How Do Compatible Observable Properties Transform Between Equations?

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SUMMARY

The discussion focuses on the transformation of compatible observable properties in quantum mechanics, specifically how equations (1), (2), and (3) relate to each other. Equation (1) represents the commutation relation, while equation (2) illustrates a more complex expression involving products of observables. The participant seeks clarification on the stepwise simplification from (1) to (2) and from (2) to (3), emphasizing that the transformations are grounded in established mathematical properties rather than any mystical principles.

PREREQUISITES
  • Understanding of quantum mechanics and observable properties
  • Familiarity with commutation relations in quantum theory
  • Knowledge of algebraic manipulation of operators
  • Experience with tensor products and their properties
NEXT STEPS
  • Study the derivation of commutation relations in quantum mechanics
  • Explore the properties of tensor products in operator algebra
  • Learn about the significance of compatible observables in quantum systems
  • Investigate simplification techniques for complex operator expressions
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Students of quantum mechanics, physicists working with operator algebra, and anyone interested in the mathematical foundations of quantum theory.

alui
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Homework Statement


I have problem with properties of compatible observable

Homework Equations


[AB,CD]=AB CD – CD AB ………. (1)
ABCD+ACBD-ACBD-ACDB+ACDB+CADB-CADB-CDAB ………. (2)
A{C,B}D-AC{D,B}+{C,A}DB-C{D,A}B ... (3)

The Attempt at a Solution


How can the equation (1) become (2), and eq (2) become (3), someone please explain to me about that properties
 
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Try to check the other direction: verify that (3) is equal to (2) and then see if you can simplify (2) to (1).
There is no magic going on.
 

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