2
votes
1answer
50 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: ...
1
vote
0answers
23 views

Asymptotics of the Wigner 6j Symbol

So, in doing some numerical computations in QFT, I've run into the following Wigner 6j-Symbol: $ \left\{ \begin{array}{ccc} x & J_1 & J_2 \\ \frac{N}{2} & \frac{N}{2} & \frac{N}{2} ...
1
vote
1answer
84 views

What is the physical importance of the commutation relations of angular momentum?

What is the physical meaning of these commutation relations: $$[L_{z},L_{\pm}]=\pm\hbar L_{\pm}\tag{1}$$ and $$[L_{+},L_{-}]=2\hbar L_{z} ~?\tag{2}$$
1
vote
1answer
83 views

The role of SO(3) and SU(2) in quantum mechanics [duplicate]

When studying the irreducible representations of SO(3) one usually looks at the irreps of the infinitesimal rotations instead, i.e. the ones of so(3), the Lie Algebra of SO(3). The Irreps of so(3) can ...
2
votes
3answers
136 views

Angular momentum eigenstates

My textbook says that if $L^2$ is the square of the angular momentum and if it's eigenstate is $|\alpha,\beta>$ then its eigenvalue is $\hbar^2\alpha$ i.e. ...
0
votes
1answer
59 views

Connection to spin 1/2 electron system?

In another Physics stack exchange thread here, Spin matrix for various spacetime fields I obtained the generator of rotations of the SO(2) rotation group for an infinitesimal rotation of 2D vectors. ...
1
vote
2answers
86 views

Given eigenvalues of $\vec l^2$ and $\vec s^2$, calculate the eigenvalue for $\vec j^2$

There was an exam question that read approximatly: Let $\vec j = \vec l + \vec s$. Given eigenvalues of $\vec l^2$ and $\vec s^2$, calculate the eigenvalue for $\vec j^2$. We came up with $$\vec ...
4
votes
1answer
120 views

From Symmetry Group to Physics Equations

To the extent that I know: There are symmetry groups like the rotation groups SO(3), the Groups of Poincare Transformations,... If the physics of a system has a symmetry group G, then it can be ...
2
votes
1answer
56 views

Formalism and representation in Quantum Mechanics

I am just curious about the formalism of basic Quantum Mechanics. Lets take for instance the system of a spin-$\frac{1}{2}$ particle. The state of the particle is described by a vector in an abstract ...
2
votes
1answer
104 views

What exactly is a coherent state and why is it interesting?

Please note that I do not have a background in physics, so if possible please refrain from a bunch of $ |x\rangle $ notations, unless clearly specifying what it symbolically means. So I have been ...
1
vote
2answers
171 views

Angular momentum - maximum and minimum values for $m_{\ell}$

I want to work out the maximum and minimum values for $m_{\ell}$. I know that $\lambda \geq m_{\ell}$, therefore $m_{\ell}$ is bounded. In the lectures notes there is the following assumption: $$ ...
1
vote
1answer
103 views

Do generalized Pauli Operators generate SU(n)?

A commonly used generalization of Pauli Operators is the "clock" and "shift" operators summarized here: http://en.wikipedia.org/wiki/Generalizations_of_Pauli_matrices Pauli Operators are generators ...
8
votes
3answers
386 views

Addition of spin angular momentum for massless particles

How do I add the spin angular momentum of massless particles, like photons, where only the transverse polarizations are allowed? If all three polarizations were allowed, this would be an easy ...
4
votes
2answers
216 views

Unitary representations of the diffeomorphism group in curved spacetime

In (special) relativistic quantum mechanics there is a standard argument that says that the (rigged) Hilbert space of states $H$ should be equipped with a projective unitary representation $U$ of the ...
1
vote
1answer
164 views

Matrix representation of a triplet state

The $SU(2)$ triplet state is typically given in the fundamental representation as a column vector, e.g. \begin{equation} \vec{\Delta} = \left( \begin{array}{c} \delta^{++} \\ \delta^+ \\ ...
7
votes
2answers
218 views

Why the lowest order of matrices in Dirac equation are 4x4 matrices?

