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The representation X  P  of the canonical commutation relations is said to be unitarily equivalent to the representation X P. The importance of (8.44) comes from the following theorem, which we state without proof. The unitary equivalence theorem of von Neumann. All representations of the canonical commutation relations of the Weyl form13 (8.42) are unitarily equivalent to the representation (8.44) on L2  . Moreover, this representation is irreducible, that is, any operator on  can be written as a function of X and P. Any operator that commutes with X (P) is a function of X (P). Any operator that commutes with X and P is a multiple of the identity I. This theorem implies that we do not have to worry about the choice of representation in (8.39), because any two choices are related to each other by a unitary transformation. In three dimensions the position and momentum operators R and P are vector operators with components X Y Z and Px, Py, Pz, which we denote collectively as Xi and Pi, i = x y z. The different components of R and P commute, and only identical components have nonzero commutation relations: Xi Pj = i ijI  (8.45) 8.3.3 The parity operator The parity operator reverses the sign of the coordinates: x → −x. It can also be viewed as a combination of reflection with respect to a plane followed by a rotation by  about an axis perpendicular to this plane. Let us take for example the xOy plane and cal&%290486gk

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💉💉਄ੰ༑ᕚ25譚敦民（資я深я電Я臺Я採Я訪Я主Я任）FiGURATऊ൬٫ٙАЗАЗАЗАዄᅊ

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