Quantum Mechanics
Particles are represented by a complex wave function.
Operators
Operators play a fundamental role in quantum mechanics. Operators are used to calculate the value of different observable quantities, such as position, momentum or energy of a particle. They are also used to calculate the evolution of the wave function in time. In a way, operators are used to access and modify the wave function.
Uncertainties
A fundamental property of waves is that they do not have definite position and momentum at the same time. This issue translates to quantum mechanics as well. A quantum particle can't have all its observables definite. What does this tell us about the operators for these observables?
Commutators
The commutator of two operators can be used to learn about various properties.
1) The commutator of two operators, $[A, B] = AB - BA$, decides wether or not the two observables can be measured simultaneously. If two operators commute, $[A, B] = 0$, then the two corresponding observables can be measured simultaneously; and if they are nonzero then the two observables cannot be measured simultaneously. Moreover, the uncertainty in these two observables is related to the commutator.
$$ \Delta A \Delta B = \bigg | \frac{1}{2i} \Big \langle [A,B] \Big \rangle \bigg | $$
1) The commutator of two operators, $[A, B] = AB - BA$, decides wether or not the two observables can be measured simultaneously. If two operators commute, $[A, B] = 0$, then the two corresponding observables can be measured simultaneously; and if they are nonzero then the two observables cannot be measured simultaneously. Moreover, the uncertainty in these two observables is related to the commutator.
$$ \Delta A \Delta B = \bigg | \frac{1}{2i} \Big \langle [A,B] \Big \rangle \bigg | $$
Sneak Peak
Stay tuned for more commutators!
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