I have the following Schrodinger equation:
i* (hbar) * partial derivative of ψ(x,t) w.r.t time
=
[(m*w^2 / 2) * x^2 * ψ(x,t) ]  (1/2m) * (hbar)^2 * (laplacian of ψ(x,t))]
m>0 is the mass
w is a positive constant
Assume that the ground state (normalizable energy eigenfunction) with the lowest possible energy E(0) is of the form
ψ(x ,t) = A * exp ((i * E(0)*t/ (hbar) )  l * x^2)
A, l are constants
Use the equation to find A and l.
My try:
I know that the Scrodinger eqn can be reduced to a time  independent form which in my case would be
E * ψ (E) = { ((hbar)^2 / 2m )* laplacian + (m*w^2 / 2) * x^2} * ψ (E)
Then I am not sure how to proceed.
For the normalization constant I know that the integral from minus infinity to infinity of ψ(x,t)^2 = 1 by Born interpretation for the probability density.
Any help would be greatly appreciated!
Thank you!
