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1 Complex Numbers in Quantum Mechanics

1 Complex Numbers in Quantum MechanicsComplex Numbers and variables can be useful in classical physics. However, theyare not essential. To emphasize this, recall that forces, positions, momenta, potentials,electric and magnetic fields are all real quantities, and the equations describing them,Newton s laws, Maxwell s equations,etc. are all differential equations involving strictlyreal Mechanics is different. Factors ofi= 1 are everywhere, for examplefrom Heisenberg we have his famous commutation relation,QP P Q=i h,and from Schr odinger his equally famous equation hi t =H .The wave function is Complex and so is practically every other quantity neededto formulate Quantum Mechanics . Granted that classical Mechanics fails at distancesof nanometer or less, why is it suddenly so necessary that everything be Complex ?Comment 1An interesting question is the distance scale below whicha Quantum treatment is essential. A nanometer seems like a reasonablechoice.

Bottom line: In quantum mechanics, complex numbers are essential. 2 Photon Polarization, Probability and Interference in Quantum Mechanics The existence of polarized light gives a wonderful example where both classical ideas and one of the most puzzling features of quantum mechanics coexist. Polarization is very familiar in daily life.

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