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I was toying with Wigner corrections to thermodynamic equilibrium. The semiclassical correction for the position probability density to second order in $\hbar$ is: $$P(x)= \text{e}^{-\beta V(x)}\left(1-\dfrac{\hbar^2 \beta^2}{12m}V^{\prime \prime}(x)+\dfrac{\hbar^2 \beta^3}{24m}\vert V^{\prime}(x) \vert ^2 \right).$$ (nice derivation of this formula are found in Landau's statistical mechanics or in the appendix to Density Functional Theory of Atoms and Molecules by Parr and Yang. The idea is that this is obtained by plugging $e^{-\beta \hat{H}}$ in the Wigner transform).

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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  • $\begingroup$ Is your high temperature (small $\beta$) limit 1 for this Wigner quasiprobability phase-space distribution of the canonical ensemble? the correction in parenthesis visibly goes to zero provided the potential in the exponential suppresses high powers of x. $\endgroup$ – Cosmas Zachos Nov 30 '18 at 17:32
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I'm not sure what you are getting wrong... Your expression, indeed, should go to 1 as β goes to 0. There must be some error in your implementation.

The Wigner transform of the canonical ensemble density, as you point out, may go negative, but not after integrating out the ps as introduced in (28) of Wigner's breathtaking 1932 paper, since it is just a conventional expectation value of positive semidefinite quantities. So, any indication of negativity is a pathology, not something chalked up to Wigner function idiosyncracies. Your expression is a marginal, a projection on the x axis after the Gaussian in p is integrated out to a β-dependent constant.

As a check of the picture, I'll summarize here the case for the quadratic potential, $x^2/2$, for which you have a closed expression for the Wigner function, (which, being a Gaussian, is positive semi-definite, a fortiori).

Let me non-dimensionalize out the useless $m=1=\omega$, absorbed in the variables, but maintain $\hbar$ visible. The oscillator harmonic hamiltonian then is just $(\hat p ^2 + \hat x ^2)/2$, with both x and p possessing dimensions of $\sqrt \hbar$, symmetrically--the convention of choice for phase-space!

In any decent phase-space quantization text (including our concise treatise) the full Wigner transform of the phase-space evolution operator for the oscillator is worked out. But, behold!, since the Bloch equation is but the Moyal evolution equation for $\beta=-it$, so $t=i\beta$, you automatically have the relevant complete, un-normalized Wigner function, the celebrated Mehler kernel, $$ e_\star ^{-\beta H/\hbar}= \frac{1}{\cosh \beta/2} ~e^{-2 H \tanh(\beta/2) /\hbar}= \frac{1}{\cosh \beta/2} ~e^{- \tanh(\beta/2) \frac{x^2+p^2}{\hbar} } . $$ Integrating out the p - dependence nets you the answer, a positive semidefinite probability density $$ P(x) \propto e^{-\frac{x^2}{\hbar} ~ \tanh(\beta/2) }= e^{-\beta \frac{x^2}{2\hbar} }~ e^{\frac{x^2}{\hbar} ~ \frac{\beta^3}{24} +... } = e^{-\beta \frac{x^2}{2\hbar} }~ \left (1 + \frac{x^2}{\hbar} ~ \frac{\beta^3}{24} +... \right ) , $$ where $\tanh (\beta/2)=\beta/2 ( 1-\beta^2/12 + ...) $. This, now, is manifestly positive semidefinite, as a bona-fide probability density should be.

To compare with your expression, you may reinstate the $\hbar \sqrt{m\omega}$ absorbed into x. But...What about the leading term $V''$? Well, it is only a constant, absorbable into the normalization above, discarded and ignored via the proportionality sign, and equivalent to multiplication by a term $(1-\beta^2 c+...)$ so properly relegated to higher orders. (After the momentum itegration above, the cosh in the denominator mutated to a square root of the sinh of the double argument.... all irrelevant and omitted.)

This is only to reassure you of the systematics of the structure, and to help catch errors. It is the reference point of any such calculation. If the nondimensionalization confused you instead of simplifying things, you may repeat all steps keeping the silly constants.

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The negativity of the Wigner function is certainly one of the most interesting feature of this approach. One can easily show that, if two states are orthogonal, $\langle \phi\vert\psi\rangle=0$, then their Wigner function satisfies $$ \int dp dx W^*_{\phi}(x,p) W_{\psi}(x,p)=0 $$ so it is not possible for $W_\psi(x,p)$ or $W_\phi(x,p)$ to be non-negative: at least one of them must have some negative regions. Negativity is quite a general feature and only Gaussian states do not have negative regions (Hudson's theorem).

Possibly the more intriguing aspect is that these regions of negativity have been "measured" for the $n=1$ states of a harmonic oscillator in a famous paper by David Wineland's group:

Leibfried, D., et al. "Experimental determination of the motional quantum state of a trapped atom." Physical Review Letters 77.21 (1996): 4281.

The full paper is freely available on the NIST website with the following figure showing the experimental results often reproduced:

enter image description here

Note that, even if there are negative regions, the marginal distributions $\int dx W_{\psi}(x,p)$ or $\int dp W_{\psi}(x,p)$ are never negative (this was a requirement built into the definition of $W$ by Wigner). Negative regions are broadly indicative of the non-classical nature of a state.

Feynmann discussed negative probabilities in this paper. There is also this quite nice paper: Leibfried, Dietrich, Tilman Pfau, and Christopher Monroe. "Shadows and mirrors: reconstructing quantum states of atom motion." Physics Today 51 (1998): 22-29 available on ResearchGate which also discusses negative probabilities and their interpretation.

The Wigner function (and more generally the Wigner transform) was the seed for a great many generalization, including extremely create work by Leon Cohen (see Cohen, Leon. "Generalized phase-space distribution functions." Journal of Mathematical Physics 7.5 (1966): 781-786) .

A great deal of work beyond physics was done using the so-called Wigner-Ville transform in signal analysis. An interesting difference between and more applied disciplines is that, whereas negative regions and interference are seen as beneficial features of the WF in physics, a lot of effort goes into doing away with them in signal analysis.

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