Canonical commutation relation

The in quantum mechanics (QM ) generic canonical commutation are:

Here denote the X is the ( Hermitian ) local operators and P is the ( Hermitian ) momentum operators of the QM, the parentheses around the operators, eg, the so-called commutator. The position and momentum operators " swap " these pairs, that is, their commutator is zero.

This means nothing else in practice, than that these metrics can be measured simultaneously ( called also observable in QM ). Does not disappear, the commutator, ie is equal to zero, a "swap " these operators do not.

The operators for position and momentum thus represent an example of non- commuting operators and describe herewith sizes in the same quantum system, which can not be measured simultaneously with arbitrary precision, their simultaneous measurement is therefore subject to a certain degree of fuzziness. This leads directly to the famous uncertainty principle of Werner Heisenberg.

Derivation and justification

Since the product (ie, the sequential execution ) of two linear operators in general not commutative (ie the order of the sequential execution can not be easily reversed ), it is the commutator (or commutation ) of two linear operators A and B as defined as follows:

Substituting the operators for position and momentum just in above equation and let this act on a wave function, as follows:

The above account for the spatial components and lead to the same result. It is interesting here is that, for example,

Reversed. The proof that the position and momentum components interchange with each other is simple. Overall, then give the above named canonical commutation.

Note:

In general, the operators are provided in the QM with a " hat ", this is omitted here for the sake of readability, so it is the case for the position operator.