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49 views

Using Lie theory to understand $U=e^{i\mathcal{H}t}$ [duplicate]

Can we use the exponential map (lie theory) to understand how the Hamiltonian $\mathcal{H}$ gives rise to the unitary, and therefore compliments an essential property of the unitary operator (ie to ...
0
votes
0answers
27 views

Calculating the Commutation Relation of the Generators of $SO(n)$ [duplicate]

I'm working through problems in the book Einstein Gravity in a Nutshell by Zee, and I'm stuck on one of the harder problems. The problem is Calculate $[J_{(mn)}, J_{(pq)}]$. We are given that $[J_{(...
1
vote
3answers
78 views

What does $X|n\rangle \propto |n+c\rangle$ mean?

$\renewcommand{\ket}[1]{\left \lvert #1 \right\rangle}$ I'm transcribing below (but see edit history for a scan) a calculation from pg 17 of this article on Lie groups and Lie algebras $$ [N,X]=cX....
7
votes
1answer
188 views

What Lie group structure is used for infinite-dimensional Unitary Groups in Quantum Mechanics?

Given an infinite-dimensional Hilbert space $H$, the set $U(H)$ of all unitary operators on $H$ forms a group, known as the unitary group. Now several subgroups of this group play an important role ...
0
votes
2answers
53 views

Lie algebra vs. position and momentum commutators

Most theoretical texts on high energy physics make statements like below: $$[A_i , A_j] = i C^k_{i,j} A_k $$ (I suppose $\hbar$ may or may not be needed) and of course they describe this as being ...
1
vote
1answer
56 views

How are Dunkl operators used in Hamiltonian mechanics?

I am currently doing a math research project on the representation theory of Cherednik (double affine Hecke) algebras, specifically the algebra $\mathcal{H}_{t,c}(\mathfrak{S}_n,\mathfrak{h})$, which ...
3
votes
2answers
88 views

Why are all transformations of quantum operators inner automorphisms?

Operators in quantum mechanics are basically related to each other through their Lie algebra i.e. the commutator $\times \frac{1}{i\hbar}$. This is then connected to the state space i.e. the Hilbert ...
0
votes
1answer
60 views

Construct components of tensor operator [closed]

I'm reading Georgi's textbook on Lie Algebras and have been struggling with this question for quite awhile. The entirety of Chapter 4 (Tensor Operators) has been much more difficult than anything I've ...
8
votes
1answer
383 views

Is there a generalised Wigner-Eckart theorem?

The Wigner-Eckart theorem gives you the matrix element of a tensor transforming according to a representation of $\mathfrak{su}(2)$, when sandwiched between vectors transforming according to another (...
1
vote
0answers
44 views

How to prove that given operators form an algebra? [closed]

I am new to Group theory and representations and I'm having trouble with this problem in an exercise: Given the two oscillator algebra $$[a, a^†] = 1$$ $$[b, b^†] = 1$$ $$[a, b] = 0$$ show that ...
2
votes
4answers
489 views

Why does the Lie algebra corresponding to the unitary group contain Hermitian operators?

I saw an awesome derivation of Schrodinger's equation on Wikipedia. Part of it relies on: We also know that when $t' = t$, we must have the unitary time evolution operator $U(t, t) = 1$. Therefore, ...
4
votes
1answer
147 views

Operators - how to motivate they must be linear ? Is this comment a hint? [duplicate]

Is there a way to motivate, retrospectively, that observables must be representable by linear operators on a Hilbert space? Specifically, there seems to be a hint to something in the accepted ...
1
vote
1answer
254 views

Symmetry of the Pauli group and its representations

The three traceless Pauli matrices $\sigma_x,\sigma_y,\sigma_z$ are arbitrary in the sense that any three operators with the appropriate commutation relations can be represented with those matrices. ...
5
votes
3answers
513 views

Source of Ladder Operators

This might be a silly question but I am curious as to where the ladder operators in quantum mechanics come from. For example, in introductory texts on quantum mechanics, they try to solve the ...
4
votes
1answer
335 views

Holstein-Primakoff Inconsistency?

