Can the Energy of a Photon be Expressed in h/s?

[DennisN]

Could you give me some examples of constants with units so that I can get in the picture?

Sure! I'll also give some dimensionless constants, so we can see the difference better.

Mathematical constants (which are dimensionless):

pi (the ratio of a circle's circumference to its diameter)
e (the base of the natural logarithm)

Physical constants that are dimensionless:

α: the fine-structure constant
μ: the proton-to-electron mass ratio

Physical constants that are not dimensionless:

h: Planck's constant
c: speed of light in vacuum
G: gravitation constant
e: the elementary charge
k_e: Coulomb's constant
m_e:the mass of the electron
m_p:the mass of the proton
etc.
etc.

To see these constants (and some more) and their units, see: Fundamental Physical Constants (HyperPhysics).

 

[BruceW]

I don't see the problem with using $h$ (or $\hbar$) as a unit. For example, the orbital angular momentum of an electron in the 'p subshell' is given by: $L=\sqrt{2} \hbar$ hmm. I guess here you guys would say that the equation is simply comparing two quantities. But If we define angular momentum as one of our dimensions, then $\hbar$ can be a unit, right? I mean, surely it just depends on how we define things.

 

[Dale]

In Planck units the value of $\hbar$ is 1. So $\hbar$ could be considered the unit of action in Planck units. However, since the same symbol is used elsewhere for the quantity, I would not call it a unit since it would lead to confusion. I would call the unit the Planck action and say that 1 Planck action = $\hbar$.

 

[BruceW]

well, in Planck units, angular momentum is dimensionless. So it doesn't need units, right?

 

[Dale]

I don't think so. I think that in Planck units angular momentum has units of (Planck mass)(Planck length)²/(Planck time).

I.e. in Planck units I think that the dimensionful constants of nature are still considered dimensionful, it is just their values which are set to 1. I think that geometrized units consider the constants of nature to be dimensionless.

 

[BruceW]

yeah, that maybe right. I am not 100% sure on this stuff. I know that people will write stuff like "The rest mass of the electron is $0.51 \ \mathrm{MeV}$ ." So in this case, they are using Planck units where speed is dimensionless.

 

[Imabuleva]

The unit is part of the number, Bobie. When we say 0.51 MeV or 5 J*s, what we mean is multiply Joule times second times 5 or 0.51 times Mega-electron-volt. 0.51 means nothing without multiplying it by Mega-electron-volt.

 

[Imabuleva]

well, in Planck units, angular momentum is dimensionless. So it doesn't need units, right?

In Planck units, we define hbar to be 1. In this case, hbar still has units of angular momentum, but we're now working in angular momentum space.

 

[dauto]

You can, if you want, define h as a unit. What you cannot do - and that's the problem with bobie's logic - is to say that h = J*s. He repeated that claim many times.

 

[dauto]

Sorry, if I misunderstood, but I thought that h=J.s? wiki:
h = 1.054571726(47)×10−34 J·s
. Can two different units have same dimensions?

You cannot simply drop the 1.054571726(47)×10−34 factor out of the equation and expect it to still be correct.

Yes different QUANTITIES may happen to have the same dimension. Compare the unit for energy with the unit for torque.

 

[BruceW]

I think what bobie meant was that h has the same dimension as J.s (although maybe he did not explain what he meant very clearly). I don't think he was actually saying that it is possible to just drop the 1.05 X 10^-34

 

[BruceW]

In Planck units, we define hbar to be 1. In this case, hbar still has units of angular momentum, but we're now working in angular momentum space.

If we're using proper terminology, that should really be "hbar still has the same dimensions as angular momentum". I'm not sure what you mean that we are working in angular momentum space. Do you mean angular momentum is also dimensionless, if we define hbar to be dimensionless? I would agree on that.

 

[dauto]

I think what bobie meant was that h has the same dimension as J.s (although maybe he did not explain what he meant very clearly). I don't think he was actually saying that it is possible to just drop the 1.05 X 10^-34

Well, that's possible, but that's then a very sloppy way to express that and it's not surprising he got all the push back that he did. I would tell him what I tell my students " The symbol = means one thing and one thing only" It can only be used to express the mathematical identity between the left side and the right side of the equation.

