Energy Uncertainty and expectation value of H

[Pikas]

Homework Statement

A particle at time zero has a wave function Psi(x,t=0) = A*[phi_1(x)-i*sin(x)], where phi_1 and phi_2 are orthonormal stationary states for a Schrödinger equation with some potential V(x) and energy eigenvalues E1, E2, respectively.
a) Compute the normalization constant A.
b) Work out Psi(x,t)
c) Compute <H> for Psi(x,t)
d) Compute delta_E, the energy uncertainty

Homework Equations

Delta_E = Sqrt(<E^2>-<E>^2)

The Attempt at a Solution

a) Set: 1 = <Psi(0)|Psi(0)> => A = 1/sqrt(2)
b) From previous: Psi(x,t=0) = 1/sqrt(2)*[phi_1 - i*phi_2]
Use the time evolution equation: Psi(x,t) = 1/sqrt(2)*[phi_1*e^(-i*E1*t/h-bar) - i*phi_2*e^(-i*E2*t/h-bar)]
c) Probability of measuring E1: P1 = |<phi_1|Psi(x,t)>|^2 = 1/2
Similarly, Probability of measuring E2: P2 = 1/2
Then <H> = P1*E1 + P2*E2 = (E1+E2)/2
d) For this one, I attempted to use the mentioned equation. However, I could not find <E^2>.
I thought it would be: (P1*E1)^2 + (P2*E2)^2, but this yields a negative values under the square root, which is not possible since the energy uncertainty is probability not imaginary.
Please help, thank you.

 

[DrClaude]

a) Set: 1 = <Psi(0)|Psi(0)> => A = 1/sqrt(2)
b) From previous: Psi(x,t=0) = 1/sqrt(2)*[phi_1 - i*phi_2]
Use the time evolution equation: Psi(x,t) = 1/sqrt(2)*[phi_1*e^(-i*E1*t/h-bar) - i*phi_2*e^(-i*E2*t/h-bar)]
c) Probability of measuring E1: P1 = |<phi_1|Psi(x,t)>|^2 = 1/2
Similarly, Probability of measuring E2: P2 = 1/2
Then <H> = P1*E1 + P2*E2 = (E1+E2)/2

That looks fine.

d) For this one, I attempted to use the mentioned equation. However, I could not find <E^2>.
I thought it would be: (P1*E1)^2 + (P2*E2)^2, but this yields a negative values under the square root, which is not possible since the energy uncertainty is probability not imaginary.

Go back to the definition of $\langle E^2 \rangle$, and see what you can get.

 

[Pikas]

Thank you very much.