I Dimensions of P and ##\omega##

[AHSAN MUJTABA]

TL;DR Summary

Dimensional analysis in quantum mechanics, of physical quantities.

I am studying polymer quantum mechanics. In it, they say that the momentum, $p$ eigenvalue, has the dimensions of $(mass)^{-1}$ and similarly $\omega$ has the dimensions of $mass$. How it is possible, please someone explain that to me. Even a little hint would work.
I don't get it. Also, I would require some assistance regarding the units of Planck's reduced mass, $M_{PI}^{2}$. How can it be measured in terms of GeV?

 

[vanhees71]

I'm not sure about the conventions in the polymer-physics community, but from what you describe, I guess they use a similar convention as we do in HEP physics, i.e., they use natural units, setting $\hbar=c=1$. In such a system of units you have only one dimension left. You can choose energies (MeV or GeV) or lengths (usually fm). Usually one uses GeV for masses, energies, momenta, frequencies, wave numbers etc. and fm for lengths and times. All you have to keep in mind to convert from GeV to 1/fm or from fm to 1/GeV is that $\hbar c \simeq 0.197 \; \text{GeV} \; \text{fm}$.

 

[AHSAN MUJTABA]

Does that imply that the dimension of momentum eigenvalue becomes inverse of mass? You say that we measure mass in terms of GeV. So, due to that unit(GeV), does the dimension of momentum become (mass)$^{-1}$?

 

[AHSAN MUJTABA]

I need to understand these dimensions as I am making some equations dimensionless for my tasks.

 

[PeterDonis]

they say

Who says? Can you give a specific reference?

 

[PeterDonis]

Does that imply that the dimension of momentum eigenvalue becomes inverse of mass? You say that we measure mass in terms of GeV. So, due to that unit(GeV), does the dimension of momentum become (mass)$^{-1}$?

Not in the "natural" units @vanhees71 described. In those units, as he said in post #2, the unit of momentum is the same as the unit of mass, energy, etc.--all are measured in a unit like GeV.

 

[vanhees71]

Does that imply that the dimension of momentum eigenvalue becomes inverse of mass? You say that we measure mass in terms of GeV. So, due to that unit(GeV), does the dimension of momentum become (mass)$^{-1}$?

In the natural system of units, where $\hbar=c=1$ the unit for mass, energy, and momentum is GeV (or any other energy unit you prefer). Lengths and times are usually measured in fm. Angular momenta and actions are dimensionless.

If you also set $k_{\text{B}}=1$ then also temperatures are measured in GeV.