Selective Entropy Based on Measurement, Conjugate Variables and Quantum Numbers

 In classical mechanics, one assumes one may measure precise momentum and x positions. One may argue, however, that a measurement involves changing the momentum of a particle (i.e. an impulse hit) and that such a change does not occur either in an instant of time or at a point x. For example, a classical particle in a potential V(x) changes its velocity in a region dx, but at the same time is said to have a velocity of v(x) to leading order as dx shrinks to zero. One may argue that dx does not shrink to zero and may not be tiny with respect to an overall length scale. Given that an interaction involves a particle and a potential or another particle, one may argue that a finite dx should be associated with both. Thus a particle with constant momentum should have a tiny dx associated with it, possibly a function of p, but at the same time there is a constant  probability for a measurement to take place at any x point.Thus we argue that there are two sets of probability schemes, one for dx and another for a larger length scale and constant probability. This should point to two entropy schemes, each associated with the type of measurement done and its analysis. For example, a two slit system may be used to measure the interaction probability. This gives rise to an interference probability. At the same time, many two slit system may be used to measure the center of mass position of the particle, ignoring the interference pattern. This gives rise to P(x)=1/L and a completely different Shannon’s entropy.

    Given that there is a dx probability scheme for a constant p associated with interactions, an associated entropy should appear when such an interaction occurs, but this now involves breaking the constant p and considering the associated dx probabilities. Thus there is both momentum and spatial entropy combined as in two slit interference.

    In the case of a bound state, there are numerous exp(ipx) eigenstates (due to interactions with V(x)), suggesting both internal p and x entropies, but overall one has a single energy eigenstate. In terms of energy one has a fixed state (although exp(-iEn t) suggests there  are frequency pockets in time pointing towards a kind of time entropy. If one considers only interactions which change a particle from one En to another, a particle in a single state at T=temperature=0 has entropy 0 (even though the state itself has momentum and spatial entropy). Thus it seems entropy is selective depending on the type of interactions present and what eigenvalues they change.

   In other words, classically one counts states when computing entropy, but in quantum mechanics one may count states by defining states in terms of quantum numbers. One may consider a bound state to be a single certain state with eigenvalue En and consider entropy being created if interactions (photon/phonon absorption/emission) break the single En. In such a case, one may ignore the breaking of exp(ipx)s within a bound state which implies momentum entropy as well as the spatial entropy linked with even a single exp(ipx). In other words, one may select various different kinds of probabilities linked with different eigenvalues and their measurements when calculating Shannon’s entropy.