clem said:
If you are saying an isolated hydrogen atom in an excited state won't decay, you are wrong.
Maybe that's not what you mean.
That's
exactly what I mean. If you write the Schrodinger equation for a single particle (without taking the environment into account) , you'll never see the decay theoretically.
Again a simple question, yet full of pitfalls:
Why does an atom emit light?
The decay totally follows from somewhere else. Let me first
prove that SE does not capture the decay of an excited state:
In the eigenstate basis the Schrodinger equation neatly decouples into the following set of equations:
i\hbar\frac{d}{dt}\left\{\psi_n \right\}= [\epsilon_n]\left\{\psi_n\right\}
which can be solved trivially:
\psi_n(t)=\psi_n(0) e^{-i\epsilon_n t/\hbar}
then comes the coup de grace:
Let's look at the probability P for finding an electron in state n:
|\psi_n(t)|^2 = P_n(0)
where P_n(0) is a constant and does not change with time.
According to the Schrodinger Equation an excited electron stays there forever... Whatever it is that causes it to decay is clearly not above.
The missing part requires some many-particle physics knowledge which I'll skip here. So,
yes, although not true, the decay never occurs according to the one-particle Schrodinger equation and thus according to HUP.
(Edit: Of course, the truth is if you start out in an excited state, you'll end up decaying to a lower level. But
that will introduce a broadening as well (the excited state will not be as sharp anymore) to protect the validity of HUP ---- but I did not take that into consideration because the excited state is not a sharp level and it is NOT the counterpart of what I stated.)
I hope it's clear.
