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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 • 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 • 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
Commutator Algebra • If A and B commute they can be measured simultaneously. The results of their measurements can be carried out in any order.
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.
Commutator Algebra • Consider the following properties: • R • R • R • R • R • R • r
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).
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.
Commutator Algebra • However,
Commutator Algebra • However, • The two observables can be measured independently.
Commutator Algebra • Some examples of other cases are shown below: • R • R • R • R • R
Commutator Algebra • The order of operators should not be changed unless you are sure that they commute.
Operators • Operators act on everything to the right unless constrained by brackets.
Operators • Operators act on everything to the right unless constrained by brackets.
Operators • Operators act on everything to the right unless constrained by brackets. • The product of operators implies successive operations.
Hermitian Operators • All operators in QM are Hermitian.
Hermitian Operators • All operators in QM are Hermitian. • They are important in QM because the eigenvalues Hermitian operators are real!
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.
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
Hermitian Operators • Evaluate
Hermitian Operators • Evaluate • From definition
Hermitian Operators • Evaluate • From definition • Replace by operators
Hermitian Operators • Evaluate • From definition • Replace by operators
Hermitian Operators • Evaluate • From definition • Replace by operators
Hermitian Operators • Therefore
