Shannon's Entropy and Two Coupled Oscillators

 In previous notes (1) we discussed the free particle classical action in the form A= -Et+px in which p,x and E and t are independent for the relative and nonrelativistic cases. This leads to a scenario in which one cannot follow a particle in time i.e. there does not exist an x(t) as in classical mechanics. Instead one may develop eigenvalue equations -i d/dx partial exp(iA) = p exp(iA) and id/dt partial exp(iA) = E exp(iA) for the free particle. The solution exp(-iEt+ipx) is similar to a classical wave (which is a different solution following form A=-Et+px) in that it describes a kind of probability resonance in all of space. If one introduces a potential V(x), this is localized and one needs to think in terms of impulse hit resonances like exp(ikx) creating V(x). In other words, both an ensemble of exp(ipx) representing the particle and V(x)=Sum over k Vk exp(ikx) are described in terms of this resonance type periodic probability which interferes to create local features. One might argue that this feature of V(x) drives the particle.

   In this note we wish to examine the coupled oscillator  H= p1p1/2 + p2p2/2 + w1w1/2(x1x1+x2x2) + .5w2w2(x1-x2)(x1-x2) (with m=1). Transforming variables leads to an equation with a separable solution, but in y=x1+x2 and y1=x1-x2. In this note we try to examine this solution in terms of the above resonance picture and also suggest a possible overall Shannon’s entropy.