Quantum problem - Calculating the expectation value of energy?

[jeebs]

Homework Statement

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 Schrödinger 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 Schrödinger equation:

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

time-dependent Schrödinger 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) |a_n|²E_n that i thought might be useful here but i really don't get how to use it - if anyone could explain to me about that, if it is relevant, i'd really appreciate it.

Thanks.

 

[buffordboy23]

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 Schrödinger equation is a linear combination of stationary states:

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

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

$$\sum_{n=1}^{\infty} \left|a_{n}\right|^{2} = 1$$

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

$$\left<H\right> = \sum_{n=1}^{\infty} \left|a_{n}\right|^{2}E_{n}$$

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

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

where $$\Psi \left(x,0\right)$$ 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 $$\psi_{n}$$ are already normalized simplify the work for this particular problem.

 

[jeebs]

thanks very much buffordboy.

 

[buffordboy23]

your welcome. =)