# Step Potential Transmission Probability as t->inf

At t=0 we have a particle incident from the left on a step potential (where V(x) = V0 for x ≥ 0 and V(x) = 0 otherwise). The particle has energy of 5/4 V0 and the question asks to find the probability that it will be on the right (ie. x > 0) as t→∞.

I understand how to solve this problem and that I'm looking for the transmission coefficient T, but what's throwing me off is the time evolution thing. I know that the wavefunction can be evolved by simply multiplying by e-iEt/ħ (ie. ψ(x,t) = e-iEt/ħφ0(x) where φ0(x) was found using the TDSE and relevant boundary conditions).

But...does this even matter when it comes to finding T? T depends on the wavenumber from either side of the barrier (k and q, if it were), which are related solely to the position part of the wavefunction, not the time part. Am I right in this? Since the potential isn't changing, wouldn't T be the same for all time?

Thanks in advance for any help! This has been bugging me all day and none of my books explicitly address this in their treatment of the step potential.

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