Dimensions of P and ##\omega##

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In summary, the conversation discusses the use of natural units in polymer quantum mechanics and how it affects the dimensions of momentum and other quantities. The conversation also delves into the measurement of mass in terms of GeV and the conversion between different units in natural units. The concept of natural units is explained as well. The conversation ends with a mention of the measurement of temperature in GeV when ##k_{\text{B}}## is also set to 1.
  • #1
AHSAN MUJTABA
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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?
 
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  • #2
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}##.
 
  • #3
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}##?
 
  • #4
I need to understand these dimensions as I am making some equations dimensionless for my tasks.
 
  • #5
AHSAN MUJTABA said:
they say
Who says? Can you give a specific reference?
 
  • #6
AHSAN MUJTABA said:
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.
 
  • #7
AHSAN MUJTABA said:
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.
 

Related to Dimensions of P and ##\omega##

What are the dimensions of P?

The dimensions of P depend on the specific quantity it represents. For example, if P represents power, its dimensions would be energy divided by time (e.g. joules per second). If P represents momentum, its dimensions would be mass multiplied by velocity (e.g. kilograms times meters per second).

What is the significance of the symbol ##\omega##?

The symbol ##\omega## is often used to represent angular frequency, which is a measure of how quickly an object rotates or oscillates. It is typically measured in radians per second.

How are P and ##\omega## related?

P and ##\omega## are related in several ways, depending on the specific context. In general, P is often proportional to ##\omega##, meaning that an increase in one will result in a corresponding increase in the other. For example, in the case of a rotating object, the power required to maintain that rotation will increase as the angular frequency increases.

Can P and ##\omega## have different units?

Yes, P and ##\omega## can have different units depending on the specific quantities they represent. For example, if P represents power in watts and ##\omega## represents angular frequency in radians per second, they have different units but are still related in terms of their dimensions.

How do P and ##\omega## affect the behavior of a system?

P and ##\omega## can have a significant impact on the behavior of a system, depending on the specific system and the relationship between these quantities. In general, an increase in P can result in an increase in the energy or momentum of a system, while an increase in ##\omega## can result in a change in the speed or frequency of oscillation. Understanding the relationship between these dimensions is crucial for studying and predicting the behavior of physical systems.

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