Commutate relation of lowering operator and sperical tensor operators

In summary, there is a commutation relation between the lowering operator (J-) and the spherical operator (T_k^q) in Shankar's QM (2ed, page 418, Eq 15.3.11). The right hand side of the equation has a minus sign, which may or may not appear depending on the convention adopted for the T operators. Both conventions are correct and can be found in different sources.
  • #1
Einsling
2
0
Hi all,

I found a commutation relation of lowering operator(J-) and spherical operator in Shankar's QM (2ed, page 418, Eq 15.3.11):
[tex][J_-,T_k^q] = - \hbar \sqrt{(k+q)(k-q+1)} T_k^{q-1}[/tex]

I wonder how the minus sign in the beginning of the right hand side come out?

I have googled some pages, some of them have that "-", e.g. :
http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/TensorOperators.htm
the formula appears at the end of this page.

and some has no "-", e.g.:
http://atoms.vuse.vanderbilt.edu/Elements/CompMeth/HF/node30.html [Broken]
Eq(116) at the beginning .

I also found there's no minus in Messiah's QM (Vol II, page 572, Eq XIII.123a)
[tex][J_-,T_q^{(k)}] = \sqrt{k(k+1)-q(q-1)} T_{q-1}^{(k)}[/tex]


So, the question is which one is correct?

Thanks :)
 
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  • #2
Both are correct. It depends on the convention adopted for the T operators. In particular, for a vector operator, it depends on whether [tex]T^{\pm 1}_1 = x\pm iy[/tex] or [tex]T^{\pm 1}_1 = \pm(x\pm iy)[/tex].
 
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  • #3


Hello,

Thank you for bringing up this interesting question about the commutation relation between the lowering operator and spherical tensor operators. The commutation relation you have found in Shankar's QM textbook is indeed correct and can be derived using standard techniques in quantum mechanics. The minus sign on the right-hand side of the equation arises due to the specific definition of the lowering operator in terms of the angular momentum operators. This definition is chosen to ensure that the lowering operator decreases the eigenvalue of the angular momentum operator by one unit, hence the minus sign appears in the commutation relation.

Regarding the difference in notation and minus signs in different sources, it is important to note that there can be variations in notation and conventions in different textbooks and resources. However, the underlying physical principles and mathematics remain the same. Therefore, it is important to understand the concepts and principles behind the equations rather than focusing on differences in notation.

I hope this helps clarify your doubts. Keep exploring and learning about quantum mechanics!
 

1. What is the commutator relation between lowering operators and spherical tensor operators?

The commutator relation between lowering operators and spherical tensor operators is given by [L^- , T^kq] = -kT^k-1q, where L^- is the lowering operator, T^kq is the spherical tensor operator, k is the rank of the tensor, and q is the projection quantum number.

2. How is the lowering operator defined in terms of angular momentum operators?

The lowering operator, L^-, is defined as L^- = Lx - iLy, where Lx and Ly are the x and y components of the angular momentum operator, respectively.

3. What is the significance of the commutator relation between lowering operators and spherical tensor operators?

The commutator relation between lowering operators and spherical tensor operators is significant because it allows us to determine the action of the lowering operator on a spherical tensor, which is useful in solving quantum mechanical problems involving angular momentum.

4. How do the quantum numbers k and q relate to the rank and projection of a spherical tensor, respectively?

The quantum number k corresponds to the rank of the spherical tensor, which determines its overall shape and symmetry. The quantum number q corresponds to the projection of the tensor along a given axis, which determines its orientation in space.

5. Can the commutator relation between lowering operators and spherical tensor operators be extended to other operators?

Yes, the commutator relation between lowering operators and spherical tensor operators can be extended to other operators that have a similar structure and obey the same commutation relations, such as raising operators and ladder operators.

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