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Potential well

  1. Mar 6, 2014 #1
    1. The problem statement, all variables and given/known data

    At the instant t=0, there's a system with 1000 particles in a box of length a. It is known that 100 have energy 4E1 and 900 have energy 225E1, where E1 is the energy of the fundamental state.

    i) Build a wave function that can represent the state of a particle

    ii) How many particles are in the right half of the well? [a/2 , a]

    2. Relevant equations

    i) [itex]\Psi(x,t) = \Sigma C_{n} \varphi_{n} (x) e^{-i n^{2}wt} [/itex] (1)

    [itex]P(E_{n}) = |C_{n}|^{2} [/itex] (2)


    ii) [itex] P(t) = \int |\Psi|^{2}| dx [/itex] (3)
    [itex]Number of particles = N_{total} P(t) [/itex] (4)


    3. The attempt at a solution

    i) [itex]\Psi (x,t) =\frac{1}{3} \sqrt{\frac{2}{a}} sin(\frac{4 \pi x}{a}) e^{i 16 wt} + \frac{3}{\sqrt{10}} \sqrt{\frac{2}{a}} sin(\frac{225 \pi x}{a}) e^{i 50625 wt} [/itex]

    ii) I need to know if the formula (4) is right


    Thanks in advance.
     
    Last edited: Mar 6, 2014
  2. jcsd
  3. Mar 6, 2014 #2

    BvU

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    i) What is the ##{1\over 3}## and the ##{3\over \sqrt{10}}## ?

    ii) Is there a way there can be an asymmetry that causes the answer to differ from 500 ?
    You know the ##\phi_n## are orthonormal, right ?​
     
  4. Mar 6, 2014 #3
    i) the squares of 1/9 and 9/10, which are the probabilities of a particle having energy 4E1 or 225E1.

    ii) I just have to integrate (3) with the limits a/2 and a, no?
     
  5. Mar 6, 2014 #4

    BvU

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    i) Yeah, well, the 1/9 is probably 1/10 . And the square of 1/9 is 1/81 (:smile: -- it pays to be a nitpicker in physics...)

    ii) Yes. But you already know that the cross term gives 0. And squaring gives you even functions, so a/2 -- a should be the same as 0 -- a/2 Or am I wrong ?
     
  6. Mar 6, 2014 #5
    i) I meant square root, sorry. I don't know where the hell I got that 1/9 for the probability from, it should be 1/10 yes.

    ii) yes you are right , that means what? there are 250 particles on each half?

    Thank you for the help so far.
     
  7. Mar 6, 2014 #6

    BvU

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    I wouold guess that with a total of 1000 particles there are 500 in the right half. PS is it a 3D box or a 1 D infinitely deep well ?

    Would be interesting to calculate the standard deviation in that 500 for an observation period of 1 femtosecond...
     
  8. Mar 6, 2014 #7
    1D infinitely deep well. I'll try to do some calculations on my own now.

    Thank you, have a good night(or day) sir.
     
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