Quantum orbital angular momentum operators (Cartesian coordinates)

The quantum orbital angular momentum operators in Cartesian coordinates are derived from the classical angular momentum components




by replacing the position and linear momentum components with their corresponding operators. Since r_x,r_y,r_z are position components with position operators \hat{r}_x,\hat{r}_y,\hat{r}_z respectively, and p_x,p_y,p_z are linear momentum components with linear momentum operators \frac{\hbar}{i}\frac{\partial}{\partial r_x},\frac{\hbar}{i}\frac{\partial}{\partial r_y},\frac{\hbar}{i}\frac{\partial}{\partial r_z} respectively (see eq4), we have

\hat{L}_x=\frac{\hbar}{i}\left (r_y \frac{\partial}{\partial r_z}-r_z\frac{\partial}{\partial r_y}\right )\; \; \; \; or\; \; \; \;\hat{L}_x=\frac{\hbar}{i}\left (y \frac{\partial}{\partial z}-z\frac{\partial}{\partial y}\right )\; \; \; \; \; \; \; \; 72

\hat{L}_y=\frac{\hbar}{i}\left (r_z \frac{\partial}{\partial r_x}-r_x\frac{\partial}{\partial r_z}\right )\; \; \; \; or\; \; \; \;\hat{L}_y=\frac{\hbar}{i}\left (z \frac{\partial}{\partial x}-x\frac{\partial}{\partial z}\right )\; \; \; \; \; \; \; \; 73

\hat{L}_z=\frac{\hbar}{i}\left (r_x \frac{\partial}{\partial r_y}-r_y\frac{\partial}{\partial r_x}\right )\; \; \; \; or\; \; \; \;\hat{L}_z=\frac{\hbar}{i}\left (x \frac{\partial}{\partial y}-y\frac{\partial}{\partial x}\right )\; \; \; \; \; \; \; \; 74

From eq70, we have

\hat{L}^{2}=\hat{L}_x^{\, \, 2}+\hat{L}_y^{\, \, 2}+\hat{L}_z^{\, \, 2}\; \; \; \; \; \; \; \; 75

Since \hat{L}^{ 2} is defined as the operator for the square of the magnitude of \boldsymbol{\mathit{L}}, each of its eigenvalues is the square of the magnitude of the orbital angular momentum of an electron. 



Can we construct an angular momentum operator using \boldsymbol{\mathit{L}}=\boldsymbol{\mathit{i}}L_x+\boldsymbol{\mathit{j}}L_y+\boldsymbol{\mathit{k}}L_z, such that

\hat{\boldsymbol{\mathit{L}}}=\boldsymbol{\mathit{i}}\hat{L}_x+\boldsymbol{\mathit{j}}\hat{L}_y+\boldsymbol{\mathit{k}}\hat{L}_z\; \; \; \; \; \; \; \; 76

which is then used to generate eigenvalues?


\hat{L}^{2}, a scalar operator, is preferred over \hat{\boldsymbol{\mathit{L}}}, a vector operator, because it easier to manipulate in quantum computations. \hat{L}^{2} commutes with its component operators, allowing us to simultaneously determine the eigenvalues of \hat{L}^{2} and say, \hat{L}_z (which is useful, for example in the verification of singlet and triplet eigenstates). It also commutes with the time-independent Hamiltonian \hat{H}, also a scalar operator, implying that we can select a common complete set of eigenfunctions for \hat{L}^{2} and \hat{H}. Note that \hat{L}^{2} has a form that is consistent with the angular portion of \hat{H} in spherical coordinates (compare eq49 with eq96). An eigenvalue of \hat{H}_{angular} is the energy associated with the angular motion of an electron, while the square root of an eigenvalue of \hat{L}^{2} is the magnitude of the orbital angular momentum of an electron in a particular state.



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