Wigner-Eckart Thm: Calc Double Bar for Electric Dipole Op

  • Thread starter jdstokes
  • Start date
  • Tags
    Theorem
In summary, vanesch says that the position operator (proportional to the position operator) is an irreducible spherical tensor operator, and that you can use the WE Theorem to find the reduced matrix element.
  • #1
jdstokes
523
1
Given a rank-k spherically symmetric tensor operator [itex]\hat{T}^{(k)}_q[/itex] (in other words a family of 2k + 1 operators satisfying [itex][J_z,T_q^{(k)}] = q T_{q}^{(k)}[/itex] and [itex]J_{\pm},T_q^{(k)}] = \sqrt{(k\pm q + 11)(k \mp q)}T_{q\pm 1}^{(k)}[/itex] for all k.

We have the Wigner-Eckart thorem

[itex]\langle j',m' |T^{(k)}_q|j,m \rangle = \frac{1}{\sqrt{2j+1}}\langle jk; mq | jk; j'm' \rangle\langle j' || T^{(k)} || j \rangle[/itex]

where the ``double bar'' is independent of m, m' and q.

I want to calculate the double bar for the electric dipole operator (proportional to the position operator). I'm expecting the answer to be proportional to [itex]\sqrt{2j+1}[/itex].

The first thing to answer is whether the theorem applies, ie is the position operator an irreducible spherical tensor operator. Secondly, how would I go about computing the double bar in this case?
 
Physics news on Phys.org
  • #2
I'm a bit rusty on this, so if I tell nonsense, I hope to be corrected.
But I would say, yes, the position operator is an irreducible spherical tensor operator of spin 1. Only, the 3 components, x, y and z, are not the "m" components. I guess you have to use something like x + iy, x - iy and z.
 
  • #3
jdstokes said:
The first thing to answer is whether the theorem applies, ie is the position operator an irreducible spherical tensor operator. Secondly, how would I go about computing the double bar in this case?

As vanesch says: you can use W-E Theorem here as long as your careful about what m value to use ([itex]r^{\pm 1}\propto x\pm iy[/itex], [itex]r^0\propto z[/itex]).

The WE Theorem does not tell you how to compute the reduced matrix element ("double-bar") - to compute that, you must go ahead and actually do the integral explicitly. The power of WE is that it allows you to RELATE several matrix elements to only one or two reduced matrix elements. So all you have to do is find the Clebch-Gordan coefficients (table) and compute one or two integrals (as opposed to tens of integrals!). In fact, sometimes you don't even have to compute them: if, for example, you are taking the ratio of matrix elements, sometimes the reduced matrix element cancels and you don't have to do a single integral (yay!).
 
  • #4
Thanks for responding guys.

Can you tell me how you knew that r was an irred spherical tensor and moreover how did you deduce that the components were [itex]x\pm iy,z[/itex].
 
  • #5
How does one evaluate [itex]\langle j'm' | z | jm \rangle [/itex]. This seems a little bit strange because z is a variable which extends to plus or minus infinity whereas the spherical harmonics only have [itex]\theta,\phi[/itex] dependence.
 
  • #6
jdstokes said:
How does one evaluate [itex]\langle j'm' | z | jm \rangle [/itex]. This seems a little bit strange because z is a variable which extends to plus or minus infinity whereas the spherical harmonics only have [itex]\theta,\phi[/itex] dependence.

That's because you forgot the "non-angular" index. Look at http://en.wikipedia.org/wiki/Wigner-Eckart_theorem
for instance. The idea is that you have a complete basis, which as a "non-angular" index n (which can consist of several indices if you want), but which are eigenfunctions of L^2 and Lz (the j and m rotation group indices).

A rotation applied to the state |n,j,m> will then only mix the m-values.

What is a spherical tensor operator ? (or better, a set of spherically symmetric operators) It is a set of operators T_k that, under rotation, transform within this set, in a linear combinations of themselves, just like a set |j,m> does.
 

1. What is the Wigner-Eckart theorem?

The Wigner-Eckart theorem is a mathematical tool used to simplify the calculation of matrix elements for operators in quantum mechanics. It relates the matrix elements of a tensor operator in one basis to those in another basis.

2. What does the Wigner-Eckart theorem tell us about electric dipole operators?

The Wigner-Eckart theorem can be used to calculate the matrix elements of electric dipole operators, which describe the interaction between an electric field and a charged particle. It allows us to determine the selection rules for transitions between different energy levels in an atom or molecule.

3. How is the Wigner-Eckart theorem used in quantum mechanics?

The Wigner-Eckart theorem is used in the calculation of matrix elements for operators, which are important in quantum mechanics for describing the behavior of physical systems. It simplifies the calculation of these matrix elements and helps to determine the selection rules for transitions between different states of a system.

4. Can the Wigner-Eckart theorem be applied to other types of operators?

Yes, the Wigner-Eckart theorem can be applied to other types of operators, such as spin operators and angular momentum operators. It is a general theorem that can be used to calculate matrix elements for any tensor operator.

5. Is the Wigner-Eckart theorem a fundamental principle in quantum mechanics?

No, the Wigner-Eckart theorem is a mathematical tool that is derived from other fundamental principles in quantum mechanics, such as the uncertainty principle and the principles of symmetry. However, it is a very useful tool for simplifying calculations and understanding the behavior of physical systems at the quantum level.

Similar threads

Replies
2
Views
956
Replies
3
Views
805
  • Advanced Physics Homework Help
Replies
17
Views
1K
  • Quantum Physics
Replies
2
Views
1K
  • Introductory Physics Homework Help
Replies
1
Views
808
  • Quantum Physics
Replies
4
Views
2K
Replies
1
Views
1K
Replies
6
Views
1K
Replies
5
Views
2K
  • Quantum Physics
Replies
1
Views
1K
Back
Top