Tagged Questions
10
votes
2answers
132 views
When are there enough Casimirs?
I know that a Casimir for a Lie algebra $\mathfrak{g}$ is a central element of the universal enveloping algebra. For example in $\mathfrak{so}(3)$ the generators are the angular momentum operators ...
3
votes
1answer
92 views
Different representations of the Lorentz algebra
I've found many definitions of Lorentz generators that satisfy the Lorentz algebra: ...
2
votes
2answers
151 views
In quantum mechanics(QM), can we define a high-dimensional “spin” angular momentum other than the ordinary 3D one?
Inspired by my previous question Questions about angular momentum and 3-dimensional(3D) space? and another relevant question How to define angular momentum in other than three dimensions? , now I get ...
2
votes
2answers
124 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 ...
3
votes
1answer
90 views
Mathematically, how do we deduce that angular momentum is bounded?
So, how do we know $J_{+}|j,(m=j)\rangle =|0\rangle$?
I.e. that m is bounded by j.
We know that $J_{+}|j,(m=j)\rangle =C|j, j+1\rangle$, but how do I know that gives zero? Is it by looking at its ...
3
votes
3answers
203 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
1answer
194 views
Holstein-Primakoff and Dyson-Maleev representation
In Holstein-Primakoff and Dyson-Maleev representation, spin operators are represented by bosonic operators. Roughly speaking, a state with $S^z=S-m$ corresponds to a state containing $m$ bosons. In ...
1
vote
1answer
247 views
Wigner-Eckart projection theorem
I'm following the proof of Wigner-Eckart projection theorem which states that:
$$\langle \bf{A} \rangle ~=~ \frac{\langle \bf{A} \cdot \bf{J} \rangle}{\langle {\bf{J}}^2 \rangle} \langle \bf{J} ...
8
votes
2answers
364 views
Lie bracket for Lie algebra of $SO(n,m)$
How does one show that the bracket of elements in the Lie algebra of $SO(n,m)$ is given by
$$[J_{ab},J_{cd}]
~=~ i(\eta_{ad} J_{bc} + \eta_{bc} J_{ad} - \eta_{ac} J_{bd} - \eta_{bd}J_{ac}),$$
...
2
votes
1answer
479 views
General procedure for Clebsch-Gordan expansions
I'm wondering if the Clebsch-Gordan series generalize to any orthonormal set of basis functions? If so, how would one go about deriving an expression for an arbitrary set of basis functions (perhaps ...
3
votes
1answer
525 views
Angular momentum coupling-calculation of Clebsch–Gordan coefficients
I am facing problem in calculating the value of given Clebsch–Gordan coefficients representing the coupled angular momenta of two-particle system. For example
$$\begin{pmatrix}2 & 1 & 2 \\ 1 ...
3
votes
3answers
270 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 ...
3
votes
1answer
334 views
Why is the value of spin +/- 1/2?
I understand how spin is defined in analogy with orbital angular momentum. But why must electron spin have magnetic quantum numbers $m_s=\pm \frac{1}{2}$ ? Sure, it has to have two values in ...
5
votes
0answers
186 views
Coupling Coefficients in SO(4)
I have two equations (from two distinct authors) for the decomposition of a coupling coefficient of SO(4) (i.e. Wigner 3j-symbol for SO(4)). In the first:
...
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 ...
