Wigner distribution in phase space

[aim1732]

This is about a specific property of the Wigner distribution in phase space. My professor mentioned the other day that the Wigner distribution treats all functions of momentum and space on the same footing as momentum itself or at least that's what I recall.He mentioned a specific problem where we had to show the following:

Considering the operator qcosθ + psinθ = q_θ
and -qsinθ + pcosθ = p_θ

such that q_θ |q_θ> = q_θ|q_θ>

We had to show that ∫ dp_θW'(q_θ,p_θ) = |<q_θ|ψ>|²
where W is the Wigner dis.
You may find the problem a little hazy(so do I). Even my professor said that it he did not recall it exactly and it might not be properly defined.If anyone has seen this or might have a useful suggestion please help me.I would be very thankful.

 

[samalkhaiat]

This is about a specific property of the Wigner distribution in phase space. My professor mentioned the other day that the Wigner distribution treats all functions of momentum and space on the same footing as momentum itself or at least that's what I recall.He mentioned a specific problem where we had to show the following:

Considering the operator qcosθ + psinθ = q_θ
and -qsinθ + pcosθ = p_θ

such that q_θ |q_θ> = q_θ|q_θ>

We had to show that ∫ dp_θW'(q_θ,p_θ) = |<q_θ|ψ>|²
where W is the Wigner dis.
You may find the problem a little hazy(so do I). Even my professor said that it he did not recall it exactly and it might not be properly defined.If anyone has seen this or might have a useful suggestion please help me.I would be very thankful.

If the system is in a pure state with wave function $\Psi ( x )$, the Wigner function is defined by
$$W( x , p ) = \frac{ 1 }{ \pi } \int dy \ \Psi^{ * } ( x + y ) \Psi ( x - y ) e^{ 2 i y p }$$
or, in terms of the density matrix $\rho ( x , \bar{ x } )$:
$$W( x , p ) = \frac{ 1 }{ \pi } \int dy \ \rho ( x + y , x - y ) e^{ 2 i y p }$$
OK, can you now convert the Wigner function back to the following density matrix?
$$\rho ( x , \bar{ x } ) = \int dp \ W \left( \frac{ x + \bar{ x } }{ 2 } , p \right) e^{ - i p ( x - \bar{ x } ) }$$
$\rho ( x , x )$ is your answer.

 

[aim1732]

Yes I can.I will Fourier invert the expression for W(x,p) after assuming 2y=y'(say).Then I will make the appropriate substitution to get the expression for the density matrix.However I am still not sure what to make of it.My doubt primarily revolves around what function I have to employ as in:
∫ dp_θW'(q_θ,p_θ) = |<q_θ|ψ>|²

Here the expression is trivially reduced to a marginal if I consider taking the Wigner function W'(qθ,pθ) as being defined in terms of the wave function in the vector space of eigenfunctions of qθ.Is this what I have to show?I mean that would be trivially true because we have already shown that the marginal of W(p,q) reduces to the probability distribution in the vector space of the other variable.

Please specify exactly what you had in mind when you said that.I might be missing something obvious because I am after all a lost undergrad who can not locate appropriate sources to get a hang of this distribution.
Regards
Aiman.

 

[andrien]

try this
http://en.wikipedia.org/wiki/Wigner_quasi-probability_distribution