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Probability of finding particle in 1D finite potential well

  1. Jun 10, 2012 #1
    1. The problem statement, all variables and given/known data

    ψx is the function of postion for a particle inside a 1D finite square well. Write down the expression for finding the particle a≤x≤b. Do not assume that ψx is normalised.

    2. Relevant equations



    3. The attempt at a solution

    This is to check I'm not going insane:

    P = ∫ lψxl2 dx = 1

    Is that about right?
     
  2. jcsd
  3. Jun 10, 2012 #2

    vela

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    So you're saying you will always find the particle between x=a and x=b? That's what a probability of 1 means.
     
  4. Jun 10, 2012 #3
    Err... not sure why I typed in the "= 1" bit since I don't know what the probability is.
    But you understood that the probability of finding the particle between a and b is expressed through my equation. So I rest my case.
     
  5. Jun 10, 2012 #4

    vela

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    Well, I wouldn't say I understood that's what you meant. I guessed that is what you probably meant. For all I know, you could have been writing down the normalization condition. What you wrote is too vague to be meaningful, and it isn't correct. Can you flesh it out a bit?
     
  6. Jun 10, 2012 #5
    P = ∫ lψxl2 dx (integral from a to b)

    This is the formal expression for probability of finding a particle within a boundary.

    For normalisation, we say that the particle must be somewhere since we take the integral from -∞ to ∞.

    1 = ∫ lψxl2 dx (integral -∞ to ∞)

    Two totally separate equations.
     
  7. Jun 10, 2012 #6

    vela

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    Both of those equations hold only if the wave function is normalized. If the wave function isn't normalized, what would be the expression for the probability?
     
  8. Jun 10, 2012 #7
    ...the awkward moment when I don't know the answer has arrived.
    Please illiterate.
     
  9. Jun 10, 2012 #8
    Consider finding the average value of a function over its domain.

    ave[f(x)] = ⌠f(x)dx / ⌠dx

    Food for thought.
     
  10. Jun 10, 2012 #9

    HallsofIvy

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    If you are given that the object is in an infinite square well, then of course there is a probabilty of 1 that it is in the square well. I would have expected the problem to ask for the probability distribution of its position in that well. You can get that by solving Shrodinger's equation.

    Oh, dear! I just reread the title and realized it says finite well, not infinite"!
     
    Last edited: Jun 11, 2012
  11. Jun 11, 2012 #10
    So I'm getting... do a double integral or solve the schödinger equation or do a barrel roll.
    Care to go a tiny bit further for me please?
     
  12. Jun 11, 2012 #11

    vela

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    How do you normalize a wave function?
     
  13. Jun 11, 2012 #12
    square and integrate it. So... double integration and power of 4?
     
  14. Jun 11, 2012 #13
    That's given that the particle is between a and b and that isn't specified
     
  15. Jun 11, 2012 #14

    vela

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    I don't know what you're trying to say. Show your work so that your meaning is clear.
     
  16. Jun 11, 2012 #15
    ∫∫ llψxl2l2 dx dx (inner from a to b, outer from -∞ to ∞)

    seems strange...but that's what i'm imagining at the minute
     
  17. Jun 11, 2012 #16

    vela

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    Let's say you have the unnormalized wave function ##\psi(x)##, and let ##\phi(x)## be its normalized counterpart. How are ##\psi## and ##\phi## related?
     
  18. Jun 12, 2012 #17
    -∞ to ∞ ∫ lψ(x)l2 dx = ϕ(x)
     
  19. Jun 12, 2012 #18

    vela

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    The lefthand side of your equation is a definite integral, so the result is a number, not a function of x. Try again. Surely your textbook has an example.
     
  20. Jun 12, 2012 #19
    This is why I'm here, I've looked through 3 books and numerous sites and I can't seem to find this particular example where the wave function is NOT normalised.
     
  21. Jun 12, 2012 #20

    vela

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