Comparing Wigner, Husimi and Bohmian distributions: which one is a true probability distribution in phase space?

Abstract

The Wigner distribution function is a quasi-probability distribution. When properly integrated, it provides the correct charge and current densities, but it gives negative probabilities in some points and regions of the phase space. Alternatively, the Husimi distribution function is positive-defined everywhere, but it does not provide the correct charge and current densities. The origin of all these difficulties is the attempt to construct a phase space within a quantum theory that does not allow well-defined (i.e. simultaneous) values of the position and momentum of an electron. In contrast, within the (de Broglie–Bohm) Bohmian theory of quantum mechanics, an electron has well-defined position and momentum. Therefore, such theory provides a natural definition of the phase space probability distribution and by construction, it is positive-defined and it exactly reproduces the charge and current densities. The Bohmian distribution function has many potentialities for quantum problems, in general, and for quantum transport, in particular, that remains unexplored.

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Fig. 1
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Notes

  1. 1.

    It is obvious that new mathematical elements open the path for different conceptual interpretations. The authors’ point of view is that Bohmian mechanics and orthodox quantum mechanics are indeed two different theories, while the Wigner distribution function is a new formalism belonging to the orthodox school. In any case, such distinction is not clear at all and it is not relevant for the conclusions of this work.

  2. 2.

    Strickly speaking, the phase space of a system of 2 interacting particles is not \(\{x,p\}\) but \(\{x_1,x_2,p_1,p_2\}\).

  3. 3.

    In the text, all the integration limits are from \(-\infty \) to \(\infty \) and thus we will not write them explicitly.

  4. 4.

    For simplicity, we avoid the explicitly consideration of the charge q of an electron in Eqs. (5) and (6). We notice also that the words charge or current densities can be misleading when a wave packet is partially transmitted or reflected, and the charge is in fact either fully transmitted or fully reflected, not both, when measured.

  5. 5.

    It must be said that, due to the measurement processes, position and momentum are not accessible simultaneously in a laboratory in a single experiment as seen in expression (2), but the Bohmian theory supports the (ontological) definition of both quantities.

  6. 6.

    We notice that the (mathematical) variables x and p in the Husimi (quasi) phase space \(\{x,p\}\) are a smoothed version of the x and p variables of the Wigner (quasi) phase space.

  7. 7.

    We notice that the variables x and p in the Bohmian phase space \(\{x,p\}\) are directly defined in the theory itself. They are part of the ontology of Bohmian mechanics. For this reason, the Bohmian phase space is a natural space, without mathematical tricks.

  8. 8.

    Let us emphasize that the different \(x_i(t)\) and \(p_i(t)\) are not associated to different particles (as we have said, for simplicity, throughout the paper we only deal with single-particle one-degree-of-freedom problems), but to different realizations of the same experiment. The probability obtained from the wave function \(\psi (x,t)\) has exactly the same (ensemble) meaning.

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Acknowledgments

This work has been partially supported by the “Ministerio de Ciencia e Innovación” through the Spanish Project TEC2012-31330 and by the Grant agreement no: 604391 of the Flagship initiative “Graphene-Based Revolutions in ICT and Beyond”. Z. Z acknowledges financial support from the China Scholarship Council (CSC).

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Correspondence to X. Oriols.

Appendices

Appendix 1: Q(x) and J(x) derivations

In this appendix, we will develop the calculus to obtain the charge and current densities (Eqs. (5) and (6)) for each of the three analysed distributions.

Wigner distribution

The charge distribution is straightforwadly found out:

$$\begin{aligned}&Q_W(x,p) = \frac{1}{h} \int dp\int \psi \Big (x+\frac{y}{2}\Big ) \psi ^{*}\Big (x-\frac{y}{2}\Big )e^{i\frac{py}{\hbar }} dy \nonumber \\&\quad = \int \delta (y) \psi \Big (x+\frac{y}{2}\Big ) \psi ^{*}\Big (x-\frac{y}{2}\Big )e^{i\frac{py}{\hbar }} dy = |\psi (x)|^2. \end{aligned}$$
(24)

