# What is the total angular momentum quantum number?

##### 1 Answer

**total angular momentum**, which is just a value that collapses

*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

It's the ** simplest** example that isn't

**simple:**

*too*

**DETERMINING TOTAL SPIN ANGULAR MOMENTUM**

To determine

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

(Had there been two or more electrons,

**DETERMINING TOTAL ANGULAR MOMENTUM**

Finally,

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

Here, we have:

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

but

#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:

- The number of unpaired electrons is
#1# , because#2S + 1 = 2# , so#S = "1/2"# . - Because
#S = "1/2"# ,#J - S = L = "3/2" - "1/2"# , so#L = 1# , and we are looking at a#p# orbital. - 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# . - 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

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