Quantum Velocity Direction, Information and Negative Wavefunction Values

  In classical mechanics a constant velocity may be defined by dx/dt where x is in the direction of motion. Thus two dimensions are required and the concept of positive/negative x. Quantum mechanics of a free particle involves a v=x/t, but also independent periodic fluctuations of  x and t i.e. exp(-iEt) and exp(ipx) may be considered separately. One, however, cannot lose information we argue so two dimensions are still required to describe motion only using x as well as  a concept of negative and positive. (The notion of “only using x” applies to a bound problem.) If the two dimensions only depend on x, then there is no a priori dimension which may be strictly positive such as t. Both dimensions should allow for positive and negative values so that somehow overall the direction of velocity may be defined.

   Quantum mechanics is concerned with probabilities, but classical probabilities do not introduce negative values. As argued in previous notes, it is possible to have two layers of probabilities P(x) (classical) and P(p/x) linked to exp(ipx) (quantum). The second result contains the two required dimensions and the negative/positive quality.

    As argued in (1) a quantum free particle has a velocity v=x/t, but the classical action = -Et+px is unchanged if t=1/E and x=1/p. Thus we argue that there may be periodic fluctuations about the classical motion v=x/t. In other words there are uncoupled t and x fluctuations which are linked to p (i.e. v) and are systematic and periodic. These define spatial density.  v=x/t defines velocity and has two dimensions and the notion of positive and negative x. We argue that the fluctuation scheme must also define velocity and so requires two dimensions and a notion of positive/negative. As a result the mathematical result of a wavefunction component  cos(px) or sin(px) taking on a negative value accounts for information which cannot be lost. For a particle with a single p in a box with infinite walls, a positive or negative hump represents the same probability (sin(px)sin(px) C), but one does not account for the direction of motion. Even using one dimension cos(px) as in a bound state problem, the information of negative and positive values must be retained (in cos(px) over x) because the negative/positive property is used to define overall velocity (even if it is 0 on average for a particle in a box). Thus the concept of interference arises because one cannot lose information in the quantum probabilistic scheme. As noted before it seems one defines new “density” and kinetic energy dimension exp(ipx) and pp/2m exp(ipx) etc.