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In quantum mechanics, Ehrenfest's theorem states that $\langle p_x\rangle = m \frac{d}{dt}\langle x\rangle$. My question is, does there exist a similar relationship between $\langle L_z\rangle$, the expectation value of the z-component of the orbital angular momentum operator, and the time derivative of $\langle\theta\rangle$, the expectation value of the position operator $\hat{\theta}$ in spherical coordinates?

If not, is there any way to relate $\langle L_z\rangle$ to the time derivatives of expectation values of one or more operators? For instance if you knew what $\langle x\rangle$, $\langle y\rangle$, $\langle z\rangle$, $\langle p_x\rangle$, $\langle p_y\rangle$, and $\langle p_z\rangle$ are as a function of time, would that give you enough information to determine $\langle L_z\rangle$?

And are the answers to these questions affected at all by whether the particle has spin or not? By the way, this question was inspired by the comment section of this answer.

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Keshav Srinivasan
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2 Answers2

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Thanks to @udrv, I found the answer in this journal paper. Let's work in cylindrical coordinates $(r,\theta,z)$. Let $\cos\hat{\theta}$ and $\sin\hat{\theta}$ be defined by Taylor series, and let $\hat{L}_z = m\hat{r}^2 \hat{\omega}_z$. Then we can write the result in two forms:

$$\frac{d}{dt}\langle \cos\hat{\theta} \rangle = \langle -\frac{1}{2}(\hat{\omega}_z \sin\hat{\theta}+ \sin\hat{\theta}\hat{\omega}_z))\rangle$$

$$\frac{d}{dt}\langle \sin\hat{\theta} \rangle = \langle\frac{1}{2}(\hat{\omega}_z \cos\hat{\theta}+ \cos\hat{\theta}\hat{\omega}_z))\rangle$$

The reason why why we can't simply use the operator $\hat{\theta}$ is that $\hat{L}_z$ is only a Hermitian operator if its domain is restricted to periodic functions, and $\hat{\theta}$ maps periodic functions to non-periodic functions. So if we want to keep things within the domain of $L_z$ we need to work with an operator $f(\hat{\theta})$ where $f$ is a periodic function. And the simplest periodic functions which make $f(\hat{\theta})$ a Hermitian operator are sine and cosine. (It needs to be Hermitian if we want our Ehrenfest result to be between observable quantities.)

EDIT: The paper also provides a more general result for arbitrary periodic functions $f$ with period $2\pi$:

$$\frac{d}{dt}\langle f(\hat{\theta}) \rangle = \langle \frac{1}{2}(\hat{\omega}_z f'(\hat{\theta})+ f'(\hat{\theta})\hat{\omega}_z)\rangle$$

where again $f(\hat{\theta})$ and $f'(\hat{\theta})$ are defined via Taylor series.

Note that while this formula is true for all such functions $f$, in order it to be a result between observable quantities $f(\hat{\theta})$ needs to be a Hermitian operator. I posted a question here to find out what functions $f$ make $f(\hat{\theta})$ Hermitian.

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Keshav Srinivasan
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Yes, indeed, it is possible to derive such a result... although it might look somewhat trivial, if compared to position and momentum in an arbitrary potential.

Hamiltonian of angular momentum in magnetic field is: $$ H=-\frac{\mu_B g_l}{\hbar} \mathbf{L}\cdot\mathbf{B}. $$ We can write Heisenberg equations of motion for the components of the angular momentum, $L_x,L_y,L_z$: $$ \frac{d}{dt}L_\alpha = \frac{1}{i\hbar}\left[L_\alpha, H\right], $$ calculate the commutators, average the equations, and obtain the usual equation for the Larmor precession: $$ \frac{d}{dt}\langle\mathbf{L}(t)\rangle=\gamma \langle\mathbf{L}(t)\rangle\times\mathbf{B}, $$ (where $\times$ stands for vector product) as it appears in well-known Bloch equations. The reason why I called this result "trivial" is because the equations are linear to begin with, as is in the case of Harmonic oscillator for $x,p$. That is, one does not have to deal with the average of the potential not being equal to the potential of the average, inherent in the Ehrenfest formulation: $\langle V(x)\rangle\neq V(\langle x\rangle)$ (see this answer for more discussion.)

Perhaps, one could have some more interesting cases, e.g., in the context of Landau–Lifshitz–Gilbert equation, but this one is phenomenological, so the original Ehrenfest's point of relating quantum and classical mechanics is lost.

Roger V.
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