The commutation relations between the components of position and momentum in two dimensions are:
$$[x, y] = [y, x] = 0$$
$$[p_x, p_y] = [p_y, p_x] = 0$$
$$[x, p_x] = -[p_x, x] = i\hbar$$
$$[y, p_y] = -[p_y, y] = i\hbar$$
They can be though as a kind of postulate, according to Why did physics go quantum.
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Author of the notes: Antonio J. Pan-Collantes
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