I Units analysis for this phase shift problem using a differential equation

[jhonconnor]

TL;DR Summary

I'm doing unit analysis to understand phase shift using a differential equation. That phase shift depends on some constants and a sine function that includes the phase shift. My problem is when I simplify units of constants, I have one unit and not “radians” or adimensional unit. Is necessary expressing that phase in radians, or can I express as I get?

I'm trying to solve an ED numerically, but before to doing it I try to understand the system physically according to nuclear scale. In most books and articles use MeV for energy and mass energy and fm to represent distances and phase shift of wave functions are in radians or degree. But when I analyze the problem I found that phase have 1/fm unit and not radians or adimensional unit. I can "arrange" the expresion using V in MeV fm units but don't make to much sense to me. Someone could explain to me what phase shift means physically?

 

[Baluncore]

Someone could explain to me what phase shift means physically?

A phase shift is usually specified mathematically as an angle in radians.
Technicians may convert that to angular degrees, if that is more convenient.

For a wave of period T, a phase shift can be specified as a time, t.
For a wave of wavelength λ metres, a phase shift can be specified as a length.
Those are proportional, to one period, to one wavelength, or to one full cycle.

Phase shift can therefore have units of time, length or angle, depending on how it is specified in your derivation.

If the phase shift was specified by time, t, then;
Phase shift = 2π ⋅ t / T , radians.

In your case, if the shift is specified by, fm = femtometre, a length, then;
Phase shift = 2π ⋅ fm / λ , radians; where λ is also specified in units of fm.

 

[sophiecentaur]

Phase shift can therefore have units of time,

Carefull here. A time delay is the same for all frequencies but the phase shift φ for a uniform time delay t will be a function of frequency. So Δφ = fΔt. In practice, a general waveform that's delayed by Δt remains the same shape but becomes distorted (in time) by the same phase shift of Δφ for all frequencies.

I'd hesitate to try messin' with the basics just for the sake of it seeming to make sense. I know that many branches of Physics use their own private set of units for mass, energy etc. but that's just for convenience and relates to the particular measurement method and they all justify what they do for the sake of scales on graphs etc.. But nothing changes fundamentally, (IMO).

 

[mjc123]

I'd check if you have copied your first equation correctly. δ must be dimensionless, as it is an argument of sin. But the RHS has dimensions of 1/L.