What are some examples of non-commutating observables in quantum mechanics?

[anorlunda]

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?

 

[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.

 

[anorlunda]

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.

 

[anorlunda]

Thank you kith for helping me phrase the question correctly.

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

 

[kith]

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

I don't know of a complete list. This is essentially a question about classical Hamiltonian mechanics. Do you know what conjugated variables are?

 

[anorlunda]

Conjugate variables are Fourier transform duals. I see now why my question should not be under quantum physics. It can be answered with Fourier analysis of classical variables.

But thanks again, double checking Wikipedia on conjugate variables led me to an article with a more complete list of examples. That's what I was seeking in the OP, so thanks again; question answered.