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Consider noncanonical operators $\hat{x}_1,\hat{x}_2,\hat{p}_1,\hat{p}_2$ satisying the following condition in the $q_1,q_2$ - basis ($\psi=\langle q_1,q_2|\Psi\rangle$)(similar to wave mechanics):

$$\hat{x}_1\psi=q_1\psi-ia\hbar\frac{\partial}{\partial q_2}\psi$$

$$\hat{x}_2\psi=q_2\psi-ib\hbar\frac{\partial}{\partial q_1}\psi$$

$$\hat{p}_1\psi=-i\hbar\frac{\partial}{\partial q_1}\psi+cq_2\psi$$

$$\hat{p}_2\psi=-i\hbar\frac{\partial}{\partial q_2}\psi+dq_1\psi+i\hbar\gamma\frac{\partial}{\partial q_1}\psi$$

The commutation relations matrices are (similar to Hilbert-space quantum mechanics):

$[x_i,x_j]=i\hbar\left(\begin{array}{cc} 0 & (b-a) \\ (a-b) & 0 \end{array}\right)$
$[p_i,p_j]=i\hbar\left(\begin{array}{cc} 0 & (c-d) \\ (d-c) & 0 \end{array}\right)$

$[x_i,p_j]=i\hbar\left(\begin{array}{cc} 1 -ac & -\gamma \\ 0 & 1-bd \end{array}\right)$

Because $\hat{x}_1,\hat{x}_2$ are not commute so I have to introduce 2 other parameter $q_1,q_2$.

My question is: is it true or is it not true that all wave mechanics theories with the same commutation relations will predict the same results?

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closed as unclear what you're asking by Emilio Pisanty, Ali, Colin McFaul, Jim, BMS Jul 24 at 15:12

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I think there is not enough information here: On what space do these operators act? What are $q_1,q_2$? What constitutes a "wave mechanics" theory for you? –  ACuriousMind Jul 14 at 13:43
    
The example you present is just playing around with multiplication and derivation operators. Commutation relations are also a very delicate object when dealing with unbounded operators. You should be considering commutation of unitary groups (provided the operators are self-adjoint). Then you might be able to say whether a representation of a group is unique in some sense. For example the canonical commutation relations (generators of the Heisenberg group) are uniquely represented (up to isomorphisms) by a multiplication and a derivation operator on some $L^2$ space. –  yuggib Jul 15 at 9:16