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A question from QM by Messiah.

  1. Aug 30, 2009 #1

    MathematicalPhysicist

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    1. The problem statement, all variables and given/known data
    With the wave function [tex]\psi(r)[/tex] of a particle, one forms the function:
    [tex]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}[/tex], which is the density in phase of a classical statistical mixture associated with this wave function, Show that:
    1. [tex]\int D(R,P)dP=|\psi (R)|^2[/tex] [tex]\int D(R,P)dR=|\psi (P)|^2[/tex].
    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.


    2. Relevant equations
    I think one of them is dirac delta function:
    [tex]\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)[/tex]


    3. The attempt at a solution
    1. So far by using the above equality I get that:
    [tex]\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[/tex], 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!
     
    Last edited: Aug 31, 2009
  2. jcsd
  3. Sep 2, 2009 #2
    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.
     
  4. Oct 2, 2009 #3

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    Can someone help me with questions 2,3?
     
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