Lorentz Invariant -Et+px Vs. Nonrelativistic Oscillator Scaling and Quantum Wavelength

 In a previous note (1) we argued that the notion of a quantum wavelength follows from the comparison of the classical energy conservation equation for an oscillator with the time-independent Schrodinger equation. In particular for the classical oscillator pp/2m + k/2 xx = E transforms into cos(wt)cos(wt) + sin(wt)sin(wt) = 1 which shows the oscillatory nature of the system explicitly. The time independent Schrodinger equation must also fit this form in the sense that kinetic energy and potential energy must become functions of a variable y such that there are no coefficients. E on the RHS is multiplied by a factor and cannot depend on parameters such as k or m and so is proportional to sqrt(k/m) i.e. the angular frequency.

As a result momentum squared scales as 1/xx and momentum as 1/x. Thus momentum is of the form of 1/wavelength.These arguments seem to rely only on nonrelativistic physics. 

   In a previous note (2) we argued that the quantum wavelength follows from p scaling x in the Lorentz invariant -Et+px. This is strictly a relativistic argument so how can it be compatible with the nonrelativistic oscillator argument suggested above? 

    We use the result of (3) in which we argue that for a relativistic Klein-Gordon oscillator near the turning points velocity must be very low and then become 0. Thus a relativistic oscillator equation should become a nonrelativistic one in the neighbourhood of turning points. In other words the time-independent Schrodinger equation should hold in such a neighbourhood. Classically the form cos(wt)cos(wt)+sin(wt)sin(wt) should also hold. Thus p behaving as 1/x must apply both in the relativistic scenario because one has the Klein-Gordon equation, but also due to the classical oscillator scaling situation because the time-independent Schrodinger equation holds in the vicinity of the endpoints. Thus both arguments for p behaving as 1/x should apply and be consistent with each other.