 

[BruceW]

very true. I also don't like it when people write multiple equalities like $expression1=expression2=expression3=expression4...$ When they are trying to derive something. Even though this is correct, it makes it more difficult to see the reason for the logical steps. It's much better for there to be a short sentence giving a reason for each equality, instead of just writing down a string of equalities. (sorry for going off on a tangent).

 

[Imabuleva]

If we're using proper terminology, that should really be "hbar still has the same dimensions as angular momentum". I'm not sure what you mean that we are working in angular momentum space. Do you mean angular momentum is also dimensionless, if we define hbar to be dimensionless? I would agree on that.

Yes, that should say, "hbar still has the same dimension as angular momentum." By angular momentum space, I mean that our quantities will be divided by a factor of h. If the energy of a photon was once h*f, it will now just be 2*pi*f. So energy has dimension of inverse time, whereas it usually has the dimension of mass*length squared / time squared.

 

[BruceW]

ah right, I understand you now :)

 

[bobie]

I think what bobie meant was that h has the same dimension as J.s (although maybe he did not explain what he meant very clearly). I don't think he was actually saying that it is possible to just drop the 1.05 X 10^-34

I was not dropping nor thinking of dimensions, I have explained that I was just moving s from the right to the left side and did not know it is forbidden.
I did not see the necessity of dimensions and you confirmed they are not indispensable.

I do not know if I made it clear that the reason is that for a beginner 1.6*10^-19 J desn't mean much, whereas 2.4 *10¹⁴ Hz, B (or the infamous) h/s gives a clear, immediate picture of the real energy of a photon , in the region of visible light etc.
I was deceived by the fact that many times I have seen even mass converted to Hz , here:
http://en.wikipedia.org/wiki/Kilogram, under the picture "unit conversions" a Kg is converted to Hz
I am also muddled by the fact that two different entities can have same dimensions.

I want to express my sincere gratitude to all of you who have contributed to make this textbook thread about Planck constant.

Can you tell me if this concept of action is found anywhere else in physics, or is just an ad hoc creation for this particular instance?

 

[BruceW]

I was not dropping nor thinking of dimensions, I have explained that I was just moving s from the right to the left side and did not know it is forbidden.
I did not see the necessity of dimensions and you confirmed they are not indispensable.

it's not forbidden as far as I know. But it seems that people like to keep 'units' as a separate concept from 'quantities', although I don't see any fundamental difference.

I was deceived by the fact that many times I have seen even mass converted to Hz , here:
http://en.wikipedia.org/wiki/Kilogram, under the picture "unit conversions" a Kg is converted to Hz

In this case, they are also redefining dimensions, not just converting between units. I think it is sometimes called 'nondimensionalization'. This is the same thing as the electron's rest mass being given the dimensions of energy, and time being given the dimension of inverse energy, which is common among the particle physicists (I think). I mean, I think they like to write everything as having dimensions of energy to some power.

I am also muddled by the fact that two different entities can have same dimensions.

what is confusing, specifically? Or is it just surprising that angular momentum has the same dimensions as energy x time ?

 

[f95toli]

I was deceived by the fact that many times I have seen even mass converted to Hz , here:
http://en.wikipedia.org/wiki/Kilogram, under the picture "unit conversions" a Kg is converted to Hz

They did not "convert" mass to Hz, if you read the note it says that this is the frequency of a photon with the same energy (via the famouse E=mc^2 relation) as 1 kg of mass.

Expressing mass as energy or even temperature as is very common, and is often very convenient; even though it is NOT something one should do without keeping in mind what it is you are really doing.
You can for example if you want you can express nearly everything as temperature, since Boltzmann's constant kB allow you to go from energy to temperature. Hence, if you want you could express mass in Kelvin, but it wouldn't be very useful.
This to some extent "cultural" and depends on the field, particle physicists express nearly everything in energy (electron Volts), people in low temperature physics like Kelvin, in my area we tend to use Hz etc.
But again, none of these are "proper" conversions, they are just used because it simplifies some calculations since you don't need to keep track of some constant and/or it makes the numbers more managable.

 

[BruceW]

I guess you need to keep in mind exactly what you did to obtain the 'natural' equations. i.e. $c=1$ to get the natural equation $E^2=p^2+m^2$. Since if you forget the choice you made and forget the original equation, then you can't get back the original equation.

