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Quantum problem - Calculating the expectation value of energy?

  1. Dec 14, 2008 #1
    1. The problem statement, all variables and given/known data

    Hi all,
    i have a problem:

    i am given a time-dependent wavefunction, Ψ(x,t), and i am asked to calculate the expectation value of total energy E(t) and potential energy V(t).
    Ψ(x,t) = (1/sqrt2)[Ψ0(x).e-[i(E0)t/h] + Ψ1(x)e-[i(E1)t/h]],
    where Ψ0,1(x) are the ground and first excited normalised eigenstate of the linear harmonic oscillator, n=0,1.

    What i have tried is <E(t)> ∫Ψ*(x,t).E^.Ψ(x,t) dx, where E^ is E(hat), the total energy operator (the hamiltonian?). the integrals are between x = minus and plus infinity by the way.

    i have been through a hell of a lot of integration (2 sides of small handwriting) and came up with an answer i am not happy with. So, my first query is, am i using the correct energy operator, E^ = -(h2/2m).(d2/dx2) + V(x), or is it something else?

    the reason i ask this is because i just lifted it straight from the time-independent schrodinger equation, but my Ψ(x,t) is clearly time-dependent, so does this change the operator i need?

    also, i have done the same thing for the potential with V^ = V(x), but i did not know what i was supposed to do with V(x) when it came to applying it to Ψ(x,t) to the right of it. So, i just called it V(x) and got on with it. After this was worked out my answer was <E(t)> ∫Ψ*(x,t).V.Ψ(x,t) dx = V.

    So my second query is, for a linear harmonic oscillator, is there something more specific i am meant use than just V^ = V(x) = V ??

    other than that, am i approaching this problem all wrong or what? i'm quite new to quantum mechanics so i'm never really sure if my solutions even resemble the actual answer. i don't even feel as if i've asked these questions very well.

    Thanks.

    PS. just incase, time-independent schrodinger equation:

    [-(h2/2m).(d2/dx2) + V(x)]Ψ(x) = EΨ(x)

    time-dependent schrodinger equation:

    Ψ(x,t) = Ψ(x)e-iEt/h

    the other thing was there was something in my notes about <E> that says <E> = (∑n) |an|2En that i thought might be useful here but i really dont get how to use it - if anyone could explain to me about that, if it is relevant, i'd really appreciate it.

    Thanks.
     
    Last edited: Dec 14, 2008
  2. jcsd
  3. Dec 14, 2008 #2
    Hi Jeebs,

    You could calculate the expectation value of energy by computing the integral. This will likely take quite a bit of time to do, so let's look at an alternative method that you already hinted at.

    First, the general solution to the time-dependent Schrodinger equation is a linear combination of stationary states:

    [tex] \Psi\left(x,t\right) = \sum_{n=1}^{\infty} a_{n}\psi_{n}e^{-iE_{n}t/\hbar} [/tex]

    You can think of the coefficient [tex] a_{n} [/tex] as the amount of [tex] \psi_{n} [/tex] that exists in [tex] \Psi [/tex]. What this means is that [tex] \left|a_{n}\right|^{2} [/tex] is the probability of finding the particle in the nth state. This requires

    [tex] \sum_{n=1}^{\infty} \left|a_{n}\right|^{2} = 1 [/tex]

    Note that these coefficients have no time dependence. This leads to the expectation value of energy (or any other observable).

    [tex] \left<H\right> = \sum_{n=1}^{\infty} \left|a_{n}\right|^{2}E_{n} [/tex]

    You should already know the energies of the harmonic oscillator, so you can use this equation easily. To determine the coefficients we use the formula

    [tex] a_{n} = \int_{-a}^{a} \psi_{n}\Psi\left(x,0\right) dx [/tex]

    where [tex] \Psi \left(x,0\right) [/tex] is the initial wave function. You may be thinking, wait, I need to compute numerous integrals now to solve the problem. This is true but the fact is that orthogonality principles and the fact that the [tex] \psi_{n} [/tex] are already normalized simplify the work for this particular problem.
     
  4. Dec 14, 2008 #3
    thanks very much buffordboy.
     
  5. Dec 14, 2008 #4
    your welcome. =)
     
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