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Homework Statement
I have the following expressions for angular momentum components: [itex]L_1 = x_2\frac{\partial}{\partial x_3} - x_3\frac{\partial}{\partial x_2}[/itex], [itex]L_2 = x_3\frac{\partial}{\partial x_1} - x_1\frac{\partial}{\partial x_3}[/itex], [itex]L_3 = x_1\frac{\partial}{\partial x_2} - x_2\frac{\partial}{\partial x_1}[/itex], and I simply need to work out [itex]L^2 = L_1^2 + L_2^2 + L_3^2[/itex].
Homework Equations
N/A
The Attempt at a Solution
Well, the way I expand it gives [tex]L_1^2 = (x_2\frac{\partial}{\partial x_3} - x_3\frac{\partial}{\partial x_2})(x_2\frac{\partial}{\partial x_3} - x_3\frac{\partial}{\partial x_2}) = x_2\frac{\partial}{\partial x_3}x_2\frac{\partial}{\partial x_3} - x_2\frac{\partial}{\partial x_3}x_3\frac{\partial}{\partial x_2} - x_3\frac{\partial}{\partial x_2}x_2\frac{\partial}{\partial x_3} + x_3\frac{\partial}{\partial x_2}x_3\frac{\partial}{\partial x_2} = -x_2\frac{\partial}{\partial x_2} - x_3\frac{\partial}{\partial x_3},[/tex] and similarly [tex]L_2^2 = -x_1\frac{\partial}{\partial x_1} - x_3\frac{\partial}{\partial x_3}[/tex] and [tex]L_3^2 = -x_1\frac{\partial}{\partial x_1} - x_2\frac{\partial}{\partial x_2},[/tex] so that [tex]L^2 = -2x_1\frac{\partial}{\partial x_1} - 2x_2\frac{\partial}{\partial x_2} - 2x_3\frac{\partial}{\partial x_3}[/tex] But this is not the expression for [itex]L^2[/itex] that I'm supposed to get! So I must be doing something wrong. Can anyone help?