Eigenvalues of total angular momentum

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Discussion Overview

The discussion revolves around the calculation of eigenvalues of total angular momentum and the associated matrix elements between different quantum states. Participants explore the notation and methods for expressing these states, particularly in the context of three-particle systems.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant asks for the formula to calculate the matrix element of total angular momentum between two states, specifically .
  • Another participant questions the clarity of the notation used, suggesting that the problem may be trivial if the quantum numbers are correctly defined.
  • A third participant provides a response indicating that the matrix element can be expressed with delta functions, implying a straightforward solution.
  • One participant expresses confusion about the notation and suggests that the eigenkets should be represented in a different form, noting that the eigenvalue of J² is given by \hbar² j(j+1) rather than j².
  • A later reply discusses the complexity of calculating the matrix element if using a specific notation for three particles, mentioning the need to fix the projection and the combinations of single particle states.
  • The same participant proposes a formula for calculating the matrix element and describes the process of diagonalizing the resulting matrix to find eigenstates of total angular momentum.
  • There is an acknowledgment of a poorly formed initial question, indicating a shift in focus to a more complex scenario involving three particles.

Areas of Agreement / Disagreement

Participants express differing views on the clarity and complexity of the problem, with some suggesting it is trivial under certain conditions while others highlight the complications that arise in multi-particle systems. No consensus is reached on a definitive approach or solution.

Contextual Notes

Participants reference specific quantum states and their representations, indicating that assumptions about notation and definitions may impact the clarity of the discussion. The complexity of the calculations for three particles remains unresolved.

stefano
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Who knows the formula to calculate the eigenvalues of total angular momentum between two different states? In particular, what is the matrix element of

<S, L, J, M_J | J^2 | S', L', J', M'_J> ?

Thank's...
 
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Eh? You've given the states here with j being one of the quantum numbers, which makes the problem trivial. Can you clarify?
 
And so? What is the trivial answer?
 
stefano said:
And so? What is the trivial answer?

J^2 delta_(J,J') delta(s,s') delta(l,l') delta(m,m')

cheers,
Patrick
 
stefano said:
Who knows the formula to calculate the eigenvalues of total angular momentum between two different states? In particular, what is the matrix element of

<S, L, J, M_J | J^2 | S', L', J', M'_J> ?

Thank's...
I don't understand your notation. Your eigenkets should be either

[tex]|j_1 j_2; m_1 m_2\rangle[/tex]

or

[tex]|j_1 j_2;j m\rangle[/tex]

If you're using the second option, the problem is trivial, as Vanesch said, but the eigenvalue of J² is

[tex]\hbar^2 j(j+1)[/tex]

not j².

If you're using the first option, it gets much more complicated. In principle, you can calculate your matrix element if you first expand one of the kets in eigenkets of the second kind. The coefficients of the expansion are called Clebsch-Gordan coefficents.
 
I am agree with you. In fact I made bad the question, because my original problem is that I need to write eigenstate of J for three particles. Then J=j_1+j_2+j_3; to do this I have to fix the projection (for example M=1/2) and the three sets of single particles for which M=1/2 are (5/2,1/2, -5/2), (3/2,1/2, -3/2) and (5/2, -1/2, -3/2) named A, B, C. Now I have to build the matrix U of elements of J, that I arrive at the original question: for example U_12 is <A|J^2|B> where |A> and |B> are
the states written before, but they aren't eigenstates of J^2.
I found a formula that may be mine answer:

<A| J |B> = delta(j_a, j_b)*delta(m_a+-1,m_b)*sqrt ((j_b+-m_b)*(j_b-+j_b+1))

with |a> = !j_a, m_a>
I need to perform all this matrix elements and then I have to diagonalize this matrix to have the three coefficients to combine |A>, |B> and |C>, in order to have eigenstates of J.

Sorry for my first question that was badly formed!
 

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