Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

Most sources say that Wigner distribution acts like a joint phase-space distribution in quantum mechanics and this is justified by the formula

$$\int_{\mathbb{R}^6}w(x,p)a(x,p)dxdp=<\psi|\hat{a}\psi>$$ as this allows us to compute the expectation value for the operator $\hat{a}$ corresponding to the physical quantity $a$. However, without the knowledge of this formula, how did Wigner come up with this definition? $$w(x,p)=\frac{1}{(2\pi)^3}\int_{\mathbb{R}^3}\psi(x-\frac{v}{2})\psi^*(x+\frac{v}{2})e^{iv.p}dv.$$ I would be greatly indebted for any mathematical motivation that anyone could provide. Similarly, I also wonder at the mathematical motivation for the Weyl quantization.

share|improve this question
    
If you like this question you may also enjoy reading this post. –  Qmechanic May 11 '13 at 12:47

1 Answer 1

up vote 3 down vote accepted

Let us for simplicity work in 1D with $\hbar=1$. (The generalization to higher dimensions is straightforward.) Moreover, let us for simplicity take an operator $\hat{f}(\hat{X},\hat{P})$ without any ordering ambiguities, i.e., each monomial term in the symbol $f(x,p)$ depends only on either $x$ or $p$, but not on both. Then one possible motivation of Wigner's phase space distribution $$ \tag{1} w(x,p)~:=~ \int_{\mathbb{R}}\! {dy\over2\pi}e^{ipy}\psi^{*}(x+\frac{y}{2}) \psi(x-\frac{y}{2}). $$

goes as follows. The expectation value of the operator $\hat{f}(\hat{X},\hat{P})$ in the Schrödinger position representation

$$\tag{2} \hat{X} ~\longrightarrow~x, \qquad \hat{P} ~\longrightarrow~ -i \frac{\partial}{\partial x}, $$

reads

$$\langle\psi| \hat{f}(\hat{X},\hat{P})|\psi \rangle ~\stackrel{(2)}{=}~ \int_{\mathbb{R}} \!dx~ \psi^{*}(x) f\left(x, -i \frac{\partial}{\partial x}\right) \psi(x) \qquad\qquad $$ $$ ~\stackrel{\begin{matrix}\text{substitute}\\ x=x^{\pm}\end{matrix}}{=}~ \int_{\mathbb{R}^{2}}\! dx^{+}dx^{-}~ \delta(x^{+}-x^{-})\psi^{*}(x^{+}) f\left(\frac{x^{+}+x^{-}}{2}, -i \frac{\partial}{\partial x^{-}}\right) \psi(x^{-}) $$ $$ ~\stackrel{\delta\text{-fct}}{=}~ \int_{\mathbb{R}^{3}}\! {dx^{+}dx^{-}dp\over2\pi}~ e^{ip(x^{+}-x^{-})}\psi^{*}(x^{+}) f\left(\frac{x^{+}+x^{-}}{2}, -i \frac{\partial}{\partial x^{-}}\right) \psi(x^{-}) $$ $$ ~\stackrel{\text{int. by part}}{=}~\int_{\mathbb{R}^{3}}\! {dx^{+}dx^{-}dp\over2\pi}~ e^{ip(x^{+}-x^{-})}\psi^{*}(x^{+}) f\left(\frac{x^{+}+x^{-}}{2}, p\right) \psi(x^{-}) $$ $$ ~\stackrel{\begin{matrix}\text{substitute}\\ x^{\pm} = x\pm \frac{y}{2}\end{matrix}}{=}~\int_{\mathbb{R}^{3}}\! {dxdydp\over2\pi}~ e^{ipy}\psi^{*}(x+\frac{y}{2}) f(x, p) \psi(x-\frac{y}{2}) $$ $$ \tag{3} ~\stackrel{(1)}{=}~\int_{\mathbb{R}^{2}}\! dxdp~w(x,p) f(x, p) .$$

That's the motivation! The above manipulations make sense for a sufficiently well-behaved function $f(x, p)$.

For more general operators $\hat{f}(\hat{X},\hat{P})$, we leave it for OP to show that if $f(x,p)$ is interpreted as the Weyl-symbol of the operator $\hat{f}(\hat{X},\hat{P})$, then the equation

$$\tag{3'} \langle\psi| \hat{f}(\hat{X},\hat{P}) |\psi\rangle ~=~\int_{\mathbb{R}^{2}}\! dxdp~w(x,p) f(x, p)$$

continues to hold [at least for a sufficiently well-behaved function $f(x, p)$].

One important virtue from a physics perspective of Weyl-ordering (as opposed to other ordering prescriptions) is that the operator $\hat{f}(\hat{X},\hat{P})$ formally becomes Hermitian for real functions $f:\mathbb{R}^2 \to\mathbb{R}$ and two Hermitian operators $\hat{X}$ and $\hat{P}$. Recall that Hermitian operators correspond to physical observables in quantum mechanics. OP might also find this Phys.SE post useful.

share|improve this answer
    
So Wigner was aware of the Weyl Quantization rule? –  Vivek May 11 '13 at 22:15
    
Weyl quantization (1927) predates Wigner's phase space distribution (1932) by five years, so it seems likely. –  Qmechanic May 11 '13 at 22:35
    
I am not a physics student so I was wondering whether physics students at the graduate level are familiar with Weyl Quantization? –  Vivek May 13 '13 at 14:17
1  
That seems to be a good question for our chat room. My impression is that students generally have heard that one is supposed to symmetrize the $\hat{X}$s and $\hat{P}$s. Whether they have pondered how to do this systematically depends on their specialization. –  Qmechanic May 13 '13 at 20:31
1  
I updated the answer. –  Qmechanic May 13 '13 at 21:30

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.