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Probability of finding a particle in the ground state

  1. Jun 1, 2010 #1
    1. The problem statement, all variables and given/known data

    A particle is prepared in the state [tex]\psi (x) = \frac{1}{\sqrt{L}}[/tex] in a region [tex]0 < x < L[/tex] between two hard walls (particle in a box). Calculate the probability that the particle is found in the ground state when its energy is measured.

    2. Relevant equations

    This question is worth 10 marks, so I presume it's not as simple as squaring the wave function to find the probability. However I'm just not sure what else to do, a hint in the right direction would be greatly appreciated.


    3. The attempt at a solution

    [tex]\left| \psi (x) \right|^{2} = \frac{1}{L}[/tex]

    That's all I can think of doing. I've checked the given wave function is normalised, which it is.
     
  2. jcsd
  3. Jun 1, 2010 #2
    You need to use Fourier Series. You know the normalised solution to Schrodingers equation is root(2/L)sin(nPix/L)

    Let the initial wave function be f(x)=1/root(L)

    So f(x)=root(2/L)[sum of (c_n)sin(nPix/L)

    So c_n=root(2/L)Integral 0 to L of f(x)sin(nPix/L) by Fourier methods

    The probability of a particular state is the square of c_n if the wave function is normalised. I make your answer 8/(Pi)^2

    Does that make sense?
     
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