Wigner-Eckart theorem and selection rules

In summary, the Wigner-Eckart theorem is a mathematical theorem used in quantum mechanics to calculate matrix elements between states of different total angular momentum. Selection rules are a set of rules based on conservation laws that determine allowed transitions between quantum states. The Wigner-Eckart theorem and selection rules are closely related, with the former providing a way to calculate matrix elements used in determining allowed transitions. They are important in understanding atomic and molecular systems, particularly in fields such as spectroscopy and quantum chemistry. However, there are exceptions to the selection rules, which occur when there is a violation of a conservation law. In these cases, the selection rules may be relaxed.
  • #1
JohanL
158
0
If you have an operator which in spherical tensor language

[tex]
T^k_q
[/tex]

are

[tex]
V=T^2_2 + T^2_{-2} + T^2_0
[/tex]

you get a selection rule for j'

[tex]
abs(j-k)=< j' <= j+k
[/tex]

in my case i start with angular momentum j=1 and k=2 from above so
the possible new states are

[tex]
1=< j' <= 3
[/tex]

But the operator is even under parity and the angular momentum states have parity (-1)^j=-1 in my case.
What does this mean for the possible states j=1 can jump to?

from the selection rule above you get that

j=1 to j'=2
j=1 to j'=3

is possible but does the parity consideration remove any of these possibilities?
 
Last edited:
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  • #2


I would like to clarify some points and provide an explanation for the forum post.

Firstly, the operator mentioned in spherical tensor language, T^k_q, is a mathematical representation of an operator that acts on a quantum system. In this case, the operator is a combination of T^2_2, T^2_{-2}, and T^2_0. These are known as spherical tensor operators, which are used to describe the angular momentum of a system.

The selection rule mentioned in the post is a result of the properties of the spherical tensor operators. It states that the possible new states, represented by j', must satisfy the condition abs(j-k)=< j' <= j+k. This means that the new states can have angular momentum values ranging from j-k to j+k.

In this case, the starting angular momentum is j=1 and the operator has k=2. This means that the new states can have angular momentum values ranging from 1-2 to 1+2, which gives us the range of 1=< j' <= 3.

The parity consideration mentioned in the post is related to the concept of parity in quantum mechanics. Parity is a property that describes the symmetry of a system under spatial inversion. In this case, the angular momentum states have parity values of (-1)^j=-1. This means that the possible new states must also have parity values of -1.

Applying this condition to the selection rule, we can see that j=1 can only jump to states with j'=2 or j'=3, as these are the only values that satisfy both the selection rule and the parity consideration.

In summary, as a scientist, I would say that the parity consideration does not remove any of the possibilities mentioned in the post, but it narrows down the options to only j'=2 or j'=3 for the new states that j=1 can jump to. This is because of the properties of the spherical tensor operators and the concept of parity in quantum mechanics.
 

1. What is the Wigner-Eckart theorem?

The Wigner-Eckart theorem is a mathematical theorem that relates the matrix elements of a tensor operator in quantum mechanics. It allows for the calculation of matrix elements between states of different total angular momentum, making it a powerful tool in studying atomic and molecular systems.

2. What are selection rules?

Selection rules are a set of rules that determine which transitions between quantum states are allowed and which are forbidden. They are based on conservation laws, such as conservation of energy, momentum, and angular momentum.

3. How are selection rules related to the Wigner-Eckart theorem?

The Wigner-Eckart theorem provides a way to calculate the matrix elements of a tensor operator, which are used to determine the allowed transitions between quantum states. Therefore, the selection rules are closely related to the Wigner-Eckart theorem in determining which transitions are allowed.

4. What is the significance of the Wigner-Eckart theorem and selection rules?

The Wigner-Eckart theorem and selection rules are important in understanding the behavior of atomic and molecular systems. They allow us to predict which transitions will occur in a system and the relative strengths of those transitions. This information is crucial in fields such as spectroscopy and quantum chemistry.

5. Are there any exceptions to the selection rules?

Yes, there are some exceptions to the selection rules. These exceptions occur when there is a violation of one of the conservation laws, such as when an external electric or magnetic field is present. In these cases, the selection rules may be relaxed, allowing for transitions that would normally be forbidden.

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