How to Simplify Commutators Using Levi-Civita Symbol?

In summary, a commutator in mathematics is the difference between a product of two variables and the reverse of that product. It is represented by square brackets [X,Y] and is significant in determining the uncertainty principle in quantum mechanics. The Cevi Levita symbol, also known as the Levi-Civita symbol, is a mathematical symbol used to represent the sign of a permutation and is related to commutators through the concept of antisymmetry.
  • #1
KostasV
21
0
image.jpg

Homework Statement


The problem statement can be seen in the picture i uploaded.

Homework Equations


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The Attempt at a Solution


The attempt to the solution can be seen in the picture i uploaded.
I reached to the A and i don't know how to proceed to the solution that is given below. How does the minus and δkj disappear?
If i do the double summation on k and j I think that every term gets zero either because of εijk (levi-cevita) or δkj (kronecker)
 
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  • #2
You used the index j twice: once for the index on x and then again as a dummy summation index when writing out the angular momentum in terms of the levi-civita symbol.
 
  • #3
TSny said:
You used the index j twice: once for the index on x and then again as a dummy summation index when writing out the angular momentum in terms of the levi-civita symbol.
I can't see why this is wrong ... :/
 
  • #4
TSny said:
You used the index j twice: once for the index on x and then again as a dummy summation index when writing out the angular momentum in terms of the levi-civita symbol.
Moreover , if i use , let's say the index m on x (not on x that comes from angular momentum , yes on x that is alone) , then i still have the minus on ih bar ... My solutions say that it should not be there ...
 
  • #5
image.jpg
Ok i think i understand why i can't have the same index on these two .
Moreover i think i found how i get rid of this minus ...
I must use the fact that εijk=-εikj wright ?
Is now the solution correct ? (Uploaded photo)
 
  • #6
Yes. That looks very good.
 
  • #7
TSny said:
Yes. That looks very good.
Thank you very much for your help :D
 

FAQ: How to Simplify Commutators Using Levi-Civita Symbol?

1. What is a commutator in mathematics?

A commutator is a mathematical operation that involves taking the difference between a product of two variables and the reverse of that product. In other words, it's the difference between XY and YX. This operation is used in various mathematical fields such as group theory and linear algebra.

2. How is the commutator represented in notation?

The commutator is typically represented by square brackets [X,Y]. The variables X and Y can represent any mathematical object such as numbers, matrices, or vectors.

3. What is the significance of commutators in quantum mechanics?

In quantum mechanics, commutators play a crucial role in determining the uncertainty principle. The uncertainty principle states that certain pairs of physical properties, such as position and momentum, cannot both be known precisely at the same time. This is due to the non-commutative nature of these properties.

4. What is the Cevi Levita symbol?

The Cevi Levita symbol, also known as the Levi-Civita symbol, is a mathematical symbol used to represent the sign of a permutation. It is commonly used in vector calculus and differential geometry to denote the direction of a cross product between two vectors.

5. How is the Cevi Levita symbol related to commutators?

The Cevi Levita symbol is related to commutators through the concept of antisymmetry. In mathematics, a commutator is considered antisymmetric, meaning that switching the order of the variables results in the negative of the original value. Similarly, the Cevi Levita symbol is also antisymmetric, making it a useful tool in calculating commutators in various mathematical operations.

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