The computation of \(J_W (x)\) needs a more detailed discussion:

$$\begin{aligned}&J_{W} (x)= \int p F_W(x,p) dp\nonumber \\&\quad = \frac{1}{h}\int p \int \psi \Big (x+\frac{y}{2}\Big ) \psi ^{*}\Big (x-\frac{y}{2}\Big )e^{i\frac{py}{\hbar }} dy dp. \end{aligned}$$
(25)

Using the chain rule for a function F(xy) derivable and zero-valued at \(y \rightarrow \pm \infty \), we can use the following relation

$$\begin{aligned} \int dy e^{-ipy/\hbar } \frac{\partial }{\partial y} F(x,y) = \frac{i}{\hbar }p \int dy e^{-ipy/\hbar } F(x,y) \end{aligned}$$
(26)

and using the polar form of the wave function (\(\psi (x)=R(x)e^{iS(x)}\)), then rewrite Eq. (25) as:

$$\begin{aligned}&J_{W} (x) = \frac{-i}{2\pi } \int dp \int dy e^{-ipy/\hbar } \cdot \left[ R\Big (x+\frac{y}{2}\Big )R\Big (x-\frac{y}{2}\Big )\right. \nonumber \\&\quad \left. \cdot \, e^{\frac{i}{\hbar }\left[ S(x+\frac{y}{2})-S(x-\frac{y}{2})\right] } \frac{i}{\hbar }\left( \frac{\partial S(x+\frac{y}{2})}{\partial y} -\frac{\partial S(x-\frac{y}{2})}{\partial y} \right) \right. \nonumber \\&\quad \left. + \, e^{\frac{i}{\hbar }\left[ S(x+\frac{y}{2})-S(x-\frac{y}{2})\right] } \frac{\partial }{\partial y} \left( R\Big (x+\frac{y}{2}\Big )R\Big (x-\frac{y}{2}\Big ) \right) \right] . \;\; \end{aligned}$$
(27)

To proceed, let us focus on the following term:

(28)

which can be used to rewrite Eq. (27) as

$$\begin{aligned}&J_W (x) = \int dy \delta (y) \left[ R\Big (x+\frac{y}{2}\Big )R\Big (x-\frac{y}{2}\Big ) e^{\frac{i}{\hbar }\left[ S(x+\frac{y}{2})-S(x-\frac{y}{2})\right] }\right. \nonumber \\&\quad \left. \cdot \, \frac{1}{2}\left( \frac{\partial S(x+\frac{y}{2})}{\partial x} +\frac{\partial S(x-\frac{y}{2})}{\partial x} \right) - i\hbar e^{\frac{i}{\hbar }\left[ S(x+\frac{y}{2})-S(x-\frac{y}{2})\right] } \right. \nonumber \\&\quad \left. \cdot \, \frac{\partial }{\partial y} \left( R\left( x+\frac{y}{2}\right) R\left( x-\frac{y}{2}\right) \right) \right] = R^{2}(x)\frac{\partial S(x)}{\partial x}. \end{aligned}$$
(29)

Equation (29) is the expression for the current density distribution for the Wigner function.

In Eq. (29) we have used the following property:

$$\begin{aligned} \int dp e^{-ipy/\hbar } = 2\pi \hbar \delta (y), \end{aligned}$$
(30)

and the fact that the second term within the integral in Eq. (29) is zero:

$$\begin{aligned}&\frac{\partial }{\partial y} \left( R\Big (x+\frac{y}{2}\Big )R\Big (x-\frac{y}{2}\Big ) \right) \Big |_{y=0} \nonumber \\&\quad = \frac{1}{2}\left[ \frac{\partial R(x+\frac{y}{2})}{\partial x} R\Big (x-\frac{y}{2}\Big ) - R\Big (x+\frac{y}{2}\Big )\frac{\partial R(x-\frac{y}{2})}{\partial x} \right] \Bigl |_{y=0} \nonumber \\&\quad = 0. \end{aligned}$$
(31)

