1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

1D Particle in a box - locations at set probability

  1. Mar 18, 2009 #1
    1. The problem statement, all variables and given/known data

    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

    2. Relevant equations

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

    3. 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:

    [tex]\psi^2(x)=0.5[/tex]

    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
     
  2. jcsd
  3. Mar 18, 2009 #2

    Ygggdrasil

    User Avatar
    Science Advisor

    The maximum value of ψ2(x) is not 1.
     
  4. Mar 19, 2009 #3
    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?

    [tex]\frac{d\psi^2(x)}{dx}=2\sin\left(\frac{n\pi x}{L}\right)\cos\left(\frac{n\pi x}{L}\right)=0[/tex]
     
  5. Mar 19, 2009 #4

    Ygggdrasil

    User Avatar
    Science Advisor

    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.
     
  6. Mar 19, 2009 #5
    Hmmm.... okay, let's simplify this: 2/L is just a constant, and so's [tex]\frac{n\pi}{L}[/tex], so that gives us:

    [tex]\psi^2(x)=A\sin^2\left(ax)[/tex]

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

    [tex]\sin^2\left(ax) = \sin\left(ax)\sin\left(ax)[/tex]

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

    Hey, I'm a chemist - I'm amazed I've managed this much
     
  7. Mar 19, 2009 #6

    Ygggdrasil

    User Avatar
    Science Advisor

    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)!
     
  8. Mar 20, 2009 #7
    The maximum of sin(x) is 1
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: 1D Particle in a box - locations at set probability
  1. Particle in a box (Replies: 8)

  2. Particle in a Box (Replies: 4)

Loading...