How to Show the Evolution of a Free Particle's Statistical Mixture?

[MathematicalPhysicist]

Homework Statement

With the wave function $$\psi(r)$$ of a particle, one forms the function:
$$D(R,P)=\frac{\int exp(-\frac{i}{\hbar} P\cdot r)\psi^*(R-r/2)\psi (R+r/2) dr}{(2\pi \hbar)^3}$$, which is the density in phase of a classical statistical mixture associated with this wave function, Show that:
1. $$\int D(R,P)dP=|\psi (R)|^2$$ $$\int D(R,P)dR=|\psi (P)|^2$$.
2. if the particle is free, the evolution in time of the mixture is strictly that of a statistical mixture of free classical particles of the same mass;
3. find the spreading law of a free wave packet.

Homework Equations

I think one of them is dirac delta function:
$$\int \frac{exp(-\frac{i}{\hbar} P\cdot r)}{(2\pi \hbar)^3}dr=\delta(r) \ \int \frac{exp(-\frac{i}{\hbar} P\cdot r)}{(2\pi \hbar)^3}dP=\delta(P)$$

The Attempt at a Solution

1. So far by using the above equality I get that:
$$\int D(R,P)dP=\int \frac{\int exp(-\frac{i}{\hbar} P\cdot r)\psi^*(R-r/2)\psi (R+r/2) dr}{(2\pi \hbar)^3} dP=\int \psi^*(R-r/2)\psi (R+r/2) dr$$, I don't see how this becomes the amplitude of the wave function in the R space squared?

2. Don't know what I need to show here.

3. The same as with 2.

Any hints?
Not answers!

 

[fantispug]

Just to help you with 1, your equations are wrong - both of them depend only on r on one side of the equation and P on the other side. Fix them and redo the integration.

 

[MathematicalPhysicist]

Can someone help me with questions 2,3?