In "J.M. Ziman. Electrons and Phonons: The Theory of Transport Phenomena in Solids" the author formally introduces the position (displacement) operator and then defines the momentum operator with the correct commutation relation [ˆul,ˆpl′]=iℏδl,l′ between these two. Is such approach formally correct?
Edit: This suggestion is wrong: Can this lead to some degenerate form of "momentum operator"? First that comes to my mind is ˆpl=−iℏ∂∂ulifu<u0ˆpl=−iℏ1ulifu≥u0. Putting this theoretical construction aside, the core of the question is that if generally such definition of an operator can be done?
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