Orbital Angular Momentum Operators
In quantum mechanics, angular momentum is represented by a set of operators rather than by classical vectors. For a particle moving in three dimensions, the orbital angular momentum operator \( \hat{\mathbf{L}} \) is defined as
\[ \hat{\mathbf{L}} = \mathbf{r} \times \hat{\mathbf{p}}, \]
where \( \mathbf{r} = (x,y,z) \) is the position operator and \( \hat{\mathbf{p}} = -i\hbar\nabla \) is the linear momentum operator. Because \( \hat{\mathbf{p}} \) involves derivatives, the components of \( \hat{\mathbf{L}} \) are differential operators acting on wavefunctions.
Cartesian Components
Expanding the cross product gives the three Cartesian components of the orbital angular momentum operator:
\[ \hat{L}_x = -i\hbar\!\left( y\frac{\partial}{\partial z} - z\frac{\partial}{\partial y} \right), \] \[ \hat{L}_y = -i\hbar\!\left( z\frac{\partial}{\partial x} - x\frac{\partial}{\partial z} \right), \] \[ \hat{L}_z = -i\hbar\!\left( x\frac{\partial}{\partial y} - y\frac{\partial}{\partial x} \right). \]
Each component generates an infinitesimal rotation of the wavefunction about the corresponding Cartesian axis. For example, \( \hat{L}_z \) generates rotations about the \( z \)-axis.
Commutation Relations
Unlike the components of a classical angular momentum vector, the operators \( \hat{L}_x, \hat{L}_y, \hat{L}_z \) do not commute with one another. Their fundamental commutation relations are
\[ [\hat{L}_x, \hat{L}_y] = i\hbar \hat{L}_z, \qquad [\hat{L}_y, \hat{L}_z] = i\hbar \hat{L}_x, \qquad [\hat{L}_z, \hat{L}_x] = i\hbar \hat{L}_y. \]
These relations imply that it is impossible to simultaneously measure more than one Cartesian component of orbital angular momentum with arbitrary precision. If one component is known exactly, the other two are fundamentally uncertain.
Derivation: Explicitly showing cancellation in \( [\hat L_x,\hat L_y] \)
Recall \( \hat L_x=-i\hbar\!\left(y\frac{\partial}{\partial z}-z\frac{\partial}{\partial y}\right) \) and \( \hat L_y=-i\hbar\!\left(z\frac{\partial}{\partial x}-x\frac{\partial}{\partial z}\right) \). We apply each product to an arbitrary function \( \psi(x,y,z) \).
Step 2: Evaluate both products (color shows terms that cancel)
\[ \hat L_x(\hat L_y\psi) = -\hbar^2\Big( y\,\frac{\partial\psi}{\partial x} + yz\,\frac{\partial^2\psi}{\partial x\,\partial z} - xy\,\frac{\partial^2\psi}{\partial z^2} - z^2\,\frac{\partial^2\psi}{\partial x\,\partial y} + xz\,\frac{\partial^2\psi}{\partial y\,\partial z} \Big) \]
\[ \hat L_y(\hat L_x\psi) = -\hbar^2\Big( x\,\frac{\partial\psi}{\partial y} + yz\,\frac{\partial^2\psi}{\partial x\,\partial z} - xy\,\frac{\partial^2\psi}{\partial z^2} - z^2\,\frac{\partial^2\psi}{\partial x\,\partial y} + xz\,\frac{\partial^2\psi}{\partial y\,\partial z} \Big) \]
The second-derivative terms appear in both expressions and therefore cancel when the difference is taken to derive the commutator.
