1D Particle in a box - locations at set probability

AI Thread Summary
The discussion revolves around finding the locations in a one-dimensional box where the probability of locating an electron is half its maximum value, given a state n = 3. The wavefunction squared, ψ²(x), is used to determine the probability density, and participants explore setting ψ²(x) equal to 0.5 to solve for x. They discuss the necessity of differentiating the wavefunction to find maxima and emphasize the importance of understanding sine functions in this context. The conversation highlights the interplay between chemistry and mathematics in solving quantum mechanics problems.
Nykrus
Messages
4
Reaction score
0

Homework Statement



An electron is confined to the region of the x-axis between x = 0 and x = L (where L = 1nm). Given a state n = 3, find the location of the points in the box at which the probability of finding the electron is half it's maximum value

Homework Equations



\psi^2(x)=\frac{2}{L}\sin^2\left(\frac{n\pi x}{L}\right)

The Attempt at a Solution



I understand that the wavefunction squared (above) gives the probability at location x, and its integration gives the probability over set regions between x = 0 and x = l. However, the only way I can see of finding x from a given probability is to assume:

\psi^2(x)=0.5

and try to manipulate the equation to give it in terms of x. Is this the right method? If so, how do I take out the sine term?

Cheers
 
Physics news on Phys.org
The maximum value of ψ2(x) is not 1.
 
Ah, I see - half the maximum value. I'm guessing it'd only be 1 if we were considering the entire box, not one point.

Alright, so you determine the maxima by differentiation?

\frac{d\psi^2(x)}{dx}=2\sin\left(\frac{n\pi x}{L}\right)\cos\left(\frac{n\pi x}{L}\right)=0
 
The rigorous way to determine the maxima would be by differentiation, as you said. However, ψ2 has a fairly simple formula and from what you know about sine functions, you should be able to see the maximum value by inspection.
 
Hmmm... okay, let's simplify this: 2/L is just a constant, and so's \frac{n\pi}{L}, so that gives us:

\psi^2(x)=A\sin^2\left(ax)

Sine functions go to 1 when ax = 90, and since sin2(ax) is :

\sin^2\left(ax) = \sin\left(ax)\sin\left(ax)

This means... I'm seriously clutching at straws

Hey, I'm a chemist - I'm amazed I've managed this much
 
What is the largest value sin(x) can give?

Being a chemist is no excuse for not knowing math (unless of course, you're a biochemist :p)!
 
The maximum of sin(x) is 1
 
Back
Top