How to Construct Quantum Operators for a Two-Particle System?

[gshock]

Homework Statement

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

Homework Equations

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

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

 

[Jhero]

Thank you for your question. In order to find the QM operator for this potential, we need to first understand the form of the potential itself. The potential given in your post is a combination of two terms: the first term is a constant and the second term is a function of position and momentum operators. Therefore, we can write the potential as V = V_0 + V_1, where V_0 is the constant term and V_1 is the function of position and momentum operators.

Now, let's consider the second term V_1 in more detail. It can be rewritten as V_1 = (\hat{r1}*\hat{r2}/R2)((\hat{P1}*\hat{P1}/2m1)+(\hat{P2}*\hat{P2}/2m2)). As you correctly pointed out, the inner product of two vectors is a scalar, so we can rewrite this as V_1 = (1/R2)((\hat{r1}*\hat{r2})(\hat{P1}*\hat{P1})/2m1 + (\hat{r1}*\hat{r2})(\hat{P2}*\hat{P2})/2m2).

Now, we can use the commutation relation between position and momentum operators, [\hat{r},\hat{p}] = i\hbar, to rewrite the term (\hat{r1}*\hat{r2})(\hat{P1}*\hat{P1}) as [\hat{r1},\hat{r2}]\hat{P1}\hat{P1}+ \hat{r1}[\hat{r2},\hat{P1}]\hat{P1}+ \hat{r2}[\hat{r1},\hat{P1}]\hat{P1}. Using this, we can rewrite V_1 as V_1 = (1/R2)([\hat{r1},\hat{r2}]\hat{P1}\hat{P1}+ \hat{r1}[\hat{r2},\hat{P1}]\hat{P1}+ \hat{r2}[\hat{r1},\hat{P1}]\hat{P1})/2m1 + (1/R2