Transcription of 1 Complex Numbers in Quantum Mechanics
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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 ?
Quantum mechanics is different. Factors of i = √ −1 are everywhere, for example from Heisenberg we have his famous commutation relation, QP −PQ = i¯h, and from Schr¨odinger his equally famous equation ¯h i ∂tΨ = HΨ. The wave function Ψ is complex and so is practically every other quantity needed to formulate quantum mechanics.
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