# What is Orbital Angular Momentum?

1. Feb 21, 2013

### Johnleprekan

This is a very basic question, but what is orbital angular momentum actually? I know it can be quantized as zero, 1, etc. What is it in layman's terms? I tend to understand it in terms like the earth orbiting the sun. Spin on the other hand is either right or left spin. So what do the different numbers of orbital angular momentum mean? If it were the earth rotating around the sun, would it be like different ones orbiting at different angles to the axis of the sun?

2. Feb 21, 2013

### Staff: Mentor

In Bohr's model of the atom as extended by Sommerfeld, different values of l (the orbital angular momentum quantum number) corresponded to different amounts of ellipticity of the orbit. For l = 0 the "orbit" is a straight line back-and-forth through the nucleus. For larger values the orbit becomes more and more circular.

Different values of the quantum number m (for the same l) give you different orientations of the orbit in three dimensions.

But keep in mind that the Bohr-Sommerfeld model is dead, dead, dead as a doornail. You should not attempt to visualize the electron as actually moving in a distinct path like a circle, ellipse, straight line, etc. l = 0 gives you a spherical "probability cloud" with the nucleus at the center. Other values of l give you different shape "clouds" which you can find pictures of with a Google search for something like "hydrogen orbitals."

3. Feb 21, 2013

### Darwin123

First, one should understand the classical analogue of angular momentum

Classically, the orbital angular momentum of a body is the component of angular momentum associated with translation of the center of mass of the body. The spin orbital momentum is the component of momentum associated with the the rotation of the body about its center of mass. The total angular momentum is the vector sum of the spin angular momentum and the orbital angular momentum.

Consider the earth relative to the sun. Ignore the earth's moon just for purposes of this illustration.

The orbital angular momentum of the earth relative to the sun is calculated from the motion of the center of mass of the earth relative to the sun. The center of mass of the earth moves in an ellipse about the sun about once every 365 days.

The spin orbital angular momentum of the earth has nothing to do with the position of the sun. The spin angular momentum is calculated from the motion of all the particles that make up the earth relative to the center of mass of the earth. The particles of the earth come back to their original positions, in a coordinate system centered on the earth CM,about once a day.

The total angular momentum of the earth is about the spin angular momentum and the orbital angular momentum. Of course, there are very small components of angular momentum associated with the suns motion in the galaxy and the galaxies motion in the galactic cluster. However, these terms are negligible for most purposes.

The total angular momentum of the earth is precisely conserved, according to classical mechanics. The separate components of angular momentum do not have to be conserved. Some interactions between earth and sun can cause a transfer of angular momentum from one component to the other. For instance, the tides that the sun cause on the earth speed up the spinning of the earth and slow down the orbital motion of the earth. Therefore, there is a spin-orbit interaction acting on the earth due to tides.

Because the axis of the orbit and the axis of the spin don't line up, the tidal force has to cause the axis of the earth's spin to change. This is called precession. The rate at which the direction of the axis changes will be much greater than the rate at which the magnitude of the earth's spin angular momentum changes.

The magnitude of the spin angular momentum also changes due to tides, but much more slowly than the direction. Eventually, the spin-orbit interaction will cause the spin period of the earth to equal the orbital period of the earth around the sun. However, this would take a long time. The moon would change the spin of the earth long before the sun did. Therefore, the spin-orbit interaction between sun and earth due to tides is usually considered negligible. However, the spin orbit interaction is there.

I didn't use the earth-moon analogy because this asymptotic state has been reached. due to tidal interaction, the spin period of the earth precisely equals the spin period of the moon.

The extension of these concepts to quantum mechanics is slightly challenging. The electron in an atom also has a spin angular momentum, an orbital angular momentum, and a total angular momentum. Only the total angular is precisely preserved. However, magnetic forces cause interchange of angular momentum between spin and orbit. Because of some quantum mechanical "magic", the magnitude of the spin of an electron can't change ever. However, the direction of the spin of an electron can change.

The quantum picture of the electron in a atom maps onto this earth-sun analogy roughly like this. Be very cautious with this analogy, as classical mechanics is not precisely the same as classical mechanics. The earth's spin is analogous to the electron spin (be VERY cautious with spin). The earth's angular momentum is analogous with the electrons orbital angular momentum. The spin-orbit interactions of the electron are analogous to the interaction caused by the solar tide on the earth. The mixing of states is analogous to the precession of the earths axis.

4. Feb 21, 2013

### Johnleprekan

Thank you.

About the electron's angular momentum. The earth's axis is at a slight tilt when orbiting the sun. Is an electron tilted too when orbiting the nucleus?

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