# Question of quantum mechanics

1. Oct 24, 2009

### gshock

1. The problem statement, all variables and given/known data

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

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

3. The attempt at a solution

$$\hat{r1}$$=x1x+y1y
$$\hat{r2}$$=x2x+y2y
$$\hat{P1}$$=-i$$\hbar$$/2m1(d/dx)+d/dy))
$$\hat{P2}$$=-i$$\hbar$$/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+($$\hat{r1}$$*$$\hat{r2}$$)/R2) is constant. Because the inner product of two vectors is scalar.

I have the question about the 2nd term:

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

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

thx