Commutator Relation: What is [\hat{A}\hat{B}, \hat{C}\hat{D}] Equal to?

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    Commutator Relation
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SUMMARY

The commutator [\hat{A}\hat{B}, \hat{C}\hat{D}] can be expressed as (ab)(cd) - (cd)(ab). The discussion emphasizes the application of the commutation rule [A,BC]=B[A,C]+[A,C]B to simplify the expression. Participants confirm the correct interpretation of the commutation rules, ensuring clarity in the mathematical manipulation involved. This foundational understanding is essential for further exploration of quantum mechanics and operator algebra.

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  • Understanding of quantum mechanics principles
  • Familiarity with operator algebra
  • Knowledge of commutation relations
  • Basic proficiency in mathematical proofs
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  • Study the implications of commutation relations in quantum mechanics
  • Learn about the significance of the Heisenberg uncertainty principle
  • Explore advanced topics in operator theory
  • Practice proving commutation relations through exercises
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Students of quantum mechanics, physicists specializing in quantum theory, and mathematicians focusing on operator algebra will benefit from this discussion.

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What is the commutator [\hat{A}\hat{B}, \hat{C}\hat{D}] equal to? How to distribute what's inside?
 
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=(ab)(cd)-(cd)(ab)
 
Use the rule [A,BC]=B[A,C]+[A,C]B. It's very easy to prove, so I suggest that you do that as an exercise.
 
Fredrik said:
Use the rule [A,BC]=B[A,C]+[A,C]B. It's very easy to prove, so I suggest that you do that as an exercise.

Do you mean [A,BC]=[A,B]C+B[A,C]?
 
DocZaius said:
Do you mean [A,BC]=[A,B]C+B[A,C]?
Ah, yes of course. Thanks.
 

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