Husimi distribution

The derivation of the Husimi charge and current densities are quite similar to the derivation realized for the Wigner distribution, with the only difference that now, the Wigner distributions are smoothed with a Gaussian function. In first place, we derive the charge distribution:

$$\begin{aligned}&Q_H(x,p)= \frac{1}{\pi \hbar }\int \int \int \frac{1}{h} \int \psi \Big ( x'-\frac{y}{2}\Big )\psi ^*\Big (x'+\frac{y}{2}\Big ) e^{i\frac{p'y}{\hbar }}dy \nonumber \\&\qquad \cdot \, e^{-\frac{(x-x')^2}{2s^2}}e^{-\frac{(p-p')^22s^2}{\hbar ^2}}dx'dp'dp = \frac{1}{\pi \hbar }\frac{1}{\sqrt{(2\pi s^2)}} \nonumber \\&\qquad \cdot \int \int \int \psi \Big ( x'-\frac{y}{2}\Big )\psi ^*\Big (x'+ \frac{y}{2}\Big ) e^{i\frac{p'y}{\hbar }}dy e^{-\frac{(x-x')^2}{2s^2}}dx'dp'\nonumber \\&\quad =\frac{1}{\sqrt{(2\pi s^2)}}\int \int \psi ( x'- \frac{y}{2})\psi ^*(x'+ \frac{y}{2}) \delta (y)dy e^{-\frac{(x-x')^2}{2s^2}}dx' \nonumber \\&\quad =\frac{1}{\sqrt{(2\pi s^2)}}\int |\psi ( x')|^2 e^{-\frac{(x-x')^2}{2s^2}}dx'. \end{aligned}$$
(32)

After that, we derive the current density. For this, Eqs. (26), (28), (30) and (31) will be used again:

$$\begin{aligned}&J_H (x)= \frac{1}{\pi \hbar }\int \int \int p F_H(x',p')e^{-\frac{(x-x')^2}{2s^2}}e^{-\frac{(p-p')^2}{2\sigma _p^2}}dx'dp' dp\nonumber \\&\quad =\frac{1}{\pi \hbar }\int \int \int p \frac{1}{h} \int \psi \Big ( x'-\frac{y}{2}\Big )\psi ^*\Big (x'+ \frac{y}{2}\Big ) e^{i\frac{p'y}{\hbar }}dy e^{-\frac{(x-x')^2}{2s^2}}\nonumber \\&\qquad \cdot \, e^{-\frac{(p-p')^22s^2}{\hbar ^2}}dx'dp' dp =\frac{1}{h}\frac{1}{\sqrt{(2\pi s^2)}}\int p'\psi \left( x'- \frac{y}{2}\right) \nonumber \\&\qquad \cdot \, \psi ^*\Big (x'+\frac{y}{2}\Big ) e^{i\frac{p'y}{\hbar }}dy e^{-\frac{(x-x')^2}{2s^2}}dx'dp' \nonumber \\&\quad =\frac{1}{\sqrt{(2\pi s^2)}}\int R^{2}(x')\frac{\partial S(x')}{\partial x'}e^{-\frac{(x-x')^2}{2s^2}}dx' \end{aligned}$$
(33)

Therefore, we can clearly see that these results for the charge and current densities are different from the Wigner results.

Bohmian distribution

The charge distribution is the following:

$$\begin{aligned}&Q_B(x,t)=\int \lim _{N \rightarrow \infty }\frac{1}{N}\sum _{i=1}^N\delta (x-x_i(t))\delta (p-p_i(t))dp \nonumber \\&\quad = \lim _{N \rightarrow \infty }\frac{1}{N}\sum _{i=1}^N\delta (x-x_i(t))=|\psi (x,t)|^2. \end{aligned}$$
(34)

The last equality in Eq. (34) is due to the quantum equilibrium hypothesis, which states that the charge distribution at time t is equal to the modulus squared of the wave function (for a more detailed discussion see [41]).

In order to calculate the current density, we need to express the wave function in the polar form (in the same way as done for the Wigner distribution) and we proceed in the following way:

$$\begin{aligned}&J_B(x,t)=\lim _{N \rightarrow \infty }\frac{1}{N}\sum _{i=1}^Np\delta (x-x_i(t))\delta (p-p_i(t))dp \nonumber \\&\quad = \lim _{N \rightarrow \infty }\frac{1}{N}\sum _{i=1}^Np_i\delta (x-x_i(t))=R^{2}(x)\frac{\partial S(x)}{\partial x}. \end{aligned}$$
(35)

Apart from using again the quantum equilibrium hypothesis, we have also used that the momentum in Bohmian mechanics in a single-particle case is [41]:

$$\begin{aligned} p(x,t) = \text {Im} \frac{\nabla \psi }{\psi }\equiv \frac{\partial S(x)}{\partial x}. \end{aligned}$$
(36)

Let us notice that (36) is just (17) multiplied by the electron mass m.