Why the lowest order of matrices in Dirac equation (Relativistic Quantums) are 4x4 matrices (and can not be 2x2 matrices)? How to prove it?
1
vote
1answer
585 views

Understanding Triplet And Singlet States

We know, $2\otimes 2=3\oplus 1$. Thus we have a spin triplet of states and a spin singlet. Can we regard these states as the spin part of wavefunction for the excited states and the ground state of ...
2
votes
1answer
145 views

How does $SU(2)$ group enters quantum mechanics?

What is the reason that $SU(2)$ group enters quantum mechanics in the context of rotation but not $SO(3)$? What really rotates and which space it rotates? It cannot be the physical electron that ...
3
votes
2answers
157 views

Why is $U(\Lambda)^{-1} = U(\Lambda^{-1})$ for a unitary representation?

This is from the beginning of Srednicki's QFT textbook, where he writes (approximately): In QM we associate a unitary operator $U(\Lambda)$ to each proper orthochronous Lorentz transformation ...
2
votes
1answer
135 views

How to find a particular representation for the gamma matrices?

I asked this question as a subquestion in another thread, but got the answer below and thought it deserved a thread of its own. Two well-known representation of the gamma matrices are the Weyl and ...
2
votes
3answers
207 views

How do you find a particular representation for Grassmann numbers?

This question is more general in the sense that I want to know how one finds a particular (say matrix) representation for any object. For the case of Grassmann numbers we have from Wikipedia the ...
3
votes
1answer
129 views

What is the Weyl algebra of a confined bosonic particle?

The abstract Weyl Algebra $W_n$ is the *-algebra generated by a family of elements $U(u),V(v)$ with $u,v\in\mathbb{R}^n$ such that (Weyl relations) $$U(u)V(v)=V(v)U(u)e^{i u\cdot v}\ \ Commutation\ ...
0
votes
0answers
30 views

Transformation from waves to matter

Let $|\psi \rangle$ represents a wavefunction through and $\langle \psi |$ represent the dual vector. Now there are things such as matter waves, could $|\psi \rangle$ represents a matter wavefunction, ...
1
vote
0answers
82 views

Dirac representation between matter and anti-matter

If $|\psi \rangle$ represents a wavefunction through a column matrix and $\langle \psi |$ represents the dual vector in a row matrix and $|\psi \rangle \langle \psi | = \rho $ is the probability ...
0
votes
1answer
96 views

Triangle inequality Clebsch-Gordan coeffcients

The Clebsch-Gordan 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 ...
0
votes
2answers
127 views

Finding possible values of $L_x$ given $L^2$

Here's a homework problem I'm working on. I am not asking for the answer, but any guidance or comments on the approach are appreciated. Given that a measurement of $L^2$ for a free particle has ...
5
votes
2answers
894 views

Can one show that ${\gamma^5}^\dagger = \gamma^5$ directly from the anticommutation relations?

Is it possible to show that ${\gamma^5}^\dagger = \gamma^5$, where $$ \gamma^5 := i\gamma^0 \gamma^1 \gamma^2 \gamma^3,$$ using only the anticommutation relations between the $\gamma$ matrices, $$ ...
1
vote
1answer
160 views

Matrix operations on Quantum States in a composite quantum system

Intro (you may skip this if you're an expert, I'm including this for completeness): Say I have two bases for two systems, The first is a spin-1/2 system $|+\rangle = \left(\begin{array}{c} 1\\0 ...
0
votes
0answers
58 views

Rotation matrix for a coupled spin system

For an angular momentum basis with magnitude $F$ and magnetic numbers $m_F\in [-F,F]$, the unitary matrix that will perform the Euler rotations is the Wigner-D matrix of order $F$. I have applied the ...
1
vote
1answer
131 views

Shouldn't the addition of angular momentum be commutative?

