A particle of mass m is in the state
Psi(x,t) = Ae^(-a[(mx^2)+it])
where A and a are positive real constants.
a) Find A
b) For what potential energy function V(x) does Psi satisfy the Shrodinger equation?
c) Calculate the expectation values of x, x^2, p, and p^2
d) Find sigma_x and sigma_p. Is their product consistent with the uncertainty principle?
Intetgral of p(x)dx = 1 (from negative infinity to infinity)
<x> = Integral of Psi* (x) Psi dx
<p> = Integral of Psi* (hbar/i partial derivative with respect to position) Psi dx
sigma_x times sigma_p > or = hbar/2
The Attempt at a Solution
I managed to get solutions for parts a and b, and most of part c. Using the equations listed above, I got 0 for <x> and <p> and 1/4 hbar/(am). The problem I had was figuring out how to solve <p^2>. I tried squaring hbar/i and the partial derivative term but that lead to some problems.
The partial derivative is A^2 (-2amx/hbar) e^(-amx^2/hbar - iat).
When I squared only the (-2amx/hbar) part, I managed to get a solution of hbar*am, and when I solved for the standard deviations and put them into the Heisenberg Uncertainty Principle I got exactly hbar/2. However, a friend of mine informed me that e^(-amx^x/hbar - iat) is also part of the derivative, and when I tried squaring that as well I ran into problems with simplification.
Is hbar*am the correct answer? And if so, why don't I have to square e^(-amx^2/hbar - iat)? Or if I'm completely wrong, what am I supposed to do to calculate <p^2>?
Thanks for your help!