The Imaginary Number in quantum mechanics: Why the noncommutative phase is the origin of time-frequency uncertainty

 What is essentially different in quantum mechanics is that it deals with complex quantities (e.g. wave functions and quantum state vectors) of a special kind, which cannot be split up into pure real and imaginary parts that can be treated independently. Furthermore, physical meaning is not attached directly to the complex quantities themselves, but to some other operation that produces real numbers (e.g. the square modulus of the wave function or of the inner product between state vectors).

https://www.ind.ku.dk/english/research/didactics-of-physics/Karam_AJP_Complex_numbers_in_QM.pdf 

not until Dirac formulated his bra-ket notation that it became clearer that the complex quantities of quantum mechanics were of a different kind than the ones commonly used in classical physics.

 If we should have no information about position,
Shankar argues, the probability distribution should be “flat”, meaning that it is equally likely to find it anywhere...Is there a function that has a wavelength and, yet, a constant absolute value? The complex exponential has this property....

 

 So the position of time as the phase shift being a negative frequency and reverse time is always less than any positive number as an absolute value...of a geometric dimension of zero - due to the convergence of the discrete inverse frequency!

 

 So there you get the inverse frequency

ω has units of frequency

 k, called the wave number, has units of inverse length.

  a minus sign be chosen for the inverse (frequency to time) transform

 You have a wavenumber of N/cm indicating the number of waves per unit length. Wave length is related to Wave frequency and Wave speed by Wave length [cm] = Wave speed [cm/s] / Wave freq [1/s] so the (inverse Wave length) = 1/cm = Wave number per cm = Wave frequency [1/s] / wave speed [cm/s]

 standard matrix techniques for the least squares solution of m
x n systems of equations can be used to design an inverse
frequency response matrix H + (k) for each frequency k.

You have a wavenumber of N/cm indicating the number of waves per unit length. Wave length is related to Wave frequency and Wave speed by Wave length [cm] = Wave speed [cm/s] / Wave freq [1/s] so the (inverse Wave length) = 1/cm = Wave number per cm = Wave frequency [1/s] / wave speed [cm/s]

Reference: https://www.physicsforums.com/threads/frequency-in-units-of-inverse-length.177300/