What is the Difference Between Spin and Orbital Angular Momentum?
🆚 Go to Comparative Table 🆚The main difference between spin and orbital angular momentum lies in the axis of rotation and the nature of the motion they describe.
- Spin angular momentum refers to the angular momentum about the object's center of mass. It represents the rotation of an object around an axis through its center of mass. For example, the Earth has a spin angular momentum because of its daily rotation around its polar axis.
- Orbital angular momentum is the angular momentum about a chosen center of rotation. It describes the motion of an object relative to a point, often in a periodic manner. For instance, the Earth has an orbital angular momentum due to its revolution around the Sun.
In summary, spin angular momentum is the velocity of rotation of an object around a particular axis, while orbital angular momentum is the motion of an object relative to a chosen center of rotation. The total angular momentum is the sum of both spin and orbital angular momenta.
Comparative Table: Spin vs Orbital Angular Momentum
The difference between spin and orbital angular momentum can be summarized in the following table:
| Property | Spin Angular Momentum | Orbital Angular Momentum |
|---|---|---|
| Definition | Spin angular momentum is an intrinsic property of particles, representing their internal angular momentum. | Orbital angular momentum arises from the motion of a particle around another particle or center of mass. |
| Mathematical Expression | $$S^2 | s, m_s\rangle = \hbar^2 s (s+1) |
| Operator | $$S_z | s, ms\rangle = \hbar ms |
| Hilbert Space | $${\scr H}_{S} \equiv \mathbb C^{2s+1}$$. | $${\scr H}_{L} \equiv L^2 (\mathbb R^3)$$. |
| Total Angular Momentum | Spin angular momentum is always an integer multiple of $$\hbar$$. | Orbital angular momentum is typically denoted by the quantum number $$l$$ and can take values from 0 to $$n-1$$, where $$n$$ is the principal quantum number of the electron. |
In summary, spin angular momentum is an intrinsic property representing the internal angular momentum of a particle, while orbital angular momentum results from the motion of a particle around another particle or center of mass. The mathematical expressions, operators, and Hilbert spaces for these two types of angular momentum are different.
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