- #1

CAF123

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## Homework Statement

Let ##\Psi(x,0)## be the wavefunction at t=0 described by ##\Psi(x,0) = \frac{1}{\sqrt{2}}\left(u_1(x) + u_2(x)\right)##, where the ##u_i## is the ##ith## eigenstate of the Hamiltonian for the 1-D infinite potential well.

The energy of the system is measured at some t - what are are the possible outcomes of such a measurement?

Find the average energy of the system as a function of time. Does this value depend on the initial state of the system?

## Homework Equations

Average value of energy, ##\langle H \rangle##

## The Attempt at a Solution

Since the 1D Infinite well represents a discrete spectrum of eigenvalues, I would say that the only possible values of the energy are E1 and E2. However, to see this I considered the probabilities of obtaining such energies: For E1, it is given by ##\langle u_1 | \Psi \rangle ##and this gives ##\frac{1}{2} e^{-Kt}## where K is some numerical factor. Similarly for probability of E2: ##\frac{1}{2}e^{-K't}##. This suggests at t=0, the probability of getting a particular Ei, i either 1 or 2 is 1/2, but that this decays exponentially. How should I interpret this? At some finite t, P(E1) + P(E2) is not 1, so does this mean there exists another possible energy?

I think I may compute the average energy as ##\langle H \rangle = \int dx H |\Psi(x,t)|^2##. Since it says '...find this as a function of time, this suggests there is more to the first part since if at any time t, we could get E1 or E2 with equal probability, then this would suggest the average energy would be (E1+E2)/2 ≠ f(t).

Many thanks.