Cycling and Correlations in Quantum Mixed States

In an earlier note, bound pure states were described in terms of cycling of momentum states exp(ipx) which scattered off a potential V(x) from one momentum state into another in a time of 1/E. If the average kinetic energy, defined by  [Sum over p p*p/2m fp exp(ipx)] / W(x), where W(x)=wavefunction = Sum over p fp exp(ipx) = normalization constant, satisfies a conservation of energy equation: KE ave + V(x) = E, one obtains the time-independent Schrodinger equation. It was also argued the Schrodinger equation may be written in terms of a cycling equation by writing both V(x) and W(x) as a Fourier series and equating exp(ipx) terms, namely:

Sum k+j=p  Vk fj + p*p/2m fp     = E fp ((1))

In other words, fj scatter with fk to produce fp and potential energy, which together with the kinetic energy of fp yield a constant (energy) times fp. This is of the form of an equation with a separable time portion  id/dt W(t,x) = E W(x). It should be noted that in ((1)), the fj’s are correlated with fp. In this note, we try to argue that quantum bound states demonstrate strong correlations between fp’s and also try to extend the cycling model described above to mixed states.