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I am trying to "decode"/derive an expression for the macroscopic rate constant for the tunneling of protons through a potential energy barrier that I read in a journal article: $$ k_{\rm tun}(T)=(2\pi\hbar)^{-1}\int_0^{V_{\rm max}} Q(V,T) P_{\rm tun}(V)\ dV. $$ So basically: the authors say work out the probability of tunneling at each point up the potential energy barrier (from 0 to $V_{\rm max}$), multiply this by the Boltzmann factor ($Q(V,T)$), integrating over all energies (i.e. we are taking a Boltzmann average )and then convert from energy in J to rate by multiplying by $(2\pi\hbar)^{-1}$.

The authors use $(2\pi\hbar)^{-1}$ to convert from an energy in J to a rate in per second. This implies that the relationship between energy and frequency being used is: $$ E=\hbar\omega $$ I thought this just applied to electromagnetic radiation and free particles not in a potential. Is it OK to use this relation for protons tunneling in a potential? Or should you use some info about the shape of the potential?

Many thanks


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Welcome! I fixed up the formatting of your post using the MathJax formatting standard that we use here. I think that, when you're editing a post, there's a link on the right that'll give you some formatting tips to get started. Also, if you open up this post for editing, you'll see the syntax I used. – Ted Bunn May 5 '11 at 20:48
you can also do the same (open up a post for editing to see the used syntax) even if its not your own post – lurscher May 5 '11 at 21:05
The language MathJax eats is basically LaTeX, there are many tutorials on the web. – dmckee May 5 '11 at 22:40

$E=\hbar\omega$ is a totally universal formula that holds for all particles and everywhere in quantum mechanics. Schrödinger's equation guarantees that. The same question was being answered yesterday:

Gravitational wave energy

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