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The usual angular-momentum operators we see in quantum mechanics describe the behavior of a complex-valued function on the sphere as we rotate about each of the three axes. In particular, we can write these operators in terms of spherical coordinates as \begin{aligned} L_{x} &= i\hbar\left(\sin\phi\frac{\partial}{\partial\theta}+\cot\theta\cos\phi\frac{\partial}{\partial\phi}\right), \\ L_{y} &= i\hbar\left(-\cos\phi\frac{\partial}{\partial\theta}+\cot\theta\sin\phi\frac{\partial}{\partial\phi}\right), \\ L_{z} &= -i\hbar\frac{\partial}{\partial\phi}. \end{aligned} But there is a generalization of these operators for a complex-valued function on the rotation group. We can, for example, write these operators in terms of Euler angles as[1] \begin{aligned} L_{x} &= i\hbar\left(\sin\alpha\frac{\partial}{\partial\beta} + \cot\beta\cos\alpha\frac{\partial}{\partial\alpha} - \frac{\cos\alpha}{\sin\beta}\frac{\partial}{\partial\gamma} \right), \\ L_{y} &= i\hbar\left(-\cos\alpha\frac{\partial}{\partial\beta}+\cot\beta\sin\alpha\frac{\partial}{\partial\alpha} - \frac{\sin\alpha}{\sin\beta}\frac{\partial}{\partial\gamma} \right), \\ L_{z} &= -i\hbar\frac{\partial}{\partial\alpha}. \end{aligned} Of course, spherical coordinates can be represented in terms of Euler angles with $\alpha=\phi$, $\beta=\theta$, and $\gamma=0$; plugging that into this last set of expressions simplifies to the usual version, so this is a strict generalization of the ordinary angular-momentum operators.

Due to non-commutativity of rotations, there is another operator $K$ such that \begin{aligned} K_{x} &= i\hbar\left(-\sin\gamma\frac{\partial}{\partial\beta} + \frac{\cos\gamma}{\sin\beta}\frac{\partial}{\partial\alpha} - \cot\beta\cos\gamma\frac{\partial}{\partial\gamma} \right), \\ K_{y} &= i\hbar\left(-\cos\gamma\frac{\partial}{\partial\beta} - \frac{\sin\gamma}{\sin\beta}\frac{\partial}{\partial\alpha} + \cot\beta\sin\gamma\frac{\partial}{\partial\gamma} \right), \\ K_{z} &= -i\hbar\frac{\partial}{\partial\gamma}. \end{aligned} I already know about applications of these $K_j$ operators to spin-weighted spherical functions ($-K_x - i\, K_y$ is the $\eth$ operator).

But I really think I've seen the generalized $L_j$ operators before somewhere in the physics literature; I just can't find where exactly. I feel like it had something to do with rigid-body rotation — without the factor of $\hbar$, obviously. Has anyone seen this sort of thing before, or have any ideas about specific physics applications it could be used for? I'd also be happy to see any other applications of the $K_j$ operators.


[1] It kills me to do this, because normally I avoid Euler angles like the plague; I only do this here because I think this is probably the form most people will have seen these in. More fundamentally, these operators are defined in terms of the derivatives with respect to rotation. For example, when $f(\hat{n})$ is a function on the sphere, the usual operators can be defined as \begin{equation} L_j f (\hat{n}) = i\hbar \left. \frac{\partial}{\partial \epsilon} f \left( R_j^\epsilon\, \hat{n} \right) \right|_{\epsilon=0}, \end{equation} where $R_j^\epsilon$ is the rotation about axis $j$ by an angle $\epsilon$. For $f(R)$ a function on the rotation group, we simply generalize this definition as \begin{equation} L_j f (R) = i\hbar \left. \frac{\partial}{\partial \epsilon} f \left( R_j^\epsilon\, R \right) \right|_{\epsilon=0}. \end{equation} And again, because rotations do not commute, we can further generalize to define a related operator that differs only by the order of operations: \begin{equation} K_j f (R) = i\hbar \left. \frac{\partial}{\partial \epsilon} f \left( R\, R_j^\epsilon \right) \right|_{\epsilon=0}. \end{equation}

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  • $\begingroup$ you’re using too many different notations so I’m not sure I’m right but there is a right and a left action of angular momentum operators on the group functions $D^J_{m’m}$, and this could be what you have in mind. The left action (on the left index) is usually understood as the action in the body-fixed frame, while the right action is in the space-fixed frame. The $D$-functions occurs in the description of extended solid objects such as molecules or nuclei. $\endgroup$ Aug 31, 2023 at 21:46

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You can use these operators for the quantized rotation of a rigid body (like a molecule). The Hamiltonian is given by: $$ H=\sum \frac{1}{2I_i}L_i^2 $$ along the principle axes of inertia. If you want to quantize this system, you need to impose the uncertainty principle on the phase space $T^*SO(3)$. The differential operator expression of $L$ in terms of Euler angles is to rotation as the Schrödinger representation of momentum in position basis $p=-i\nabla$ is to translation.

Since the Euler angles commute and form a complete basis of observables, the wave function $\psi$ of a rigid top can be seen as a function of these angles. Without reference to this special coordinate system, it can equivalently be seen as a complex function on $SO(3).$ The Schrödinger equation will be: $$ i\partial_t\psi= \sum \frac{1}{2I_i}L_i^2\psi $$ If you actually want to calculate the wavefunction, you’ll therefore need your expressions of the generalized angular momentum operators.

You could also calculate the canonical ensemble as well: $$ \rho\propto e^{-\beta H} $$ and similarly, you’ll need your expressions if you want to calculate expected values in the Euler angle basis.

Hope this helps.

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