_Conjugate observables_ or _incompatible observables_ are pairs of [observables](Observable.md) in quantum mechanics that do not [commute.](Commutators%20in%20quantum%20mechanics.md) What this means is that one can measure a quantity associated with [state](State%20vector), $|\psi\rangle$ corresponding with the first observable without affecting the quantity that would be measured if we were to measure the other observable. The commutation relation for any pair of conjugate observables, $\hat{A}$ and $\hat{B}$ takes the following form: $[\hat{A},\hat{B}]=i\hbar\hat{C}$ %%What is C like and is this true for ALL conjugate pairs? C I believe should commute with A and B and there are examples with angular momentum where it does, but I need to check this.%% %%The only place where I've seen this relation in this exact form is Gerry and Knights quantum optics book on page 150, where the context is quadrature sqeezing.%% %%Note also the relation between this form of the commutation relation and its use in the Groenewold-van Hove theorem. see page 201 of Woit's book.%% The notion of [conjugate observables in quantum mechanics](Conjugate%20observables%20in%20quantum%20mechanics.md) is [an adaptation of the notion of conjugate variables in classical mechanics.](Conjugate%20observables%20in%20quantum%20mechanics.md#Relationship%20between%20conjugate%20observables%20and%20conjugate%20variables) # Quantum Uncertainty Relations Pairs of conjugate variables in quantum mechanics also form [Quantum uncertainty relations.](Quantum%20uncertainty%20relations) # Relationship between conjugate observables and conjugate variables One may notice that these commutation relations are reminiscent of [Poisson brackets](Poisson%20bracket.md) that are formed between [conjugate variables](Conjugate%20variables.md) in classical mechanics. Indeed they may be thought of as the replacement of Poisson brackets in [quantum mechanics.](Quantum%20Mechanics%20(index).md) In fact the relation between Poisson brackets and corresponding commutators goes as $\{A,B\}\rightarrow i\hbar [\hat{A},\hat{B}]$ as part of the standard [quantization](Quantization.md) procedure where we also promote $A$ and $B$ to observables. ^ed9687 # Position-momentum commutators ([... see more](Position-Momentum%20Commutators.md)) #QuantumMechanics/QuantumMeasurement/QuantumObservables #QuantumMechanics/MathematicalFoundations