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Commutator Algebra

Commutator Algebra. Commutator Algebra. The commutator of two operators is denoted by where. Commutator Algebra. The commutator of two operators is denoted by where The commutator indicates whether operators commute. Commutator Algebra.

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Commutator Algebra

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  1. Commutator Algebra

  2. Commutator Algebra • The commutator of two operators is denoted by where

  3. Commutator Algebra • The commutator of two operators is denoted by where • The commutator indicates whether operators commute.

  4. Commutator Algebra • The commutator of two operators is denoted by where • The commutator indicates whether operators commute. • Two observables A and B are compatible if their operators commute, ie . Hence

  5. Commutator Algebra • If A and B commute they can be measured simultaneously. The results of their measurements can be carried out in any order.

  6. Commutator Algebra • If A and B commute they can be measured simultaneously. The results of their measurements can be carried out in any order. • If they do not commute, they cannot be measured simultaneously; the order in which they are measured matters.

  7. Commutator Algebra • Consider the following properties: • R • R • R • R • R • R • r

  8. Commutator Algebra (in QM) • If and are Hermitian operators which do not commute, the physical observable A and B cannot be sharply defined simultaneously (cannot be measured with certainty).

  9. Commutator Algebra (in QM) • If and are Hermitian operators which do not commute, the physical observable A and B cannot be sharply defined simultaneously (cannot be measured with certainty). eg the momentum and position cannot be measured simultaneously in the same plane.

  10. Commutator Algebra • However,

  11. Commutator Algebra • However, • The two observables can be measured independently.

  12. Commutator Algebra • Some examples of other cases are shown below: • R • R • R • R • R

  13. Commutator Algebra • The order of operators should not be changed unless you are sure that they commute.

  14. Hermitian Operators & Operators in General

  15. Operators • Operators act on everything to the right unless constrained by brackets.

  16. Operators • Operators act on everything to the right unless constrained by brackets.

  17. Operators • Operators act on everything to the right unless constrained by brackets. • The product of operators implies successive operations.

  18. Hermitian Operators

  19. Hermitian Operators • All operators in QM are Hermitian.

  20. Hermitian Operators • All operators in QM are Hermitian. • They are important in QM because the eigenvalues Hermitian operators are real!

  21. Hermitian Operators • All operators in QM are Hermitian. • They are important in QM because the eigenvalues Hermitian operators are real! This is significant because the results of an experiment are real.

  22. Hermitian Operators • All operators in QM are Hermitian. • They are important in QM because the eigenvalues Hermitian operators are real! This is significant because the results of an experiment are real. • Hermiticity is defined as

  23. Hermitian Operators • Evaluate

  24. Hermitian Operators • Evaluate • From definition

  25. Hermitian Operators • Evaluate • From definition • Replace by operators

  26. Hermitian Operators • Evaluate • From definition • Replace by operators

  27. Hermitian Operators • Evaluate • From definition • Replace by operators

  28. Hermitian Operators

  29. Hermitian Operators • Therefore

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