The Hamiltonian for clocks?
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I am rather a theoretician and looking for a formalism to represent biological clocks by Hermitian operators. The simplest thought model I am looking for is a formal representation of a clock (for instance 12 hours clock) by applying a Hermitian operator (e.g. Hamiltonian)? Note: a clock is a discrete modular representation As questions came up, let me try to explain: A clock has a modular arithmetic, that is a discrete oscillator, for instance: a≡b(modn) a≡b(modn) Many biological clocks work like this, and our 12 hours clock is similar (mod12)(mod12). It is possible as Raskolnikov indicated to apply an inverse Fourier type of approach and express such clocks, as discrete as they are, in the form of a sum such as (example): ψq=∑k=1qωq4π(e−i(k−1)ωqx+ei(k−1)ωqx) ψq=∑k=1qωq4π(e−i(k−1)ωqx+ei(k−1)ωqx) Both are the representation of the same type of clock. My first question is how does the parameters in the first equation relate to the second equation? My second question is then what is the Hermitian operator of the second equation? My third question is how can I find to a type of algorithm or general method to read the first equation directly into a Hermitian operator? The fact that I use this for biological clocks is marginal. The (modn)(modn) operation is independent of where we use it. I hope this helps, but if still questions, I am glad to explain.