# Torque-Free Precession in Classical & Quantum Mechanics

• espen180
In summary, the conversation discusses the concept of precession in classical mechanics and whether it can be described quantum mechanically. The individual provides a simple example and their initial idea for a Hamiltonian, but then asks for help in finding a treatment for the problem. Another individual suggests a different Hamiltonian that includes the inverse of the inertia tensor and explains how it could lead to precession. The first individual concludes by thanking the other for their suggestion.
espen180
In classical mechanics, an asymmetric rotating object will generally precess. Expressed in the body-fixed normal system of the object, we have $I_i \dot{\omega_i}=(\vec{L}\times \vec{\omega})_i$ where $L_i=I_i\omega_i$.

Choosing a simple example where $I_1=I_2$, we obtain $\dot{\omega_3}=0$ and, for $\Omega=\frac{I_1-I_3}{I_1}\omega_3$,
$\dot{\omega_1}=\Omega \omega_2$
$\dot{\omega_2}=-\Omega \omega_1$
describing the precession. Thus, $\vec{\omega}(t)=(A\cos(\Omega t) , A\sin(\Omega t), \omega_3)$.

My question is; can this motion be described quantum mechanically?

My first guess was to write the Hamiltionian as $\hat{H}=\frac12 \hat{\vec{\omega}}I\hat{\vec{\omega}}$ with $I$ being the inertia tensor. The difficulty is then to describe $\hat{\vec{\omega}}$ in terms of $\hat{x},\hat{p_x}$ etc.

Is there any treatment of this problem available? I tried searching, but all the treatments of precession I found were related to magnetic moment precession.

Any help is greatly appreciated.

Last edited by a moderator:
That would still be concidered precession by an external torque, which is not what I am interested in here. Diatomic molecules don't experience free precession. I am sorry if I worded the problem poorly.

What I am interested in is the kind of precession the rotational axis of the Earth experiences, but at the quantum level. For example, a free spinning molecule of white phosphorus (tetrahedral molecule) would experience precession.

I see what you mean.

Try the following Hamiltonian:

$\hat{H} = \frac{1}{2} \sum\limits_{ij} \hat{L}_i I^{-1}_{ij} \hat{L}_j$

where $I^{-1}_{ij}$ is the invserse of the inertia tensor. In the normal system $I^{-1}_{ij} = \delta_{ij} \frac{1}{I_i}$

The angular momentum operator L is well defined, and the moment of inertia can be taken as constant.

If I am not mistaken, then L does not commute with the Hamiltonian, so that you get precession.

Great! I'll try it.
Thank you very much!

## 1. What is torque-free precession in classical mechanics?

Torque-free precession in classical mechanics refers to the motion of a spinning object, such as a top or gyroscope, when it is not subject to any external torques. This results in the object maintaining a constant angular momentum and precessing around a fixed axis.

## 2. How is torque-free precession different in quantum mechanics?

In quantum mechanics, torque-free precession involves the precession of the spin of a particle, such as an electron or proton, in the absence of any external magnetic fields. This precession is a result of the intrinsic spin of the particle and is described by quantum mechanical equations.

## 3. What is the significance of torque-free precession in classical and quantum mechanics?

Torque-free precession is an important concept in both classical and quantum mechanics as it demonstrates the conservation of angular momentum and the behavior of spinning objects. It also has practical applications in fields such as navigation, gyroscopic stabilization, and quantum computing.

## 4. How is torque-free precession observed in experiments?

Torque-free precession can be observed in various experiments, such as using a spinning top or gyroscope. In quantum mechanics, it can be observed through experiments such as the Stern-Gerlach experiment, which demonstrates the precession of an electron's spin in a magnetic field.

## 5. Can torque-free precession be affected by external factors?

In classical mechanics, torque-free precession is only affected by external torques. However, in quantum mechanics, the spin precession can be affected by external magnetic fields or interactions with other particles. This can lead to interesting phenomena such as spin flips and entanglement.

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