Clebsch-gordan Definition and 26 Threads

Consider a system of two identical spin zero particles on a sphere. Let ##\vec{L} = \vec{L}_1 + \vec{L}_2## be the total orbital angular momentum of the two particles, and ##l_1, l_2## be the orbital angular momentum quantum numbers corresponding to particle 1 and particle 2. Consider the...

I've seen the hydrogen electron's wavefunction expressed in the basis ##\ket{n l s m_l m_s}## or ##\ket{n l s j m_j}##, but so far, never in ##\ket{n l s m_l m_j}##. My question is, are certain combinations of quantum numbers, eg, ##\ket{n l s m_l m_j}##, forbidden? If ##\ket{n l s m_l m_j}##...

I am looking at the point group C_3v described shown here. I am trying to understand the block diagonalization process. The note says that changing the basis in the following way will result in the block diagonal form. What is the rationale for choosing the new basis. Is it...

Hi, I would like to know why a particle with spin=0 can't posses a magnetic dipole moment? Using Wigner-Eckart theorem for ##\langle j,1,m,0|j,m \rangle## I get ##\langle j'|| \vec{J}|| j \rangle = \hbar \sqrt{j(j+1)} \delta_{jj'}## It seems like the right hand side is the magnetic dipole...

Summary: Different sign in the combination of two ##\textbf{1/2}## isospins with opposite third component Hello everybody! I was doing an exercise regarding isospin and I noticed something from the Clebsch-Gordan coefficients that made me think. For example, if I consider the combination...

Homework Statement [/B] I am trying to get the C-G Decomposition for 6 ⊗ 3. 2. Homework Equations Neglecting coefficients a tensor can be decomposed into a symmetric part and an antisymmetric part. For the 6 ⊗ 3 = (2,0) ⊗ (1,0) this is: Tij ⊗ Tk = Qijk = (Q{ij}k + Q{ji}k) + (Q[ij]k +...

Homework Statement I am trying to improve my understanding of the Clebsch-Gordan coefficients. I am looking at page 5 of the following document https://courses.physics.illinois.edu/phys570/fa2013/chapter3.pdf Homework Equations I have derived the result for the I = 3/2 quadruplet but am...

Homework Statement I was going through Sakurai's textbook and I tried to work out by myself the following expression for the Clebsch-Gordan coefficients (Addition of orbital angular momentum and spin 1/2) \begin{equation} \langle m-1/2,1/2|l+1/2,m \rangle = \sqrt{\frac{l+m+1/2}{2l+1}}...

Homework Statement The question asks me to calculate the non-zero Clebsch Gordan coefficients ##\langle j_1+j_2, j_s-2|j_1,m_1;j_2,m_2\rangle## Where ##j_s=j_1+j_2##. Homework EquationsThe Attempt at a Solution The ##j_s-2## part is the ##m## of a ##|j,m\rangle## and I know that m has to equal...

Homework Statement So i have to calculate the Clebsch-gordan coefficients for the state j1=3/2 and j2=1/2[/B]Homework Equations Recursiom formula, lowering and uppering operator[/B]The Attempt at a Solution I have tried to calculate the first set of the coefficients stating that: L- l2,2> =...

Why are ##<j_1j_2m_1m_2|jm>## and ##<j_2j_1m_2m_1|jm>## negative of each other when ##j_1+j_2-j## is odd as given below? I would expect ##<j_1j_2m_1m_2|jm>## and ##<j_2j_1m_2m_1|jm>## to always have the same sign since nature doesn't care which particle we label as particle 1 and which as...

On one hand, in reading Georgi's book in group theory, I comprehend the invariant tensor as a special "tensor", which is unchanged under the action of any generators. On the other hand, CG decomposition is to decompose the product of two irreps into different irreps. Now it is claimed that...

Homework Statement The state of an electron is, |Psi> =a|l =2, m=0> ⊗ |up> + Psi =a|l =2, m=1> ⊗ |down>, a and b are constants with |a|2 + |b|2 = 1 choose a and b such that |Psi> is an eigenstate of the following operators: L2, S2, J2 and Jz. The attempt at a solution I am really not sure...

I'm doing a problem where I need to know the coefficients to change from the \vec{J} = \vec{J}_1 + \vec{J}_2 + \vec{J}_3 to the {\vec{J}_1, \vec{J}_2, \vec{J}_3} for three spin-1 particles, but I'm having trouble finding a table or reference for this... surely every time someone needs to...

Hello everyone, I'm reading a bit about Clebsch-Gordan coefficients and I found two things in their general description I didn't quite understand. Can anyone help me with these questions? First, I read that the Clebsch-Gordan coefficients are zero unless the total angular momentum...

Hello, I was wondering, is there a way to derive the expression for these coefficients without the use of the general CGC formula? For example, the J=0 case (this is taken from wikipedia) The 1/sqrt term is clear but the (-1)^.. term not so much, is there a way to find this...

Hello, I'm not sure whether I should have posted this in main Quantum mechanics thread because it's not really regarding homework, but I posted it in here just to be safe. I was wondering, is there a way to derive the expression for these coefficients without the use of the general CGC...

Hi, for the deutron triplet state, I am trying to find the coefficient using the lowering operator but I kept getting 0 hope someone can help me. Using J-|j,m>=[(j+m)(j-m+1)]1/2|j,m-1> I can get from |++> to 1/\sqrt{2}(|-+> + |+->) but from here I am stuck since j+m=0 for |+->. What is...

Homework Statement I have two spin-1/2 particles and I need to calculate their Clebsch-Gordan coefficients.Homework Equations The Attempt at a Solution I followed the procedure of applying J_- to |{j,m}\rangle and J_{1-} and J_{2-} to |{m_1,m_2}\rangle and comparing them. I got correctly...

Hello all, I'm stuck on understanding part of a discussion of representations and Clebsch-Gordan series in the book 'Groups, representations and Physics' by H F Jones. I'd be grateful to anyone who can help me out. For starters, this discussion is in the SU(2) case. I don't know how to...

Homework Statement Find the Clebsch-Gordan coefficients associated with the addition of two angular momenta j_1 = 1 and j_2 = \frac{1}{2} Homework Equations The table of coefficients. The Attempt at a Solution I think I am misunderstanding something important here. I can't see...

Trying to self-teach myself more quantum mechanics. Reading Zare's "Angular Momentum- Understanding Spatial Aspects in Chemistry and Physics". I don't really understand how to calculate eigenvalues of J using CG coefficients (I mean, I know the eigenvalues of J, just trying to calculate them a...

Clebsch-Gordan Theorem?? symmetric spinor tensors are IRR of SU(2), i.e., T_{\undergroup{\alpha_1\cdots\alpha_r}} The Clebsch-Gordan theorem says, {\{j_1\}}\otimes{\{j_2\}}={\{j_1+j_2\}}\oplus{\{j_1+j_2-1\}}\oplus\cdots\oplus{\{|j_1-j_2|\}}. Can I prove this theorem by symmetrizing the...

Homework Statement My textbook takes a look at the \Delta(1232) particle It says that \left|\pi p;\frac{3}{2},\frac{3}{2}\right>=\left|\pi;1,1\right>|N;\frac{1}{2},\frac{1}{2}\right> where N stands for a nucleon and pi could be any of the three flavours of pion. They then go on by applying...

How do you read the table?

Hey there, I'm new to this forum, but I just wanted to let you know that I have put out a clebsch-gordan calculator that is a little bit better than the ones I've found out there so far, check it out. http://phys.csuchico.edu/C-G/ -Fish