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Probability of finding particle in 1D finite potential well

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  • #1
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Homework Statement



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.

Homework Equations





The Attempt at a Solution



This is to check I'm not going insane:

P = ∫ lψxl2 dx = 1

Is that about right?
 

Answers and Replies

  • #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.
 
  • #3
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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.
 
  • #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?
 
  • #5
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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.
 
  • #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?
 
  • #7
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...the awkward moment when I don't know the answer has arrived.
Please illiterate.
 
  • #8
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Consider finding the average value of a function over its domain.

ave[f(x)] = ⌠f(x)dx / ⌠dx

Food for thought.
 
  • #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"!
 
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  • #10
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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?
 
  • #11
vela
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How do you normalize a wave function?
 
  • #12
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square and integrate it. So... double integration and power of 4?
 
  • #13
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That's given that the particle is between a and b and that isn't specified
 
  • #14
vela
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square and integrate it. So... double integration and power of 4?
I don't know what you're trying to say. Show your work so that your meaning is clear.
 
  • #15
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∫∫ llψxl2l2 dx dx (inner from a to b, outer from -∞ to ∞)

seems strange...but that's what i'm imagining at the minute
 
  • #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?
 
  • #17
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-∞ to ∞ ∫ lψ(x)l2 dx = ϕ(x)
 
  • #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.
 
  • #19
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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
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I've been reading this a while now and I finally understand what you were trying to teach me. Page two of the document shows that the normalisation process, of the wave equation. I see now, where I've been going wrong.

Thank you so much!
 

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