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Commutator Algebra

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Commutator Algebra The commutator of two operators is denoted by where

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**Commutator Algebra The commutator of two operators is denoted by where**

The commutator indicates whether operators commute.

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**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

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

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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.

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Commutator Algebra Consider the following properties: R r

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**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).

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**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.

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Commutator Algebra However,

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**Commutator Algebra However,**

The two observables can be measured independently.

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

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

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**Hermitian Operators & Operators in General**

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

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

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

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Hermitian Operators

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Hermitian Operators All operators in QM are Hermitian.

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**Hermitian Operators All operators in QM are Hermitian.**

They are important in QM because the eigenvalues Hermitian operators are real!

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**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.

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**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

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Hermitian Operators Evaluate

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Hermitian Operators Evaluate From definition

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Hermitian Operators Evaluate From definition Replace by operators

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Hermitian Operators Evaluate From definition Replace by operators

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Hermitian Operators Evaluate From definition Replace by operators

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Hermitian Operators

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Hermitian Operators Therefore

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