In quantum mechanics, the angular momentum operators \(L_x\), \(L_y\), and \(L_z\) obey the following commutation relations:
\[ [L_x, L_y] = i\hbar L_z \]
\[ [L_y, L_z] = i\hbar L_x \]
\[ [L_z, L_x] = i\hbar L_y \]
To derive these, we start by defining the angular momentum operators in terms of the position \(\mathbf{r} = (x, y, z)\) and momentum \(\mathbf{p} = (p_x, p_y, p_z)\) operators:
\[ L_x = y p_z - z p_y \]
\[ L_y = z p_x - x p_z \]
\[ L_z = x p_y - y p_x \]
We then use the canonical commutation relations for position and momentum operators:
\[ [x_i, p_j] = i\hbar \delta_{ij} \]
\[ [x_i, x_j] = 0 \]
\[ [p_i, p_j] = 0 \]
For example, to derive \([L_x, L_y]\):
\[ [L_x, L_y] = [y p_z - z p_y, z p_x - x p_z] \]
Expanding this out using the commutator properties:
\[ [L_x, L_y] = [y p_z, z p_x] - [y p_z, x p_z] - [z p_y, z p_x] + [z p_y, x p_z] \]
Each term can be simplified using the fundamental commutation relations:
\[ [y p_z, z p_x] = y [p_z, z] p_x + z [y, p_x] p_z = y (i\hbar) p_x + z (0) p_z = i\hbar y p_x \]
\[ [y p_z, x p_z] = y [p_z, x] p_z = y (-i\hbar) p_z = -i\hbar y p_z \]
\[ [z p_y, z p_x] = 0 \]
\[ [z p_y, x p_z] = z [p_y, x] p_z + x [z, p_z] p_y = z (-i\hbar) p_z + x (i\hbar) p_y = -i\hbar z p_z + i\hbar x p_y \]
Combining all terms:
\[ [L_x, L_y] = i\hbar y p_x - i\hbar y p_z - i\hbar z p_z + i\hbar x p_y = i\hbar (x p_y - y p_x) \]
Recognizing the angular momentum definition, we find:
\[ [L_x, L_y] = i\hbar L_z \]
The other two commutation relations \([L_y, L_z]\) and \([L_z, L_x]\) can be derived similarly.