Today is electricity and magnetism midterm day, so I'm just going to jot down a skeleton of a thought process about the quantum mechanical phase operator research I've been reading for the last few days, and then I have to run.

In matrix rperesentation, the derivative of a polynomial can be represented as[1]:

for a third degree polynomial and extended for higher degrees. Integration looks like this[2]:

and can again be extended. In the article by Nieto[3], he quotes Louisell as saying this about the discrete cosine and sine functions in quantum mechanics.

In the Fourier domain where functions are represented by series of sine and cosine functions, derivatives are constructed simply by multiplying by i, (the square root of negative one), times frequency, and integrals are constructed by dividing by i times the frequency.

Also, in relation to the EE discrete signal analysis, these two figures from the Nieto RMP article[4], (pictures 4 and 5):

To me, this all has the feel of discrete time signal processing, so I'll leave you with a link to the discrete Fourier transform.

**References:**

2. Integration

http://demonstrations.wolfram.com/TheDerivativeAndTheIntegralAsInfiniteMatrices/

3. Nieto's article

http://arxiv.org/abs/hep-th/9304036v1z

4. Nieto RMP article

http://dx.doi.org/10.1103%2FRevModPhys.40.411

CARRUTHERS P. & NIETO M. (1968). Phase and Angle Variables in Quantum Mechanics, Reviews of Modern Physics, 40 (2) 411-440. DOI: 10.1103/RevModPhys.40.411

5. Discrete Fourier Transform

http://www.dspguide.com/CH8.PDF

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