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Calculation of probabilities in QM

  1. Sep 19, 2009 #1
    1. The problem statement, all variables and given/known data
    Let [tex]\psi(x,y,z)=\psi(\vec{r})[/tex] be the normalized wave function of a particle.Express in terms of [itex]\psi(\vec{r})[/itex] the probability for a simultaneous measurements o X y P_z to yield :
    [tex]x_1 \leq x \leq x_2[/tex]
    [tex] p_z \geq 0[/tex]



    2. Relevant equations
    [tex]<\vec{p}|\vec{r}>=\frac{1}{(2\pi\hbar)^{3/2}}e^{-i\vec{p}.\vec{r}/\hbar} [/tex]
    [tex]<\vec{p}|\psi>=\frac{1}{(2\pi\hbar)^{3/2}}\int \psi(\vec{r}) e^{-i\vec{p}.\vec{r}/\hbar} dr^3[/tex]


    3. The attempt at a solution
    I have reached the following result:
    [tex]\int_{-\infty}^{\infty}dz\int_{-\infty}^{\infty}dy\int_{x_1}^{x_2}dx \int_{-\infty}^{\infty}dp_x\int_{-\infty}^{\infty}dp_y\int_0^{\infty}dp_z <\vec{p}|\vec{r}>\psi(\vec{r})<\psi|\vec{p}> [/tex]
    I need to know two things: 1) is my result correct? 2) in case it is correct, is there any other more simple or concrete answer?
     
    Last edited: Sep 20, 2009
  2. jcsd
  3. Sep 20, 2009 #2
    Sorry, I am not answering your question. But could you explain how you arrived at the expression [tex] <\vec{p}|\vec{r}>\psi(\vec{r})<\psi|\vec{p}>
    [/tex]
     
  4. Sep 20, 2009 #3
    I evalueted de expression [itex]<\psi|P_2P_1|\psi>[/itex] where P_1 and P_2 are the proyectors:
    [tex]P_1=\int_{-\infty}^{\infty}dz\int_{-\infty}^{\infty}dy\int_{x_1}^{x_2}dx|x,y,z><x,y,z|[/tex]
    [tex]P_2=\int_{-\infty}^{\infty}dp_x\int_{-\infty}^{\infty}dp_y\int_0^{\infty}dp_z|p_x,p_y,p_z><p_x,p_y,p_z|[/tex]
    But I'm sure whether what I'm doing is correct
     
    Last edited: Sep 20, 2009
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