(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the expectation value of position as a function of time.

2. Relevant equations

This is in the latter half of a multi-part question, previously we were given that:

Eqn 1: Ψ(x, t) = A(ψ_{1}(x)e^{−iE1t/h¯}+ iψ_{2}(x)e^{−iE2t/h¯})

and in an even earlier part:

Eqn 2: ψ_{n}(x) = sqrt(2/L)sin(n*pi*x/L)

Note: h¯ = hbar

3. The attempt at a solution

As you can tell, I'm not too awesome at formatting on here so I'm going to quickly explain my method.

I said:

<x> = Integral from 0 to L of Ψ*(x, t)Ψ(x, t) dx

So I told wolframalpha (we're allowed to use it) to simplify Eqn 1, before subbing in Eqn 2.

This gave me:

Eqn 3: -2(ψ_{1}ψ_{2}sin(t(E_{1}-E_{2})/h¯) + ψ_{1}^{2}+ ψ_{2}^{2}

I then subbed in Eqn 2 into Eqn 3 and integrated.

No matter how many times I do this I always end up with a bunch of sines. These sines are all something like sin(n*pi/L) so when I sub in the limits of integration they become sin(n*pi) or sin(0), both of which are 0!

This means that I'm just left with an answer of 2, which is not time dependent.

Can anyone see where I've gone wrong?

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# Quantum Mechanics - Finding expectation value

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