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Question of quantum mechanics

  1. Oct 24, 2009 #1
    1. The problem statement, all variables and given/known data

    two particles move in a 2-dim system
    the potential is V=(1+([tex]\vec{r1}[/tex]*[tex]\vec{r2}[/tex]/R2))(([tex]\vec{P1}[/tex]*[tex]\vec{P1}[/tex]/2m1)+([tex]\vec{P2}[/tex]*[tex]\vec{P2}[/tex]/2m2))
    find the QM operator for this potential

    2. Relevant equations
    r1, r2, p1, p2 are vectors
    [tex]\vec{r1}[/tex]=(x1,y1)


    3. The attempt at a solution

    [tex]\hat{r1}[/tex]=x1x+y1y
    [tex]\hat{r2}[/tex]=x2x+y2y
    [tex]\hat{P1}[/tex]=-i[tex]\hbar[/tex]/2m1(d/dx)+d/dy))
    [tex]\hat{P2}[/tex]=-i[tex]\hbar[/tex]/2m2(d/dx)+d/dy))

    Is it correct to replace the V with the position and momentum operators directly?
    Would this be the hermitian operator?

    I think the first term (1+([tex]\hat{r1}[/tex]*[tex]\hat{r2}[/tex])/R2) is constant. Because the inner product of two vectors is scalar.

    I have the question about the 2nd term:

    Is [tex]\hat{P1}[/tex]*[tex]\hat{P1}[/tex]=(i[tex]\hbar[/tex])2*(d2/dx2+d2/dy2)=-[tex]\hbar[/tex]2*(d2/dx2+d2/dy2) --(1)


    Or [tex]\hat{P1}[/tex]*[tex]\hat{P1}[/tex]=-[tex]\hbar[/tex]2*(d2/dx2+d2/dy2+(d/dx)(d/dy)+(d/dy)*(d/dx)) --(2)

    thx
     
  2. jcsd
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