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Quantum Harmonic Oscillator problem

  1. Mar 18, 2014 #1
    1. The problem statement, all variables and given/known data

    For the n = 1 harmonic oscillator wave function, find the probability p that, in an experiment which measures position, the particle will be found within a distance d = (mk)-1/4√ħ/2 of the origin. (Hint: Assume that the value of the integral α = ∫0^1/2 x^2e^(-x2/2) dx is known and express your result as a function of α)


    2. Relevant equations

    distance from 0 to d; d = (mk)^(-1/4) *√ħ/2

    s=[km^(1/4) /ħ^2]*x

    Normalize condition: Cn = 1/ (π^2√(2^nn!)

    Harmonic Oscillator wave function for n = 1 ψ1 = C1(2s)e^-s2/2


    3. The attempt at a solution

    Is the formula that I want to use here p=ψ∗(x)ψ(x) or
    p=∫ψ∗(x)ψ(x)dx? (p stands for probability.)

    s=[km^(1/4) /ħ^2]* (mk)^(-1/4) *√ħ/2 = 1/2

    C1=1/π^2√2 so ψ1 = (1/π^2√2)e^-1/2

    so, now, do I use:

    p=ψ∗(x)ψ(x)= [(1/π^(2)√2)e^-1/2]^2= 1/2π^4 * e^-1

    I believe that this is the right way, but the hint confuses me.
     
  2. jcsd
  3. Mar 18, 2014 #2

    vela

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    I don't see how you got that expression for ##p##. What happened to the rest of the wave function?
     
  4. Mar 18, 2014 #3
    I changed C1 when n=1 and s(d) when d = (mk)^(-1/4) *√ħ/2.

    then ψ∗ψ=ψ^2 (since there is no i, ψ is real)
     
  5. Mar 18, 2014 #4
    I made i typing mistake, I am sorry.

    so ψ1 = C1(2s)e^-((s^2)/2)

    Changing C1 and s gives ψ1 =(1/π^2√2)*(2*1/2)e^-((1/2^2)/2)

    which gives ψ1 =(1/π^2√2)e^-1/8

    and ψ1*ψ1=(1/2π^4)e^-1/4
     
  6. Mar 18, 2014 #5

    vela

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    Where's ##x##?
     
  7. Mar 18, 2014 #6
    x is d.
     
  8. Mar 18, 2014 #7

    vela

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    Why are you plugging ##d## in for ##x##?

    The problem asks you to find the probability the particle is found within a distance ##d## of the origin. What if it were to ask you find the probability the particle was found within a distance ##d/2## and ##d## of the origin? Would you still just plug in ##d## for ##x##? How would the ##d/2## fit in?
     
  9. Mar 18, 2014 #8
    right... so now i see it makes more sense to use ∫ψ∗(x)ψ(x)dx... ok, i got the result. thanks
     
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