Irreducible tensor (second or higher rank)

These techniques involve considering the irreducible components of Qij and using the Wigner-Eckart theorem to determine the selection rules for different states belonging to different irreducible representations.
  • #1
ala
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When one want to find selection rules for matrix element of (for example) electric quadrupole moment tensor Qij, irreducible components of Qij are needed to apply Wigner-Eckart theorem. When symmetry group is SO(3) irreducible component can be found using what we know from addition of angular momentum. But how to do same problem if symmetry group is O (the octahedron group) or some other point group?

For example in one book I found (without explanation) following results:
1. for O group: Qxy, Qzz, Qyz are transformed by the representation F2; Qxx+kQyy+k^2Qzz, Qxx+k^2Qyy+kQzz are transformed by the representation E, where k=exp(i2Pi/3)
2. for D3d group: Qzz by A1q; Qxx-Qyy,Qxy by Eg; Qxz,Qyz by Eg.
Standard notation for irreducible representations for group O and D3d are being used.

Thanks in advance!
 
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  • #2
To determine selection rules for matrix elements of electric quadrupole moment tensor, Qij, for a particular point group, such as O or D3d, one needs to know the irreducible components of Qij. The irreducible components are found by using the group theory techniques associated with the particular point group. For example, if the point group is O, then the irreducible components of Qij can be determined by using the addition of angular momentum principle, which states that any irreducible representation of a given point group is the direct sum of all possible combinations of the irreducible representations of the subgroups that make up the point group. For O, this requires considering the subgroups C2v and C3v.In the case of D3d, the irreducible components can be determined by using the character table of the group and the Wigner-Eckart theorem. The Wigner-Eckart theorem states that the matrix element of an operator between two states belonging to different irreducible representations of a symmetry group is proportional to the product of the Clebsch-Gordan coefficient and the reduced matrix element of the operator in the same irreducible representation. This can be used to find the selection rules for the matrix elements of Qij for the D3d point group.In summary, the selection rules for matrix elements of electric quadrupole moment tensor, Qij, for a particular point group, such as O or D3d, can be found using the techniques of group theory specific to that point group.
 
  • #3


I would first start by acknowledging the complexity and importance of the topic at hand - irreducible tensors and their role in determining selection rules for matrix elements. The use of irreducible components is a powerful tool in the application of the Wigner-Eckart theorem, and it is essential to have a clear understanding of their properties and transformations for different symmetry groups.

To address the specific question raised about finding irreducible components for the electric quadrupole moment tensor Qij in different symmetry groups, I would suggest consulting established literature and resources on group theory and symmetry. This topic is extensively covered in textbooks and research articles, and a thorough understanding of the mathematical principles and techniques involved is necessary for accurate and meaningful analysis.

In general, for a given symmetry group, the irreducible components of a tensor can be determined by considering the symmetry operations of the group and their corresponding transformation matrices. For example, for the octahedral group O, the irreducible components of Qij can be found by considering the symmetry operations of the octahedron and their corresponding transformation matrices. Similarly, for the D3d group, the irreducible components can be determined by considering the symmetry operations of the group and their transformation matrices.

It is important to note that the results mentioned in the provided example may not be applicable to all cases and may vary depending on the specific problem at hand. Therefore, it is crucial to have a solid understanding of the underlying principles and techniques involved in determining irreducible components for different symmetry groups.

In conclusion, the determination of irreducible components for tensors in different symmetry groups is a complex and mathematically intensive task. I would recommend consulting established literature and resources and seeking guidance from experts in the field to ensure accurate and meaningful analysis.
 

1. What is an irreducible tensor of second or higher rank?

An irreducible tensor of second or higher rank is a mathematical object used to describe the physical properties of a system that cannot be broken down into smaller components. It is a multidimensional array of numbers that transforms according to certain rules under rotations and reflections.

2. What is the significance of irreducible tensors in physics?

Irreducible tensors play a crucial role in modern physics, particularly in the theory of relativity and quantum mechanics. They are used to describe the symmetries of physical systems and are essential for understanding fundamental concepts such as conservation laws, angular momentum, and electromagnetic fields.

3. How are irreducible tensors different from reducible tensors?

A reducible tensor can be decomposed into smaller tensors, each of which transforms independently under rotations and reflections. In contrast, an irreducible tensor cannot be broken down into smaller components and has a unique transformation behavior. This makes irreducible tensors more useful for describing physical systems with specific symmetries.

4. Can irreducible tensors be visualized?

No, irreducible tensors cannot be visualized in the traditional sense as they exist in a multi-dimensional space. However, they can be represented mathematically using matrices or tensors with specific indices, which can help in understanding their properties and transformations.

5. How are irreducible tensors used in practical applications?

Irreducible tensors are used extensively in various branches of physics, including electromagnetism, fluid mechanics, and solid-state physics. They are also used in engineering fields such as structural analysis and materials science for understanding the behavior of complex systems and designing efficient structures.

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