Speculation on the Interplay of X and Time in Quantum Bound States

 In Newtonian mechanics, one often thinks in terms of x(t) i.e. an association between x and t. Although this is part of Newtonian mechanics, the notion of motion always being linked to time (and so an internal clock being required) is not always the case. For example, two particles with different masses and different velocities such that their momenta are the same may strike a target and reflect with the same magnitude of momenta. The impulse is 2p for both even though they have different speeds. One does not need to consider their speed to discuss the impulse change and so an external clock is not needed. 

   This is further emphasized by the Lorentz invariant A = -Et+px which separates x and t as if they were independent variables (even though implicitly a speed exists in the problem i.e. v=dx/dt.). The point is that there exists a subset of physics (i.e. interactions) which is not dependent on velocity and hence the presence of a clock. The impulse hit discussed above is one example, but interference through 2 slits etc is another. Two quantum particles with the same momentum, but different rest masses and velocities produce the same interference patterns. 

  A particle with momentum, however, has energy and so both time and space are running variables, but there are problems in which one considers p without a clock. Interestingly, even if one thinks in terms of impulses without using a clock, if there is an x interval between impulses then the notion of velocity comes into play even without an external clock because there will be more impulses imparted by a fixed p with a higher velocity than a lower one in the same fixed time period. Without using the idea of a clock, this give rise to the observable called pressure in classical physics, but is also present in a quantum bound state, we argue. Pressure in classical physics is proportional to temperature which in turn is proportional to energy (and a running time). In a quantum bound state, the equivalent to pressure (namely impulse hits with a potential) also require an energy viewpoint which is linked to running time i..e exp(-i En t) where En is the bound energy.