Commutation of operators in QM

In summary, commutation of operators in quantum mechanics is a fundamental mathematical concept that describes the interaction between two operators representing physical quantities. It is important in predicting measurement outcomes and understanding quantum systems. The commutation of operators is calculated using the commutator, [A, B] = AB - BA, and when two operators commute, their commutator is equal to zero. This relates to the Heisenberg uncertainty principle, which is expressed through the commutator of position and momentum operators, [x, p] = iħ.
  • #1
jaejoon89
195
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Can somebody please explain the following?

Given the measurements of 2 different physical properties are represented by two different operators, why is it possible to know exactly and simultaneously the values for both of the measured quantities only if the operators commute?
 
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  • #2
What attempts have you made at this problem?
 

FAQ: Commutation of operators in QM

What is commutation of operators in quantum mechanics?

Commutation of operators in quantum mechanics is a mathematical concept that describes how two operators, representing physical quantities, interact with each other. It is a fundamental aspect of quantum mechanics and plays a crucial role in understanding the behavior of particles at the quantum level.

Why is commutation of operators important in quantum mechanics?

Commutation of operators is important in quantum mechanics because it allows us to determine the uncertainty between two physical quantities. It also helps in predicting the outcome of measurements and understanding the behavior of quantum systems.

How is commutation of operators calculated?

The commutation of operators is calculated using the commutator, which is defined as the difference between the product of two operators and the product of the same operators in reverse order. The commutator is denoted by [A, B] and is given by [A, B] = AB - BA.

What does it mean when two operators commute?

When two operators commute, it means that their commutator is equal to zero, [A, B] = 0. This implies that the order in which these operators act on a state does not matter and they can be measured simultaneously without affecting each other's values.

How does the commutation of operators relate to the Heisenberg uncertainty principle?

The commutation of operators is directly related to the Heisenberg uncertainty principle, which states that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa. This uncertainty is mathematically expressed through the commutator of position and momentum operators, [x, p] = iħ, where ħ is the reduced Planck's constant.

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