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Hi all,

i have a problem:

i am given a time-dependent wavefunction, Ψ(x,t), and i am asked to calculate theexpectation value of total energy E(t) and potential energy V(t).

Ψ(x,t) = (1/sqrt2),[Ψ_{0}(x).e^{-[i(E0)t/h]}+ Ψ_{1}(x)e^{-[i(E1)t/h]}]

whereΨare the ground and first excited normalised eigenstate of the linear harmonic oscillator, n=0,1._{0},_{1}(x)

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^ = -(h, or is it something else?^{2}/2m).(d^{2}/dx^{2}) + V(x)

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 justV^ = 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:

[-(h^{2}/2m).(d^{2}/dx^{2}) + 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> = (∑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._{n}) |a_{n}|^{2}E_{n}

Thanks.

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# Homework Help: Quantum problem - Calculating the expectation value of energy?

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