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Canonical Transformations, Quantization, Mutually Unbiased and Other Complete Bases

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Document pages: 19 pages

Abstract: Using ideas based on supersymmetric quantum mechanics, we design canonicaltransformations of the usual position and momentum to create generalized“Cartesian-like” positions, W, and momenta, Pw , with unit Poissonbrackets. These are quantized by the usual replacement of the classical , x Px by quantum operators, leading to an infinite family of potential “operator observables”.However, all but one of the resulting operators are not Hermitian(formally self-adjoint) in the original position representation. Using either thechain rule or Dirac quantization, we show that the resulting operators are“quasi-Hermitian” relative to the x-representation and that all are Hermitianin the W-representation. Depending on how one treats the Jacobian of thecanonical transformation in the expression for the classical momentum, Pw ,quantization yields a) continuous mutually unbiased bases (MUB), b) orthogonalbases (with Dirac delta normalization), c) biorthogonal bases (with Diracdelta normalization), d) new W-harmonic oscillators yielding standardorthonormal bases (as functions of W) and associated coherent states andWigner distributions. The MUB lead to W-generalized Fourier transformkernels whose eigenvectors are the W-harmonic oscillator eigenstates, withthe spectrum (±1,±i) , as well as “W-linear chirps”. As expected, W, Pw satisfythe uncertainty product relation: ΔWΔPw ≥1 2 , h=1.

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