Pauli matrices and the Levi-Civita tensor : commutation relations

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SUMMARY

The discussion centers on the commutation relations of Pauli matrices and their relationship with the Levi-Civita tensor. Participants suggest starting by multiplying combinations of the Pauli matrices, specifically σ1, σ2, and σ3, and factoring out constants that can be replaced with the Levi-Civita symbol. A key example provided is the commutation relation [σ1, σ2] = σ1σ2 - σ2σ1 = iσ3, highlighting the importance of correctly identifying σ3 as a Pauli matrix. The conversation emphasizes the necessity of verifying the matrices involved in these relations.

PREREQUISITES
  • Understanding of Pauli matrices (σ1, σ2, σ3)
  • Familiarity with the Levi-Civita symbol
  • Knowledge of commutation relations in quantum mechanics
  • Basic linear algebra concepts
NEXT STEPS
  • Research the properties of Pauli matrices and their applications in quantum mechanics
  • Learn about the Levi-Civita tensor and its role in physics
  • Study commutation relations and their significance in quantum theory
  • Explore examples of matrix multiplication in quantum mechanics
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Students and professionals in physics, particularly those studying quantum mechanics, as well as anyone interested in the mathematical foundations of quantum theory.

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Homework Statement


Whats up guys!

I've got this question typed up in Word cos I reckon its faster:
http://imageshack.com/a/img5/2286/br30.jpg

Homework Equations



I don't know of any

The Attempt at a Solution


I don't know where to start! can u guys help me out please?

Thanks!
 
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You can start by multiplying each possible combination of pauli matrices. Do that and factor out a 1 or -1, which can be replaced with a Levi-Cevita symbol. Use i = 1, j = 2, k = 3.
 
Sure, just check it by putting the matrices into the commutation relation. For example, show ##[\sigma_1,\sigma_2]=\sigma_1 \sigma_2-\sigma_2 \sigma_1=i\sigma_3##. But it's not going work very well until you fix ##\sigma_3##. That's not a Pauli matrix.
 

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