Question

Convert the 2-D Laplacian operator nabla^2 delta^2/delta x^2 + delta^2/delta y^2 to polar coordinates using basic trigonometric relations between x, y, r, and Phi and use this result to determine an operator for 2D rotational kinetic energy about a fixed radius Using the cross product formulation of the angular momentum operator below, find the components l^^_r, l^^_y and l^^_z. Then show (by commutator) that l^^_x and l^^_z cannot be simultaneously determined but that l^^_2 and l^^_z can l^^= r^^times p^^= |e^bar_x x^^p^^_x e^bar_y y^^p^^_y e^bar_z z^^p^^_z|, where |M| = det(M)

Answer 1

Solution:

(a) 2-D laplacian to polar coordinates, the expression are shown below,

2-D equation given,

$\triangledown ^2 u =\frac{\partial ^2u}{\partial x^2}+\frac{\partial ^2u}{\partial y^2}=0$

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