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Commutators
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Non-commuting operators imply uncertainty.
[x,p] = iħ[A, B] = AB − BA; if non-zero the observables can't be sharply defined simultaneously. [x̂, p̂] = iħ leads to the uncertainty principle.
Spin components don't commute.
Commuting observables share eigenstates.
In more detail
The commutator [A,B] = AB − BA measures whether two operators can be measured simultaneously. If the commutator is zero, the observables share eigenstates and can be known together.
Real-world uses
- Commutators lead to uncertainty relations, such as for position and momentum.
- Quantum angular momentum uses commutators to define algebra.
- Physicists use the commutator to find the uncertainty relation between position and momentum.
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Frequently asked questions
What does a non-zero commutator mean?
The two quantities cannot both have definite values at the same time.
Give an example.
Position and momentum have [x, p] = iħ, which leads to the uncertainty principle.
What does a nonzero commutator mean?
It means the two observables cannot be measured simultaneously with arbitrary precision.
Related topics in Quantum Mechanics
Schrödinger equation (time-dependent) · Schrödinger equation (time-independent) · Operators and observables · Particle in a box · Free particle · Finite potential well