what are all Quantum conjugate pairs of observables ?

[Widdekind]

[math]\Delta x \; \Delta p \ge \hbar[/math]

[math]\Delta t \; \Delta E \ge \hbar[/math]

 

what about

 

[math]\Delta \theta_{\perp} \; \Delta L \ge \hbar[/math]

 

? i.e. a wave-function confined in angle (about some axis), would be required to possess a statistical spread, in its angular momentum, about said same axis ?

[spacelike]

Maybe someone else can list a bunch of them for you if that's what you're after.

 

But a general rule is if the observables are described by two operators that do not commute with each other, then they will have a relation like that.

 

Two opperators (A and B) commute if:

AB - BA = 0

(technically since these are operators you would need to operate on a test function "f" to prove that it works, ABf - BAf = 0)

But if you find that it is not 0, then they will have some value which the product of the standard deviations must be greater than.

 

For example, the commutator of 'x' and 'p' is 'i*h-bar' where "h-bar" is plancks constant divided by 2pi.

xp - px = ih-bar

 

Therefore the product of the standard deviations of x and p is non-zero

[math]\sigma_{x}\sigma_{p}\ge\frac{\hbar}{2}[/math]

Edited July 31, 2013 by spacelike