# Total spin angular momentum meaning

1. Aug 9, 2013

### galvin452

Two questions

1) The total orbital spin angular momentum is given as L2=l(l+1). What is the source of or meaning of the (l+1).

2) Similarly for the electron the total spin angular momentum is given as S2-1/2(1/2+1) hbar2. Is the total angular momentum precessing to give sz 1/2 hbar object?

2. Aug 9, 2013

### Simon Bridge

Last edited: Aug 9, 2013
3. Aug 10, 2013

### galvin452

Or is it that the electron has two intrinsic spin values?

4. Aug 10, 2013

### Staff: Mentor

There's no such thing as "total orbital spin angular momentum."

Orbital angular momentum has magnitude $L = \sqrt{l(l+1)} \hbar$, and its component along any direction (usually we use the z-direction) is $L_z = m_l \hbar$ where $m_l = -l \cdots +l$ in steps of 1.

Spin angular momentum has magnitude $S = \sqrt{s(s+1)} \hbar$, and its component along any direction (usually we use the z-direction) is $S_z = m_s \hbar$ where $m_s = -s \cdots +s$ in steps of 1. For e.g. an electron, s = 1/2, so ms = -1/2 or +1/2.

Total angular momentum has magnitude $J = \sqrt{j(j+1)} \hbar$, and its component along any direction (usually we use the z-direction) is $J_z = m_j \hbar$ where $m_j = -j \cdots +j$ in steps of 1.

(Some books use different notation.)

For orbital angular momentum, one way to get the $l(l+1)$ is to apply the (orbital) angular momentum operator to the solutions to the Schrödinger equation for e.g. the hydrogen atom. There's probably a "deeper" way to get it which applies to all three kinds of angular momentum, but someone else will have to provide it.

5. Aug 10, 2013

### Simon Bridge

No that's pretty much it - in QM classes (here anyway) we force students to do that calculation.
The quantum number basically comes from counting the states. There are three dimensions and the surd comes from the vector sum. You actually have to crunch through the equations to see it.

One may expect that the lth L state would have L=l\hbar ... but that neglects stuff like that there are three dimensions. The x(x+1) pattern is kind-of a symmetry in angular momentum.

Last edited: Aug 10, 2013