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Quantum mechanics: one thousand neutrons in an infinite well

  1. Oct 16, 2016 #1
    I apologize in advance for not being familiar with LaTex.

    1. The problem statement, all variables and given/known data

    One thousand neutrons are in an infinite square well, with walls x=0 and x=L. The state of the particle at t=0 is :
    ψ(x,0)=Ax(x-L)

    How many particles are in the interval (0,L/2) at t=3?

    How many particles have energy E5 at t=3s? (That is the <E> at t=3s)

    2. Relevant equations

    Wavefunction of the infinite square well: ψ(x)=Sqrt[2/L]Sin[n*Pi*x/L]

    Time dependence of wavefunctions: ψ(x,t)=∑cnψn(x,t)

    cn coefficients: cn=∫ψn(x)*f(x)dx

    3. The attempt at a solution

    This problem is quite similar to this one: https://www.physicsforums.com/threads/neutrons-in-a-one-dimensional-box.242003/, except that now, one must calculate the quantities at a later time, forcing us to construct the full time-dependent solution (I suspect).

    I begin by normalizing the initial wavefunction, where I get: A=Sqrt[30/L2].

    Then, in principle, we can get ψ(x,t) by calculating those cn coefficients, and plug it in the general solution (Griffiths example 2.2 does this).

    Well, this is all very nice and beautiful, but let's not forget what the question asks: calculating a probability at a later time. When putting this in Born's postulate to find the probability, the time dependence (a complex exponential) cancels out, so we don't even have to opportunity to 'plug in 3s! Not to mention, the infinite summation would also give me trouble when trying to perform the integral. (I hope I made sense...)

    So, does anyone have any advice on how else to attack this problem?
    Thank you!

     
  2. jcsd
  3. Oct 17, 2016 #2

    DrClaude

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    Staff: Mentor

    The problem can be simplified by consider the symmetry of the situation. Also, look out for red herrings.
     
  4. Oct 17, 2016 #3

    DrClaude

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    Staff: Mentor

    Check that result also.
     
  5. Oct 17, 2016 #4
    Right, probability is not supposed to change with time for stationary states, and it doesn't matter if it's 3 seconds or 0 seconds.
    So to calculate the probability, I can use the sin wavefunction, and I suppose the initial wavefunction must then be used in the second part, since the expectation value of energy formula involves finding the cn.
    Am I missing anything else?
     
  6. Oct 17, 2016 #5

    DrClaude

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    Staff: Mentor

    The probability of being in a given state doesn't change, but different states can interfere with each other. You need to see if this affects the answer here.
     
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