7
votes
2answers
178 views

Lie algebra and Lie group about quantum harmonic oscillator

We know that in the quantum harmonic oscillator $H=a^\dagger a$, $a^\dagger$, $a$, $1$ will span a Lie algebra, where $a, a^\dagger$ are the annihilation and creation operators, and $H$ is the ...
0
votes
3answers
60 views

Commutator summation notation

I have the relation $ e^L M e^{-L}=\sum_{n=0}^\infty \frac 1{n!} [L,M]_{(n)}$ where $L$ and $M$ are operators. What does the subscript $n$ after the commutator bracket denote?
6
votes
2answers
124 views

Galilean, SE(3), Poincare groups - Central Extension

After having learnt that the Galilean (with its central extension) with an unitary operator $$ U = \sum_{i=1}^3\Big(\delta\theta_iL_i + \delta x_iP_i + \delta\lambda_iG_i +dtH\Big) + ...
11
votes
1answer
141 views

Lie group of Schrodinger Wave equation

In Ballentine's book on quantum mechanics (in 3rd chapter), he introduces the symmetry transformation of Galilean group associated with Schrodinger equation. Now the Galilean group as such has 10 ...
2
votes
0answers
63 views

Is the algebra of a differential equation invariant under transformation?

I've found that the algebra of this differential equation $$\frac{d^2y}{dz^2}-(3z^2+\gamma)\frac{dy}{dz}+(cz+\alpha)y=0$$ is in $sl(2)$ because it is possible to use the generators of the $sl(2)$ ...
0
votes
1answer
52 views

Triangle inequality Clebsch-Gordon coeffcients

The Clebsch-Gordon coefficients can only be non-zero if the triangle inequality holds: $$\vert j_1-j_2 \vert \le j \le j_1+j_2$$ In my syllabus they give the following proof: $$-j \le m \le j$$ $$-j_1 ...
4
votes
1answer
158 views

Question on conserved quantities and Noether's theorem

I have a question about Noether's theorem in the context of QM, which I'll state in the context of the weak interaction but the basic point could be generalized. According to Noether's theorem, given ...
11
votes
0answers
269 views

How to evaluate this sum of coupling coefficients?

I would like to evaluate the following summation of Clebsch-Gordan and Wigner 6-j symbols in closed form: $$\sum_{l,m} C_{l_2,m_2,l_1,m_1}^{l,m} C_{\lambda_2,\mu_2,\lambda_1,\mu_1}^{l,m} \left\{ ...
3
votes
1answer
401 views

What physical significance has the Heisenberg Group?

I read that the canonical commutation relation between momentum and position can be seen as the Lie Algebra of the Heisenberg group. While I get why the commutation relations of momentum and momentum, ...
5
votes
2answers
156 views

Problem counting spin states

I can't figure out how many different spin states I can create with a four-electron system. I think I can create a spin-zero state, three spin-one states, and five spin-two states. That gives me nine ...
4
votes
2answers
384 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 ...
3
votes
2answers
473 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
142 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
352 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
213 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, ...
1
vote
1answer
263 views

Commutators with a density matrix

The equation describing the evolution of our system is as follows: $ \dot{\rho} = u_1(t)(a^\dagger a \rho - 2a\rho a^\dagger +\rho a^\dagger a) + u_2(t)(a a^\dagger \rho - 2a^\dagger\rho a +\rho a ...
3
votes
1answer
887 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 ...
2
votes
1answer
554 views

Weinberg's way of deriving Lie algebra related to a Lie group

I was reading the second chapter of the first volume of Weinberg's books on QFT. I am quite confused by the way he derives the Lie algebra of a connected Lie group. He starts with a connected Lie ...
3
votes
1answer
772 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
1answer
446 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 ...
3
votes
0answers
174 views

Symmetries of separable potential

For separable potential, say $x^4+y^4$, its symmetry are degenerate. Is that a generic case to every separable potential? I will explain my question: The potential $x^4+y^4$ has $A_1, B_1, A_2, B_2, ...
2
votes
3answers
523 views

Hilbert space and Lie algebra in quantum mechanics

We are looking for a publication or website that explains the Standard Model in terms of Hilbert space and Lie algebra. We are reading Debnath's Introduction to Hilbert Spaces and Applications and ...
5
votes
1answer
444 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
386 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 ...
5
votes
1answer
272 views

Why the Hamiltonian near degeneracies should be proportional to Pauli matrices?

When a quantum system have a double degenerescence at one point, the Hamiltonian should be proportional to Pauli matrices near this point (also known as diabolic point) [Ref.]. But, why the ...