Ket and Bra with the momentum j,m

[brombenzol123]

Hi there!

I try to understand and to use graphical method in the theory of angular momentum. It is not very difficult to master is fundamentals excepting a few things. The most fundamental is the following.

In literature there is used the fact that Kat and Bra vectors with momentum j,m can be mutually permutated using
<jm| = (-1)^j-m |j, -m>. It is clear for integer j. But how to understand this fact, for example, in the case j = 1/2 ?

I would be glad if somebody could give me advice.

I'm absolutely novice at forums where scientific questions are discussed. So, first of all, I would be glad to see a few words about where else I can discuss such questions. What cites / forums are the most popular for these purposes?With Best Regards,
Antony Gorkh.

 

[dextercioby]

j-m in the exponential is always an integer. If you take j=3/2, then m has 4 possible values: -3/2, -1/2, 1/2 and 3/2. Evaluate j-m for all 4 possible cases.

 

[brombenzol123]

j-m in the exponential is always an integer. If you take j=3/2, then m has 4 possible values: -3/2, -1/2, 1/2 and 3/2. Evaluate j-m for all 4 possible cases.

Yes, it's obvious true. But the deal is in the other thing.

In the case of integer j every state |jm> is represented by spherical function, and the formula above is just a property of spherical functions. But in the case of semi-integer j we have the other picture. For example in the case of 1/2 |jm> is a column of 2 elements in the basis (1,0)^T, (0,1)^T, and it's hard to understand how that formula can still hold. But it is so (in some magic way) and it is used widely in literature.

I used the book "Graphical method of spin algebras" by E. Elbaz and B. Castel. The formula (1.2.5) chapter 1, paragraph 2. I use Russian edition, so there may be some shift.

 

[dextercioby]

Thinking about it, the bra's and the kets don't live in the same vector space, hence they can't be equal up to a minus 1. There's something wrong with what you wrote.

 

[brombenzol123]

You're absolutely right and there are no doubts that Kets and Bras can't be equal up to minus 1. That formula was written originally not by me but by the authors of the book I mentioned above. Being not accurate with notations and terminilogy they confused me so that I had nothing to do but ask people on forums. I hope all people that ever read that book had mastered this problem.

The authors calls the quantity (-1)^j-m |j, -m> as covector in the respect to the vector |jm> in the sense that Σ_m(-1)^j-m |j, -m> |j,m> gives the total momentum with j=0, i.e. scalar (here (-1)^j-m plays the role of Clebsh-Gordan coefficient). It was the authors' mistake to denote it as <jm| and it was my mistake to call it Bra. It has nothing common with Bra.