Determination of time-dependent coefficients (QM)

[Aeon]

Hi,

I am trying to solve the following equation:

$\Psi(r,t) = \sum_n C_n(t) e^{-i E_n t} \psi(r)$

to find the $C_n(t)$s.

The system I am modeling is benzene. I can, by Hückel's method, determine the time-independent solution. The apparently obvious transition from time-independent coefficients to time-dependent coefficients is troubling me. A simple plot of the squared coefficients indicates that my electron density vanishes, which is indicative of error.

I'm wondering how is the proper way to solve for the time-dependent coefficients...

 

[Bill_K]

Did you leave off a subscript? If you meant to write ψ(r,t) = ∑ C_n(t) exp(-iE_nt) ψ_n(r) where the ψ_n(r)'s are orthonormal, then
∫ψ(r,t) ψ_n(r) dr = C_n(t) exp(-iE_nt), which gives you C_n.

 

[Aeon]

I forgot a subscript, you are right.

Also, the problem was that my vectors were not orthogonalized. Thanks!