1

If the position operator is $\mathbf R = (\tilde X,\tilde Y,\tilde Z)$ and the momentum operator is $\mathbf P = (\tilde P_x,\tilde P_y,\tilde P_z)$ with components $$ \tilde X = X\otimes\mathbb I \otimes\mathbb I,\qquad \tilde Y = \mathbb I \otimes Y \otimes\mathbb I,\qquad \tilde Z = \mathbb I \otimes \mathbb I \otimes Z \\ \tilde P_x = P_x\otimes\mathbb I \otimes\mathbb I,\qquad \tilde P_y = \mathbb I \otimes P_y \otimes\mathbb I,\qquad \tilde P_z = \mathbb I \otimes \mathbb I \otimes P_z $$ These operators (with a twidle) are called the extended from the untwidled operators (as in Tannoudji) and they act on the tensor product of the Hilbert spaces $\scr E_x,\ \scr E_y$ and $\scr E_z$, i.e., on $\scr E = \scr E_x \otimes \scr E_y \otimes \scr E_z$

then does that the angular momentum operator $\mathbf L = (L_x,L_y,L_z)$ have components like $$ L_x = \tilde Y\tilde P_z - \tilde Z\tilde P_y = \mathbb I \otimes Y \otimes P_z - \mathbb I \otimes P_y \otimes Z \\ L_y = \tilde Z\tilde P_x - \tilde X \tilde P_z = P_x \otimes \mathbb I \otimes Z - X \otimes \mathbb I \otimes P_z \\ L_Z = \tilde X\tilde P_y - \tilde Y\tilde P_x = X \otimes P_y \otimes \mathbb I - P_x \otimes Y \otimes \mathbb I \\ $$

Physor
  • 870
  • these are right. I think @Leo may have some misunderstanding as spatial coordinates are not in a direct sum space. – Orion Yeung Dec 24 '20 at 17:42
  • I thinks so, and I don't what do spatial dimensions have to do with this. By the way, do you have some referrence that states that explicitly ? – Physor Dec 24 '20 at 17:50
  • 2
    The specification of spatial dimension is that the position is in a product space. I think this will be indirectly illuminating. https://physics.stackexchange.com/questions/528339/how-do-tensor-products-and-direct-sums-fit-into-quantum-mechanics – Orion Yeung Dec 24 '20 at 17:54
  • Some other useful verbiage to link the notion of direct sum and product. Direct sums are associated to the notion of XOR and direct products are associated to the notion of AND. A composite particle has spin information characterized by j AND m, where j=0 XOR j=1... – Orion Yeung Dec 24 '20 at 18:00
  • @OrionYeung Thanks, I'll read it – Physor Dec 24 '20 at 18:07

0 Answers0