Appendix 2: Negative values of the Wigner distribution

If we manipulate the Wigner distribution (applying a change of variable), we can see that it can be thought as a correlation function:

$$\begin{aligned}&F_W(x,p) = \frac{1}{h} \int dp\int \psi \Big (x+\frac{y}{2}\Big ) \psi ^{*}\Big (x-\frac{y}{2}\Big )e^{i\frac{py}{\hbar }} dy \nonumber \\&\quad = 2\int \psi (r) \psi ^{*}(2x-r)e^{i\frac{py}{\hbar }}e^{i\frac{2p(r-2x)}{\hbar }} dr = 2\int \psi (r)e^{i\frac{pr}{\hbar }}\nonumber \\&\qquad \cdot \, \left( \psi (2x-r) e^{-i\frac{2p(r-2x)}{\hbar }}\right) ^{*} dr = 2\int \phi (r) \phi ^{*}(2x-r) dr.\nonumber \\ \end{aligned}$$
(37)

The change of variable is the following: \(x+\frac{y}{2}=r\). In addition, we have defined \(\phi (r)=\psi (r)e^{i\frac{pr}{\hbar }}\). For simplicity we do not indicate the dependence on p

With these considerations, we can prove that the Wigner function is real and can be negative. We compute the integral of a modulus of a function \(|\phi (r) + \phi (2x-r)|^2\) which is obviously positive:

$$\begin{aligned}&\int \big (\phi (r) + \phi (2x-r)\big ) \cdot \big (\phi (r)^{*} + \phi ^{*}(2x-r)\big )dr \nonumber \\&\quad =\int |\phi (r) + \phi (2x-r)|^2dr =\int \Big (|\phi (r)|^2+|\phi (2x-r)|^2 \nonumber \\&\qquad +\,\, \phi (r)\phi ^{*}(2x-r)+\phi (r)^{*}\phi (2x-r)\Big )dr\ge 0. \end{aligned}$$
(38)

Next, we observe that the last two terms in the right hand side are identical and, by using (37), equal to the Wigner distribution function \(F_W(x,p)\). In order to see that both terms are identical, we make a change of variable \(r'=2x-r\):

$$\begin{aligned}&\int \phi (r)^{*}\phi (2x-r)dr=-\int ^{-\infty }_\infty \phi (r')^{*}\phi (2x-r')dr'\nonumber \\&\quad =\int \phi (r')^{*}\phi (2x-r')dr'. \end{aligned}$$
(39)

Therefore, the sum of both terms are exactly equal to \(F_W(x,p)\) in (37). Thus Eq. (38) becomes:

$$\begin{aligned}&\int |\phi (r) + \phi (2x-r)|^2dr = \int |\phi (r)|^2dr\nonumber \\&\quad +\int |\phi (2x-r)|^2dr + 2\int \phi (r)\phi ^{*}(2x-r)dr\ge 0.\nonumber \\ \end{aligned}$$
(40)

that we rewrite as:

$$\begin{aligned}&F_W(x,p)=\int |\phi (r) + \phi (2x-r)|^2dr \nonumber \\&\quad - \int |\phi (r)|^2dr -\int |\phi (2x-r)|^2dr \end{aligned}$$
(41)

Therefore, the value of the Wigner function in (41) must be real and it can clearly take negative values when the first integral is smaller than the sum of the other two (for example, when \(\phi (r)=-\phi (2x-r)\)). In this way, it is proved that the Wigner function is a quasi-probability distribution, but not a true one.

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Colomés, E., Zhan, Z. & Oriols, X. Comparing Wigner, Husimi and Bohmian distributions: which one is a true probability distribution in phase space?. J Comput Electron 14, 894–906 (2015). https://doi.org/10.1007/s10825-015-0737-6

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Keywords

  • Wigner distribution
  • Husimi distribution
  • Bohmian distribution
  • Phase space