Tagged Questions
1
vote
0answers
97 views
How is multiplicity given by 2S+1?
Suppose there are two electrons in an atom with $s_1 = \frac{1}{2}, l_1 = 1$ and $s2 = \frac{1}{2}, l_2 = 1$. Hence the total $S$ (of the atom) may be +1 or 0. And total $L$ is either +2,+1 or 0.
Now ...
3
votes
3answers
200 views
Quantum mechanical angular momentum and spin formalism/notation
I am currently stuck on the following notation:
$\frac{1}{2}\otimes\frac{1}{2} = 0 \text{ (antisym) } \oplus 1 \text{ (sym) }$
No matter what I tried, I couldn't derive the identity. I am sure that ...
4
votes
2answers
239 views
Number of Components of a Spinor
I'm trying to develop my understanding of spinors. In quantum field theory I've learned that a spinor is a 4 component complex vector field on Minkowski space which transforms under the chiral ...
1
vote
2answers
277 views
Tensor product decomposition of SU(2)
I have a rather trivial question. I am looking for the decomposition of $1/2\otimes 1/2\otimes 1/2$. It should give, $0,1/2$ and $3/2$. I thought one must get as the overall dimension of this space 8, ...
10
votes
3answers
831 views
Adding 3 electron spins
I've learned how to add two 1/2-spins, which you can do with C-G-coefficients. There are 4 states (one singlet, three triplet states). States are symmetric or antisymmetric and the quantum numbers ...
2
votes
1answer
203 views
How could $\textbf{S}^2$ not be a multiple of the identity?
I'm self-studying quantum mechanics with Sakurai's book (Modern Quantum Mechanics, 2nd edition) and came across the following in reference to the operator $\textbf{S}^2$:
As will be shown in ...
1
vote
1answer
167 views
Irreducible tensor representations with “covariant” indices
As a follow-up of my question on the "most general" $\mathrm{SU}(2)$-symmetric interaction of two spin 1/2 particles, I ponder the following question:
Consider an operator acting just on one particle ...
2
votes
1answer
219 views
Constructing the “most general” two-particle spin interaction with $SU(2)$ symmetry
Suppose I want to write down an interaction term for an action for spin 1/2 fermions that is $SU(2)$-symmetric.
I start from the most naive general form of such an action:
$$S_{int} ~=~ \int_{4321} ...
3
votes
3answers
269 views
The Asymmetry between Real and Imaginary in the three Pauli Spin Matrices
The Pauli spin matrices
$$
\sigma_1 ~=~ (\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}),
\qquad\qquad
\sigma_2 ~=~ (\begin{smallmatrix} 0 & -i \\ i & 0 ...
7
votes
2answers
322 views
Groups acting on physics - a clarification on electrons and spin
My first question is fairly basic, but I would like to clarify my understanding. The second question is to turn this into something worth answering.
Consider a relativistic electron, described by a ...
2
votes
3answers
323 views
Spin decomposition in general
I can turn-the-crank and show that $\frac{1}{2}\otimes \frac{1}{2} = 1\oplus 0$ etc, but what would be a strategy to proving the general statement for spin representations that $j\otimes s ...
