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Feb27-12, 02:36 PM
P: 15
I have the following Schrodinger equation:

i* (h-bar) * partial derivative of ψ(x,t) w.r.t time
[(m*w^2 / 2) * x^2 * ψ(x,t) ] - (1/2m) * (h-bar)^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/ (h-bar) ) - 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) = { (-(h-bar)^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!
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