Quantum Conservation and Invariance and id/dt and -id/dx Operators

 It was argued in (1) that Fermat’s least time principle for light (in a two dimensional (x,y) reflection or refraction scenario) minimizes time by actually varying it in terms of a spatially invariant variable i.e. x (i.e. taking d/dx) with the flat face of the mirror/medium along the x axis at y=0. This is equivalent to conserving momentum in the x direction. Thus d/dx is associated with (px) which may be associated with the Lorentz invariant -Et+p dot r. In other words, d/dx emerges as a momentum operator (up to a factor) and d/dt  as an energy operator, yet both are generators in time and space linked with invariance of constant values i.e. conservation. Furthermore an eigenvalue equation: -id/dx exp(ipx) = p exp(ipx) and id/dt exp(-iEt) = E exp(-iEt) automatically (by the nature of eigenvalue) leads to orthogonal eigenfunctions linked to conservation of energy and momentum.

    In special relativity rest mass mo represents a constant in time which may exist at any point x. In fact, it is often thought of as a point, for example a center of mass point. The actual object with mo, however, may be large with moving parts all within a “container”. What then does x,t in -Et+px mean for constant E,p? It seems that x,t are center-of-mass coordinates for the overall system. We argued in previous notes that 1/p is proportional to a wavelength so for p→0, mo may be at any x point with the same probability. Within the system with moving parts, however, there is motion and an x scale for each moving piece. For a single particle bound system, there is a single energy at each x which is like mo and a potential V(x) and KE(x).

    In (2) we argued that given the free particle relationship -EE+pp= -momo (c=1) one may retain a constant E value by replacing E with E-V(x) and pp with <pp> (and not <p><p>). How do the probabilities appear? They are linked to -Et+px with frequency hbar/E and wavelength hbar/p and are specifically manifested by exp(-iEt) and exp(ipx). For <p> and <pp> we use spatial probabilities i.e. W(x) = Sum over p a(p)exp(ipx). Each exp(ipx) is linked to the operator -id/dx and invariance, but pp is linked with the operator -d/dx d/dx and one does not use an eigenfunction of this operator. Thus the notion of conservation of momentum is retained in the exp(ipx)s and their linear combination W(x) yields a function such that an average kinetic energy exists at each x point i.e KE(x)=KEclassical = -1/2m d/dx dW/dx / W. This is combined with a V(x) to ensure constant E at each x point i.e. create a kind of mo piece linked with id/dt. We argue that E being constant at each x is a statistical equilibrium type of result. Each exp(ipx) is associated with an average motion of x=p/m t and given that exp(ipx) and exp(-ipx) are included, there is also the notion of <p> which is positive and negative in different regions and integrates to 0. prms(x) from <pp> becomes the classical p(x).

     Thus bound state quantum mechanics seems to be based on a free particle treatment which follows from the special relativistic Lorentz invariants -EE+pp=-momo and -Et+px = constant, directly linked to conservation ideas. Conservation of momentum of an internal bound particle is maintained through exp(ipx), while <pp> by itself is not linked with conservation as there is no eigenstate, but is combined with a V(x) to create a constant En linked with the entire system (and P=0) which then becomes associated with invariance in time and the generator id/dt.

   In this note, we try to argue that the notion of W*(x)W(x) (where W(x)=wavefunction= Sum over p a(p)exp(ipx)) is spatial density and P(p)=a*(p)a(p) based on spatial invariance and conservation of momentum for a single particle bound state even though there seems to be no spatial invariance or notion of conservation of momentum in W(x) which is simply a solution of the time-independent Schrodinger equation. One may move from a W(x) scenario (i.e. Schrodinger equation) to a W*(x)W(x) picture, by trying to “pick out” invariant (p only terms) using the orthogonality of the exp(ipx)s. Thus W*(x)W(x) (spatial density) emerges naturally in this approach.