Total spin angular momentum meaning

[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 s_z 1/2 hbar object?

 

[Simon Bridge]

The total orbital spin angular momentum is given as L2=l(l+1).

I don't think so - perhaps $L^2=l(l+1)\hbar^2$ ... the angular momentum.
Total angular momentum is usually $\vec{J}=\vec{L}+\vec{S}$.

See:
http://en.wikipedia.org/wiki/Angular_momentum_operator
(under quantization)

Mathematcally: the numbers come from the angular momentum operator applied to the electron wavefunctions.
See: http://quantummechanics.ucsd.edu/ph130a/130_notes/node216.html

 

[galvin452]

Two questions
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 s_z 1/2 hbar object?

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

 

[jtbell]

1) The total orbital spin angular momentum is given as L²=l(l+1).

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 m_s = -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.)

What is the source of or meaning of the (l+1).

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.

 

[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.