1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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.


    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.

    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. =)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook