# Non commutating observables

Staff Emeritus
The Wikipedia entry on uncertainty says, "Since the Robertson and Schrödinger relations are for general operators, the relations can be applied to any two observables to obtain specific uncertainty relations. A few of the most common relations found in the literature are given below."

It then continues with examples, linear momentum-position, angular momentum orthogonal components, energy-time, the number of electrons in a superconductor and the phase of its Ginzburg–Landau order parameter.

I'm curious about other pairs of observables that don't commute. Can someone point me towards a more complete list?

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kith
Non-commuting observables are the usual case, not the exception.

Usually, you look for a complete set of commuting observables which suits your problem. All other observables then are non-commuting with at least one observable from your set.

1 person
Staff Emeritus
Thank you kith. But I meant non-commuting, i.e. those which can not be measured simultaneously and thus subject to uncertainty. Another way to phrase it is: what other uncertainty relationships exist between observables beyond the four mentioned in the Wikipedia article?

kith
The uncertainty principle for two observables A and B is ΔAΔB ≥ |<C>| with C = [A,B]. You cannot expect |<C>| to yield a general value like ħ/2 for arbitrary non-commuting A and B because it is the expectation value of the operator C and thus depends on the state of the system.

A state-independent value libe ħ/2 can be given only in the case of conjugated variables like position and momentum because there, C is proportional to the identity operator, so its expectation value doesn't depend on the state.

So your question can be rephrased to what kinds of conjugated variables exist.

1 person
Staff Emeritus
Thank you kith for helping me phrase the question correctly.

So, where ca I find a more complete list of conjugated variables?

kith