Question

What is the total angular momentum quantum number?

Answer 1

`J` is the total angular momentum, which is just a value that collapses `S` and `L` into another variable.

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DISCLAIMER: This can be a tough topic, so ask questions if you need to.

ATOMIC TERM SYMBOLS

We should see this in the context of atomic term symbols, which describe:

• The type of orbital (`s`, `p`, etc)
• The number of unpaired electrons
• The possibility for spin-orbit coupling

An atomic term symbol looks like this:

`\mathbf(""^(2S + 1)L_J)`

where:

• `S` is the total spin angular momentum of all `m_s` for each individual electron in the set of orbitals; it's a fast way of telling you how many unpaired electrons there are.
• `2S + 1` is called the spin multiplicity, which basically is a more concise way of telling you what `S` tells you, and gives rise to the terminology "singlet state", "doublet state", etc. It's a formal thing.
• `L` is similar to `l`, which is the orbital angular momentum, i.e. the shape of the orbital.
• `J` is the total angular momentum, which is just a value that collapses `S` and `L` into another variable.

P1 CONFIGURATION

So, let's take an example. Let's say we had a `2p` orbital with one electron in it. That's known as a `p^1` configuration.

It's the simplest example that isn't too simple:

DETERMINING TOTAL SPIN ANGULAR MOMENTUM

To determine `S`, simply add the `m_s` that you see for each electron. The paired electrons are always going to cancel out.

You should get:

`color(green)(S) = +["1/2"] = color(green)(+"1/2")`

Determine the spin multiplicity, and you should get:

`color(green)(2S + 1) = color(green)(2)`

DETERMINING ORBITAL ANGULAR MOMENTUM

Now, since it's a `p` orbital, `l = 1`, and for the term symbol itself, just as `l` corresponds to the `p` orbital, we write out `L` as `P`.

(Had there been two or more electrons, `L` would not just be as simple as just including `P`. I chose `p^1` for a reason.)

DETERMINING TOTAL ANGULAR MOMENTUM

Finally, `J` is where things get tricky, because there can be multiple `J` values, which ranges from `|L - S|` to `|L + S|`. So, you can have:

`\mathbf(J = L + S, L + S - 1, . . . , |L - S|)`

Here, we have:

`J = 1 + "1/2", 1 + "1/2" - 1, . . . , |1 - "1/2"|`

but `1 + "1/2" - 1 = |1 - "1/2"|`, so we just have two values for `J`.

`color(green)(J = "1/2", "3/2")`

OVERALL ATOMIC TERM SYMBOLS

So, we can write out the term symbols as:

`""^(2S + 1)L_J`

`""^(2S + 1)L_(L pm S)`

`-> color(blue)(""^2 P_"1/2", ""^2 P_"3/2")`

WHAT DOES IT MEAN?

From this, we can work backwards and make the following interpretations:

1. The number of unpaired electrons is `1`, because `2S + 1 = 2`, so `S = "1/2"`.
2. Because `S = "1/2"`, `J - S = L = "3/2" - "1/2"`, so `L = 1`, and we are looking at a `p` orbital.
3. We do NOT know whether there are `1` or `5` electrons total in the three `2p` orbitals because either configuration gives one unpaired electron. But we do know that there are either `1` or `5`, so the possible electron configurations are `p^1` and `p^5`.
4. We know that in an energy level diagram, we should see two energy states: `""^2 P_"1/2"` and `""^2 P_"3/2"`, which are very close together. Because of an effect called spin-orbit coupling, the two energy levels, which would otherwise be the same, split slightly in a magnetic field (sometimes giving differences of less than `"1 nm"` in the wavelength).

As an example of why this can be important, it tells you that there are two different `"589 nm"` electronic excitations for sodium's singular `3s` electron to an empty `3p` orbital.

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Both give a yellow emission line upon relaxation, but there are two transitions, not one.