Uncertainty Principle: Commutators

Question

How are commutators the mathematical basis for uncertainty principle? What makes one say that commutators imply uncertainty principle or vice-versa?

Answer 1

Consider a commutator relation $[\hat{A},\hat{B}]=\hat{C}$. This commutator relation is preserved if you take $\hat{A} \mapsto \hat{A} - <\hat{A}>,\hat{B} \mapsto \hat{B} - <\hat{B}>$ with expectation value denoted by $<...>$.

for Operators $\hat{A},\hat{B},\hat{C}$. Then you can act a norm on it and obtain

$||\hat{C}||=||\hat{A}\hat{B}-\hat{B}\hat{A}||\le ||\hat{A} \hat{B}||+||\hat{B}\hat{A}||$ (triangle inequality).

Now you can set this norm to supremum norm and obtain $||\hat{A}\hat{B}||\le||\hat{A}||||\hat{B}||$.

Then you will arrive at the uncertainty Relations if you use invariance of the commutator relation by shift of Operators as shown above.

Set e.g. $\hat{A} = \hat{p},\hat{B}=\hat{x},C=-i\hbar$ and you will get the uncertainty relation for this variables.