Tagged Questions
0
votes
1answer
59 views
Angular Momentum Addition Theorem
If I have, for example a particle with $s = 3/2$ and $\ell = 2$, what are the allowed values of $j$?
I'm slightly confused because I know that $j = \ell + s$, so surely there is only one allowed ...
2
votes
2answers
121 views
Quantization of orbital angular momentum
Probably a very simple question, but I can't find the answer on the Internet.
I know nearly to nothing about quantum mechanics, but in statistical physics I'm confronted with the idea that the orbital ...
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
201 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 ...
6
votes
1answer
147 views
Equivalent Representations of Clifford Algebra
I'm reviewing David Tong's excellent QFT lecture notes here and am a little confused by something he writes on page 94.
We've considered the standard chiral representation of the Clifford Algebra, ...
7
votes
5answers
307 views
The role of representation theory in QM/QFT?
I need help understanding the role of representation theory in QM/QFT. My understanding of representation theory in this context is as follows: there are physical symmetries of the system we are ...
8
votes
1answer
332 views
Schwinger representation of operators for n-particle 2-mode symmetric states
A bosonic (i.e. permutation-symmetric) state of $n$ particles in $2$ modes can be written as a homogenous polynomial in the creation operators, that is
$$\left(c_0 \hat{a}^{\dagger n} + c_1 ...
3
votes
1answer
254 views
Does a spin-2 particle really return to its previous state after 180° rotation?
It is often claimed that spin-2 particles return to their previous state after $\pi$ rotation, just like spin-1/2 particles return after $4\pi$ rotation. But my calculation suggests otherwise.
Let z ...
10
votes
3answers
831 views
Adding 3 electron spins
I've learned how to add two 1/2-spins, which you can do with C-G-coefficients. There are 4 states (one singlet, three triplet states). States are symmetric or antisymmetric and the quantum numbers ...
3
votes
1answer
165 views
Why is there a phase factor when the two composite angular momentum is exchanged in Clebsch–Gordan coefficients
An identity exists for CG coefficients:
$$\langle j_1 m_1 j_2 m_2 |J M \rangle = (-1)^{j_1+j_2-J} \langle j_2 m_2 j_1 m_1|J M\rangle,$$
But why is there a phase factor $(-1)^{j_1+j_2-J}$?
It seems ...
2
votes
1answer
203 views
How could $\textbf{S}^2$ not be a multiple of the identity?
I'm self-studying quantum mechanics with Sakurai's book (Modern Quantum Mechanics, 2nd edition) and came across the following in reference to the operator $\textbf{S}^2$:
As will be shown in ...
3
votes
1answer
251 views
Angular Momentum Addition Theorem - Sanity Check
Looking back at my quantum mechanics notes, the angular momentum addition theorem is listed as:
$j=j_1+j_2,j_1+j_2-1, ..., |j_1-j_2| $ (Using conventional notation)
, but I'm a little unsure how to ...
3
votes
2answers
243 views
A question from Weinberg QFT text
In page 71 Weinberg's QFT,
$$A\Psi^{\theta }_{a,b}
~=~(a\cos{(\theta )}-b\sin{(\theta )})\Psi^{\theta }_{a,b}.$$
He says that massless particles represented by $\Psi ^{\theta }_{a,b}$ are not ...
2
votes
1answer
478 views
General procedure for Clebsch-Gordan expansions
I'm wondering if the Clebsch-Gordan series generalize to any orthonormal set of basis functions? If so, how would one go about deriving an expression for an arbitrary set of basis functions (perhaps ...
1
vote
0answers
62 views
What is an isoscalar factor?
I need to find a definition for "the isoscalar factors of 3j-symbols for the restriction $SO(n)\supset SO(n-1)$...denoted by brackets with a composite subscript $(n: n-1)$..." They are given as:
$$
...
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 ...
5
votes
2answers
197 views
Are there irreducible tensors of half integral degree in quantum mechanics?
According to Ballentine, an irreducible tensor of degree k can be defined as a set of $2k + 1$ operators $\{T_q^{\;\;(k)}:(-k \le q \le k)\}$ satisfying the following commutation relations:
$$
...
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 $j\otimes s ...
5
votes
1answer
343 views
Simultaneously commuting set
How does one determine the members of an simultaneously commuting set (of operators)? For example, I have read that for orbital angular momentum, the set is {$H,L^2,L_z$}. How does one know that these ...
5
votes
1answer
364 views
Could I see the quantum states as representations of the Galilei algebra (or Galilei group)?
In somes references of Relativistic Quantum Mechanics, the one-particle states are given by representation theory of Poincaré algebra.
Could I mimics this for the non-relativistic case? States in ...
