What is an electron's orbital angular momentum?

[edguy99]

One of the best explanations of orbital angular momentum for the electron comes from Dirac himself. At around 39:30 of this youtube video (you will need headphones, but it is well worth it), Dirac talks about the non-commutation of operators, how quantum mechanics is more general then classical mechanics and how quantum mechanics can use any functions to give equations of motion for any hamiltonian variables.

He then talks in detail about electron spin and specifically how the 3 spin axis variables s1, s2, and s3 can vary in time and are non-commuting (ie. turn on one axis followed by a turn on a different axis depends on the order you do the turns). He points out that the 3 components of spin s1, s2, and s3 satisfy the same conditions as orbital angular momentum.

It is important to remember he is talking about an intrinsic property of the electron, not its relationship with the nucleus (ie. the electron is not orbiting anything). The only classical analogue I can think of to an electrons orbital angular momentum as a precession of the spin axis of a spinning particle. The spin of the particle (left or right) leads to the spin quantum number (spin up or down), the precession of the spin axis (around an imaginary real axis) provides the orbital angular momentum quantum number. Precession of the spin axis of a spinning particle not under the influence of an outside force (gravity, magnetic, or anything) would map out some pretty weird orbitals.

 

[Bay Chadwick]

Just a heads up, when you are talking about;

The spin of the particle (left or right)

in paragraph 3, I think you are actually referring to the chirality of a particle. Chirality is the reverse of a single spatial axis of a particle. For example, say a particle's axes are as follows: x, y, z
The opposite chirality to this would be: x, -y, z or -x, y, z or x, y, -z

Chirality is an intrinsic property of any object. This means it can't be changed by its environment. The only way to change its chirality is by changing the object itself. If that's too confusing, I'll give you an example. No matter how you twist, rotate or position your right hand, you can't get it to look exactly like your left hand. An object (or in this case a particle) can have left-handed or right-handed chirality.

Also, when you mentioned;

the spin quantum number (spin up or down)

also in paragraph 3, I think you are referring to helicity. Helicity is similar to chirality, but is an extrinsic property, meaning it can be changed by its environment.

Lastly, when you talk about an electron's

orbital angular momentum quantum number

also in paragraph 3, you must be talking about spin (sometimes called intrinsic spin or intrinsic angular momentum). This is because you mentioned it as being

an intrinsic property of the electron

at the beginning of paragraph 3, and orbital angular momentum is an extrinsic property.

So, assuming that your question is now "what is an electron's spin?", the answer is ½.
This also makes the electron have various other properties, such as obeying Fermi-Dirac statistics and being subject to the Pauli Exclusion Principle.

P.S. sorry for correcting you so much.

 

[edguy99]

I did not apply the terms Chirality or Helicity to the model, so no correction needed. Dirac is talking about the 3 components of the spin axis x, y, z satisfying the same mathematical conditions as orbital angular momentum (the main one being non-commutation).

But if you consider a spinor that looks like this, with a spin axis that is precessing (blue arrow: spin of particle, green arrow: precession of spin axis). Now we can talk about intrinsic spin of the particle (blue arrow), and intrinsic angular momentum of the particle (green arrow).

In paragraph 3, I am indeed talking about spin, but it is not sometimes intrinsic spin or intrinsic angular momentum. Intrinsic spin and intrinsic angular momentum are completely independent quantum ideas that describe the particle:

1/ Principal quantum number (n)
2/ Spin quantum number (s)
3/ Azimuthal quantum number (ℓ)
4/ Magnetic quantum number (m)

Where the Azimuthal quantum number, is studied by Dirac as orbital angular momentum.

P.S. I do like to think of the Chirality and Helicity of this model. It allows you to visualize a particle concept with a distinct difference in intrinsic right-handed particles (blue and green arrow spin matches) and intrinsic left-handed particles (blue and green arrows point in opposite directions).