Why not formulate QM in terms of |ψ| squared?

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The discussion centers on the formulation of quantum mechanics (QM) and the rationale behind not expressing it solely in terms of the probability density |ψ| squared. While the wave function provides a complete description of a quantum system, it also contains phase information essential for understanding phenomena like interference and diffraction, which cannot be captured by |ψ| squared alone. The use of density matrices is suggested as a more comprehensive approach, allowing for the representation of mixed states and the calculation of expectations for various observables, including momentum. The conversation also touches on the limitations of classical probability theory in explaining quantum behavior, emphasizing the need for a more complex framework. Ultimately, the discussion highlights the intricacies of QM that necessitate the use of both wave functions and density matrices for a complete understanding.
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DrDu said:
You are obviously right, here. Nevertheless my question remains: can we quantize charge density, viewed as a field, directly?

Together with charge density, you need st least the associated charge current operaots, which complement the density to a conservation law. Quantizing this is called current algebra. There have been varous attempts and there is significant literature, but apart from the 1+1D case (where current algebra essentially turns into Kac-Moody algebra) no real breakthrough.
 

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