501
 Quote by haruspex I don't think anyone has made Science Pope yet.
We'll know when that happens because white smoke will pour from our computers.
 P: 122 They should just have the retiring Pope be science Pope, it will be easier because he's already been a Pope.

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