 

[bobie]

it's not forbidden as far as I know. But it seems that people like to keep 'units' as a separate concept from 'quantities', although I don't see any fundamental difference.
... it is sometimes called 'nondimensionalization'

That is very comforting , so you wouldn't spank me for h/s?

...it says that this is the frequency of a photon with the same energy (via the famouse E=mc^2 relation) as
... in my area we tend to use Hz etc..

That is just what I said in post #9

.. how can we express that the scalar of the frequency is always equal to the scalar of the energy of a photon?

I found this habit everywhere, even in Encyclopaedia Britannica, which is my guidebook, as I can't afford buying texts. After all, dimensions is just a convention. Probably we can say that also scientists use poetic licenses.

 

[f95toli]

After all, dimensions is just a convention. Probably we can say that also scientists use poetic licenses.

Not quite. Everything will ultimatly refers back the SI, which by design is a consistent system of units. The SI is also what use when we realize the units, "realize" in this context means the experimental implementation used for the definitions, e.g. the atomic clocks that are used to realize the second.
While people use a bunch of other conventions, this is just because we are all a bit lazy and want to simplify equations etc. Moreover, everyone who uses these other conventions (e.g. expressing mass in MeV) knows how to convert back the SI (and frequently do so when there is a need).

Hence, if you are a beginner it is generally best to stick the SI, simply because there is less room for confusion.

Note also that it is currently strickly speaking impossible to express mass in eV or Hz (or whatever else you want) if you want an exact result. The reason is that we do not have an exact values for h or e; meaning any conversion is approximate. This is only of academic interest to most people, but is sometimes important in high-precision metrology.

 

[bobie]

I 'll follow in your steps , I'll stick to Hz, no poetic licenses

 

[BruceW]

you mean joules? joules is the SI unit of energy.

 

[BruceW]

Also, about the non-dimensionalized units, it does make sense really, there's no poetic license being used. As long as the person says what kind of system of units they are using, you'll be fine. It is only if they don't say how they have defined things, this could cause problems. For example, they could use $c=1$ or $v_{sound}=1$ where $v_{sound}$ is the speed of sound in air. The first convention means speed will be a dimensionless number as a fraction of the speed of light, and the second convention means speed will be a dimensionless number that is called the Mach number. So if they give you an equation for some velocity $V$ like $V=0.45$ without telling you which convention they are using, this could either mean the velocity is 0.45 times the speed of light, or the velocity is 0.45 times the speed of sound. (or some other convention - you don't know until they tell you what convention they have used).

edit: actually that's not quite right, because Mach number uses the local speed of sound. But you can imagine some 'standard' speed, i.e. speed of sound in still air, at a certain temperature and pressure, with a certain composition.

 

[bobie]

what is confusing, specifically? Or is it just surprising that angular momentum has the same dimensions as energy x time ?

I think dimensions have a sense if they are the fingrprint, DNA of an entity, if the relation is not biunivocal seems more like red tape, a burden; if J.s is the DNA of unrelated conceps like action and momentum I find it confusing. But surely it is my fault.

As to the action, unless someone enlightens me, I cannot imagine to what it corresponds in reality.
I see the sense of *s if an entity has been divided by s to restore its meaning : v = S/T *T = space, distance travelled, P = E/ T * T = E. Likewise, I would be muddled by S*T , what does it mean?
where do you use action, apart from h? that would help me understand!

 

[Dale]

I think dimensions have a sense if they are the fingrprint, DNA of an entity, if the relation is not biunivocal seems more like red tape, a burden

The relation is most definitely not biunivocal. I don't know why you would expect it to be. It isn't even univocal.

 

[BruceW]

Action is a different concept. The time integral of the Lagrangian of the system is the action. But again, action has the same dimensions as energy x time and angular momentum.

Also, J.s is not the S.I. units for momentum. The S.I. units for momentum are J.s/m, or more simply: kg.m/s

 

[Dale]

As to the action, unless someone enlightens me, I cannot imagine to what it corresponds in reality.
...where do you use action, apart from h?

Action is used extensively in Lagrangian mechanics (e.g. the principle of least action):
https://en.wikipedia.org/wiki/Lagrangian_mechanics

 

[bobie]

Physical constants that are not dimensionless:
h: Planck's constant
c: speed of light in vacuum

If c is not dimensionless, how come α is considered dimensionless although it is just 0.00729 c?