Step 3: Form the commutator
\[ [\hat L_x,\hat L_y]\psi = -\hbar^2\!\left( y\,\frac{\partial\psi}{\partial x} - x\,\frac{\partial\psi}{\partial y} \right) \]
\[ = \hbar^2\!\left( x\,\frac{\partial\psi}{\partial y} - y\,\frac{\partial\psi}{\partial x} \right) = i\hbar\,\hat L_z\psi, \]
\[ \boxed{[\hat L_x,\hat L_y]=i\hbar\,\hat L_z} \]
Similarly, cyclic permutation of the Cartesian components gives
\[ [\hat L_y,\hat L_z] = i\hbar\,\hat L_x, \qquad [\hat L_z,\hat L_x] = i\hbar\,\hat L_y. \]
Total Angular Momentum
The square of the total orbital angular momentum operator is defined as
\[ \hat{L}^2 = \hat{L}_x^2 + \hat{L}_y^2 + \hat{L}_z^2. \]
Importantly, \( \hat{L}^2 \) commutes with each Cartesian component:
\[ [\hat{L}^2, \hat{L}_x] = [\hat{L}^2, \hat{L}_y] = [\hat{L}^2, \hat{L}_z] = 0. \]
As a result, a quantum state can be chosen to have definite values of \( \hat{L}^2 \) and one component of angular momentum (by convention, \( \hat{L}_z \)). This structure underlies the quantization of angular momentum and leads directly to the appearance of the spherical harmonics as angular eigenfunctions.
Key idea: The non-commuting nature of \( \hat{L}_x, \hat{L}_y, \hat{L}_z \) is not a mathematical curiosity—it encodes the rotational symmetry of three-dimensional space and determines the allowed angular momentum quantum numbers used throughout atomic and molecular spectroscopy.
Angular Momentum in Spherical Polar Coordinates
For the rigid rotor (and for central-force problems more generally), it is most natural to use spherical polar coordinates \( (r,\theta,\phi) \). In these coordinates, the orbital angular momentum operators simplify dramatically. In particular, the operator \( \hat L_z \) becomes a single derivative with respect to the azimuthal angle \( \phi \).
The \( \hat L_z \) operator
\[ \hat L_z = -\,i\hbar\,\frac{\partial}{\partial \phi}. \]
This form reflects the physical meaning of \( \phi \): changing \( \phi \) corresponds to a rotation about the \( z \)-axis. Thus \( \hat L_z \) is the generator of rotations about \( z \).
The \( \hat L^2 \) operator
The total orbital angular momentum operator \( \hat L^2 \) contains only angular derivatives (no radial derivatives). In spherical polar coordinates,
\[ \hat L^2 = -\hbar^2\left[ \frac{1}{\sin\theta}\frac{\partial}{\partial\theta} \left(\sin\theta\,\frac{\partial}{\partial\theta}\right) +\frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2} \right]. \]
This operator depends only on the angles \( \theta \) and \( \phi \), which is why the rigid rotor problem separates cleanly into an angular equation.
Spherical harmonics as eigenfunctions
The spherical harmonics \( Y_\ell^{m}(\theta,\phi) \) are special functions that arise naturally as simultaneous eigenfunctions of \( \hat L^2 \) and \( \hat L_z \). They satisfy the eigenvalue equations:
\[ \hat L^2\,Y_\ell^{m}(\theta,\phi) = \hbar^2\,\ell(\ell+1)\,Y_\ell^{m}(\theta,\phi), \] \[ \hat L_z\,Y_\ell^{m}(\theta,\phi) = \hbar\,m\,Y_\ell^{m}(\theta,\phi). \]
Here, \( \ell = 0,1,2,\ldots \) is the orbital angular momentum quantum number and \( m = -\ell, -\ell+1, \ldots, +\ell \) is the magnetic quantum number.
Why \( [\hat L^2,\hat L_z]=0 \)
Because the spherical harmonics form a set of functions that can be chosen to have definite values of both \( \hat L^2 \) and \( \hat L_z \) simultaneously, the operators must be compatible (they can be diagonalized at the same time). This is expressed by the commutator:
\[ \boxed{[\hat L^2,\hat L_z]=0.} \]
Key idea: In rigid-rotor language, the quantum number \( \ell \) fixes the total rotational angular momentum, while \( m \) fixes its projection on the laboratory \( z \)-axis. The spherical harmonics provide the angular wavefunctions that carry these quantum numbers.