I have angular momenta $S=\frac{1}{2}$ for spin, and $I=\frac{1}{2}$ for nuclear angular momentum, which I want to add using the Clebsch-Gordan basis, so the conversion looks like: $$ \begin{align} ...
1
vote
1answer
90 views

Commutation of abstract $O(3)$ generators and vectors

I've been given the following problem, and I'm quite lost with it. Let $L_1$, $L_2$, and $L_3$ denote the abstract o(3) algebras. You are given that $\vec{A} = (A_1, A_2, A_3)$ and $\vec{B} = (B_1, ...
1
vote
1answer
434 views

Total angular momentum - single electron

I have been dealing with total angular momentum of the single electron which is outside the closed shells in which sum of the angular momentums is zero. My book says that total atomic angular ...
5
votes
1answer
341 views

Intuitive understanding of the irreps like Wigner-D matrix?

Wikipedia defines Wigner D-matrix as an irreducible representation of groups SU(2) and SO(3). What is a good way to visualize this representation? Is there any physical system which can be kept in ...
0
votes
1answer
187 views

Quantization of Electron Spin

Why is electron spin quantized? I've seen the derivation for the Hydrogen atom's energy levels, but my professor jumped to electrons having spin 1/2 or -1/2 as experimental. Why do electrons obey the ...
5
votes
2answers
193 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 ...
13
votes
1answer
520 views

What really are superselection sectors and what are they used for?

When reading the term superselection sector, I always wrongly thought this must have something to do with supersymmetry ... DON'T laugh at me ... ;-) But now I have read in this answer, that for ...
3
votes
1answer
153 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 ...
10
votes
2answers
705 views

Dimension of Dirac $\gamma$ matrices

While studying the Dirac equation, I came across this enigmatic passage on p. 551 in From Classical to Quantum Mechanics by G. Esposito, G. Marmo, G. Sudarshan regarding the $\gamma$ matrices: ...
3
votes
1answer
432 views

Matrix order in Dirac equations

The trace of matrix is always sum of its eigen values , which can be seen if $\hat{U}$ transforms the matrix $\alpha_i$ into it's diagonal form . $$ \begin{pmatrix} A_1 & 0 & \cdots & 0 ...
3
votes
1answer
236 views

Question on Sakurai's treatment of the Harmonic Oscillator:

In Section 2.3 of the second edition of Modern Quantum Mechanics (which discusses the harmonic oscillator), Sakurai derives the relation $$Na\left|n\right> = (n-1)a\left|n\right>,$$ and states ...
2
votes
2answers
191 views

Multiplicity of eigenvalues of angular momentum

Reading Dirac's Principles of Quantum Mechanics, I encounter in ยง 36 (Properties of angular momentum) this fragment: This is for a dynamical system with two angular momenta $\mathbf{m}_1$ and ...
8
votes
3answers
573 views

Why does spin have a discrete spectrum?

Why is it that unlike other quantum properties such as momentum and velocity, which usually are given through (probabilistic) continuous values, spin has a (probabilistic) discrete spectrum?
4
votes
1answer
183 views

Eigenvalue of $L_z$

In section 4.3 of Griffths' "Introduction to Quantum Mechanics", just below Figure 4.6, the sentence begins Let $\hbar \ell$ be the eigenvalue of $L_z$ at this top rung... Why is this valid? ...
11
votes
2answers
2k views

Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate?

The standard treatment of the one-dimensional quantum simple harmonic oscillator (SHO) using the raising and lowering operators arrives at the countable basis of eigenstates $\{\vert n \rangle\}_{n = ...
11
votes
1answer
189 views

Majorana-like representation for mixed symmetric states?

Is there a generalization of the Majorana representation of pure symmetric $n$-qubit states to mixed states (made of pure symmetric $n$-qubit)? By Majorana representation I mean the decomposition of ...
21
votes
5answers
175 views

Which symmetric pure qudit states can be reached within local operations?

There are two pure symmetric states $|\psi\rangle$ and $|\phi\rangle$ of $n$ qudits. Is there any known set of invariants $\{I_i:i\in\{1,\ldots,k\}\}$ which is equal for both states iff ...