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Energy Uncertainty and expectation value of H

  1. Nov 4, 2014 #1
    1. The problem statement, all variables and given/known data
    A particle at time zero has a wave function Psi(x,t=0) = A*[phi_1(x)-i*sin(x)], where phi_1 and phi_2 are orthonormal stationary states for a Schrodinger equation with some potential V(x) and energy eigenvalues E1, E2, respectively.
    a) Compute the normalization constant A.
    b) Work out Psi(x,t)
    c) Compute <H> for Psi(x,t)
    d) Compute delta_E, the energy uncertainty

    2. Relevant equations

    Delta_E = Sqrt(<E^2>-<E>^2)

    3. The attempt at a solution
    a) Set: 1 = <Psi(0)|Psi(0)> => A = 1/sqrt(2)
    b) From previous: Psi(x,t=0) = 1/sqrt(2)*[phi_1 - i*phi_2]
    Use the time evolution equation: Psi(x,t) = 1/sqrt(2)*[phi_1*e^(-i*E1*t/h-bar) - i*phi_2*e^(-i*E2*t/h-bar)]
    c) Probability of measuring E1: P1 = |<phi_1|Psi(x,t)>|^2 = 1/2
    Similarly, Probability of measuring E2: P2 = 1/2
    Then <H> = P1*E1 + P2*E2 = (E1+E2)/2
    d) For this one, I attempted to use the mentioned equation. However, I could not find <E^2>.
    I thought it would be: (P1*E1)^2 + (P2*E2)^2, but this yields a negative values under the square root, which is not possible since the energy uncertainty is probability not imaginary.
    Please help, thank you.
     
  2. jcsd
  3. Nov 5, 2014 #2

    DrClaude

    User Avatar

    Staff: Mentor

    That looks fine.

    Go back to the definition of ##\langle E^2 \rangle##, and see what you can get.
     
  4. Nov 8, 2014 #3
    Thank you very much.
     
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