Assume that a particle in a one-dimensional box is in its first excited state. Calculate the expectation values [x], [p], and [E], and the uncertainties
delta(x), delta(p), and delta(E). Verify that delta(x)*delta(p)>=h_bar/2.
Psi=sqrt(2/a) cos(pi*x/a) e^(-i*E*t/h_bar)
[x]=Int(Psi_star x Psi, -a/2, a/2)
[p]=Int(Psi_star (-i*h_bar*d/dx) Psi, -a/2, a/2)
[E]=Int(Psi_star (i*h_bar*d/dt) Psi, -a/2, a/2)
The Attempt at a Solution
After evaluating the above integrals, I get:
[E]=h_bar*pi^2*n^2 / 2*m*a^2
I am trying to calculate the quantities delta(x), delta(p), and delta(E) but I am having trouble doing that. Can you please suggest some hints on how to proceed. Thank you.