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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 '14 at 15:12

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question.If this question can be reworded to fit the rules in the help center, please edit the question.

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 '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 '14 at 9:16