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gshock
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Homework Statement
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
Homework Equations
r1, r2, p1, p2 are vectors
[tex]\vec{r1}[/tex]=(x1,y1)
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