I am trying to prove a simple relation involving the Holstein-Primakoff Transformation. Here is the transformation: $$S_+ = \sqrt{2S-b^{\dagger}b} b$$ $$S_- = b^{\dagger}\sqrt{2S-b^{\dagger}b}$$ $$S_z ...
2
votes
1answer
1k views

Cross Products and Commutators

For non-commuting vector operators we have in general $$\mathbf{A}\times \mathbf{B}= - \mathbf{B}\times \mathbf{A} +\sum_{ijk} \epsilon_{ijk} \big[ A_i,\,B_j\big]\hat{\mathbf{k}} $$ For the case $\...
0
votes
0answers
152 views

Norm of Classical (Poissonian) Hamiltonian Operator

In the Poissonian formulation of classical mechanics, one finds that the time evolution of the phase space vector $\eta = (q_1,q_2\cdots q_n ; p_1, p_2\cdots p_n)^T$ can be put in terms of the ...
1
vote
1answer
374 views

Does Operator Product Expansion form an algebra?

The operator product algebra in CFT is defined as $$\mathcal{O}_i(z,\bar{z})\mathcal{O}_j(\omega,\bar{\omega}) = \sum_{k} C^k_{ij}(z-\omega,\bar{z}-\bar{\omega})\mathcal{O}_k(\omega,\bar{\omega}).$$ ...
0
votes
2answers
521 views

Commutator relationships and the exponential

I am currently trying to prove that the two following commutator relationships are equivalent (for an operator $\hat{A}(s)$ that depends on a continuous parameter $s$), so if one holds the other one ...
4
votes
1answer
177 views

Nonabelian global symmetries, $SO(N)$ charges in terms of creation and annihilation operators

Consider an $SO(N)$ symmetric theory of $N$ real scalar fields,$$\mathcal{L} = {1\over2} \partial_\mu \Phi^a \partial^\mu \Phi^a - {1\over2} m^2 \Phi^a \Phi^a - {1\over4} \lambda (\Phi^a \Phi^a)^2.$$...
0
votes
1answer
147 views

Calculate mean number of particles of time evolution coherent state [closed]

I seem to be missing some identities. I know you need to calculate P_n = |<n|alpha_t>|^2 and mean number of particles is the infinite sum of nP_n. However I ...
1
vote
1answer
128 views

Relationship between those two “exponentials”

Let $G$ be a Lie group and $L(G)$ it's Lie algebra. We know that every left-invariant vector field $X$ in $G$ is complete, and so one can consider the integral curve defined for all $t\in \mathbb{R}$ ...
3
votes
4answers
101 views

Uniqueness of expression of a Lie group element

Just take the SU(2) group as an example. The three generators are $J_z$, $J_+$, and $J_-$. For an element $ g $, sometimes we want to express it as $$ g = e^{i a J_+} e^{i b J_z} e^{i c J_-} . $$ ...
10
votes
2answers
335 views

$su(1,1) \cong su(2)$?

The three generators of $su(2)$ satisfy the commutation relations $$ [J_0 , J_\pm] = J_\pm , \quad [J_+, J_- ] = +2J_0 .$$ The three generators of $su(1,1)$ satisfy the commutation relations $$ [...
2
votes
3answers
1k views

Deriving cross product from angular momentum algebra

Is it possible to derive: \begin{equation} \hat{L}=\hat{r}\times \hat{p} \end{equation} from the angular momentum algebra: \begin{equation} [\hat{L}_i,\hat{L}_j]=i\ \hbar\ \epsilon_{ijk}\hat{L}_k\ ? \...
1
vote
2answers
300 views

Why are the spin operators defined as they are?

$$\begin{align*}S_z &= \frac{\hbar}{2} \left(\left|+\right>\left<+\right| - \left|-\right>\left<-\right|\right)\\ S_y &= i\frac{\hbar}{2} \left(\left|-\right>\left<+\right| - ...
4
votes
1answer
460 views

Is there a connection between Lie Groups and observable quantities in physics?

Good evening everybody. I have some questions about the relation between Lie groups and observables in physics. Indeed, taking the example of spin formalism of Quantum mechanics I know that Pauli's ...
0
votes
1answer
444 views

Eigenvalues of Angular Momentum in Quantum Mechanics

The eigenvalue equation of the $L^2$ operator is given by $$L^2f_l^m = \hbar ^2l(l+1)f_l^m$$ Side: So a determinate state for some observable $Q$ is a state where every measurement of $Q$ returns ...
2
votes
1answer
324 views

Why does the raising and lowering operator not affect total angular momentum?

My notes define: $$ L_{\pm} = L_{x} \pm i L_{y} $$ and states: $$ [L_{z},L_{\pm}] = \pm \hbar L_{\pm} $$ I'm fine with this as it's easy to show the result with some ugly algebra. It then says: ...
0
votes
3answers
282 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?
5
votes
1answer
2k 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 ...
5
votes
1answer
705 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 ...