I was toying with Wigner corrections to thermodynamic equilibrium. The semiclassical correction for the position probability density to second order in ℏ is: P(x)=e−βV(x)(1−ℏ2β212mV′′(x)+ℏ2β324m|V′(x)|2).
I took this expression and applied to typical potential energies of chemical reactions involving protons, in which there are barriers of around 0.5 eV. I am confused because for a lot of situations I obtain negative values for the correction factor (and thus for the probability density), even for high temperatures of T=300. I don't believe higher order terms can be important at such temperatures. I even think that such corrections should be extremely small in such high temperatures. What can be the issue here? Formally, it is clear that this is a probability.
edit: I wrote a relative probability, of course. A convenient normalization factor is missing above.
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