Determination of time-dependent coefficients (QM)

Aeon
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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...
 
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Did you leave off a subscript? If you meant to write ψ(r,t) = ∑ Cn(t) exp(-iEnt) ψn(r) where the ψn(r)'s are orthonormal, then
∫ψ(r,t) ψn(r) dr = Cn(t) exp(-iEnt), which gives you Cn.
 
I forgot a subscript, you are right.

Also, the problem was that my vectors were not orthogonalized. Thanks!
 
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I am not sure if this belongs in the biology section, but it appears more of a quantum physics question. Mike Wiest, Associate Professor of Neuroscience at Wellesley College in the US. In 2024 he published the results of an experiment on anaesthesia which purported to point to a role of quantum processes in consciousness; here is a popular exposition: https://neurosciencenews.com/quantum-process-consciousness-27624/ As my expertise in neuroscience doesn't reach up to an ant's ear...

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