Speculation on the Correspondence Principle and X and T Independence

   Traditionally a high energy solution of the time-independent Schrodinger equation is compared with a classical result. In particular, the quantum spatial density W(x)W(x) where W(x) is the bound wavefunction has humps and zero regions, but an envelope through the peaks is roughly equal to C/v(x) where v(x) is the classical velocity. W(x)W(x) is the spatial density and C/v(x) is proportional to dt, the time a particle spends in a fixed dx length.

    In this note we make use of (1) namely that the free particle classical action A (relativistic or nonrelativistic) may be written with v=x/t and x and t varied independently yielding A= -Et + px. A is a Lorentz invariant, but with x and t independent this implies Et and px are constant. This suggests a characteristic length proportional to 1/p and a characteristic time proportional to 1/E as suggested in a previous note. A probability distribution is required to establish 1/p units in space such that it goes to 0 at the start and endpoints of 1/p. This implies a periodic probability function, but the particle has equal probability to be at any x so a two dimensional form with two periodic functions such that the modulus is 1 is needed.

   We argue that the basic humps and troughs in a quantum bound state are due to Lorentz invariance and x and t independence. Classical mechanics on the other hand is based on x(t). Thus the two are incompatible. The constraint:  KE average + V(x) = En is enough to allow for relative peak heights to be in the ratio v(x2) / v(x1), but the spatial density humps and the independence of x and t are uniquely quantum mechanical. This begs the question: Are these humps physical? It seems that experiments suggest that large objects (e.g. 70 kg) (2) may have quantum motion suggesting that x(t) may be an approximation.