Consequences of X-Time Independence Within a Quantum Wavelength

We have argued in previous notes (1) that x and t are independent within a wavelength. This we will argue follows from special relativity applied to uniform motion i.e. -Et+px. In particular, v=x/t on average, not dx/dt, a mathematical abstraction which breaks down within the wavelength, because one cannot follow the particle in time. We argue that the notion of independent x and t within a wavelength implies a probability distribution of x i.e F(p,x) = the probability to find the particle with momentum p at x. From special relativity we suggest that E is associated with t and so an E(x) suggests also a t(x). (In fact, for KE(x), there is an associated time at each x, but for KE(x)+V(x)=E all x are the same and there is no need for a distinguishing variable time.) What about the spatial distribution F(p,x)? For a free particle F(p,x) is a function of x, but one which must map into a P(x)=constant. (For a bound state there is a spatial distribution and so the idea of time should exist in that a particle spends “more time” in a high probability region. This is linked to the idea of a p(rms) or velocity(rms)). Even for a bound quantum state we argue that for x and t to be independent, E must be constant at each x. One may argue that a constant E with a p(x) and an introduced V(x) is already present in special relativity i.e. (E-V(x))(E-V(x)) = pp + momo (c=1) which it is, but we argue that x-t independence within a wavelength also follows from special relativity so this seems to be consistent. In earlier notes we suggested that E constant at each x is an example of equilibrium being linked with a minimum amount of information. Here we show that x-t leads to this minimum amount of information in describing E. (One may note that given a spatial density, there is quite an amount of information present there, but it is linked with the -id/dx operator which is associated with momentum and KE(x) which has time information.) A second related point which follows from x-t independence is the following. Consider an ensemble of classical bounds states. At an instant of time and a given x, the probability to find the particle moving in the forward and backward directions is the same. If x and t are independent there is no instant of time and F(p,x) and F(-p,x) are not the same. In fact, F(p,x) must show how the particle moves in a forward direction and F(-p,x), motion in a negative direction. As a result, one has constant E at each x in a quantum bound state (i.e. locality), but F(p,x) and F(-p,x) being different suggests that there is nonlocality associated with momentum as it is linked with d/dx and hence x and a wavelength (which is nonlocal i.e. does not exist at a point). Thus conservation of momentum i.e. an average momentum of zero in a bound state may only be established by integrating over space - it does not hold at a point on average like in a classical system where P(p,x) = P(-p,x). There are in a sense two average momenta in a quantum bound state. The first is linked with F(p,x) not equalling F(-p,x), but E being constant at each x. This is a nonlocal average momentum and includes both p and -p with no time present. We argued above that spatial density varies in x and so one may assign a time to the system. This time is associated with a prms(x) which follows form KE(x) = prms(x)prms(x)/2m used in KE(x)+V(x)= En. Thus there are two “velocities” or momenta. One is a prms (which is classical and associated with time and spatial density) and the other which is nonlocal and linked to x-t being independent meaning that E must be constant at each x