Why fundamental quantization of energy is hv?

[janakiraman]

Now this might seem to be a very stupid question. But neverthless, I don't understand why the fundamental quantization of energy must be hv? why not any value lower or higher like hv^2 or h/v^2. Is it possible to prove that this value of quantization is most favourable than any other value?

I'm not sure if I had put my question in an understandable form. But it would be great if someone could enlighten me on the same

 

[Pengwuino]

I don't really know what you mean by "fundamental quantization of energy" but first of all, hv^2 or hv^3 isn't even dimensionally correct. I think the answer you're looking for is simply that's how it is. We didn't ask for it to be that but that's what it is. Energy is proportional to the frequency and h was a proportionality constant to be determined.

 

[janakiraman]

Ok that's how i always get started off with, very dubious :). Well ok I can put it this way. I know E is proportional to frequency. Accepted. And why is the quantized part needs to be hv? Why can't I have some different value for 'h'? I think when Planck and Einstien proposed it, they had calculated the Planck's constant based upon the existing experimental results of black body radiation and photo electric effect and so the value of 'h' perfectly fitted with the curves. My question is can we prove that indeed this is the most favorable value for energy to be quantized and not any energy above or below it?

 

[Fredrik]

Isn't Planck's constant defined as the number that makes $E=h\nu$?

 

[Pengwuino]

Yes, that is the experimentally verified value. That's it. It's like gravity being approximately 9.8 m/s^2 near the surface of Earth. You can't ask it to be something else, that's simply what nature gave us.

 

[Count Iblis]

In natural units this is:

energy = frequency

We also have that:

momentum = wavevector

If you combine the two relations into one relation between four-vectors, you get:

(energy, momentum) = four-wavevector

Can we have a different relation? You could speculate that at high energies gravity will become important. The Schwarzschild radius is proportional to the energy, while the characteristic length scale in quantum mechanics is the Compton wavelength which is inversely proportional to the total (rest) energy.

 

[Bob S]

Isn't Planck's constant defined as the number that makes $E=h\nu$?

The present most precise value of the Planck constant is based on a measurement at NIST using a torsion balance, kilogram mass, a superconducting coil, and an induced voltage based on Faraday's Law. See
http://www.aip.org/png/html/planck.htm

 

[sokrates]

The present most precise value of the Planck constant is based on a measurement at NIST using a torsion balance, kilogram mass, a superconducting coil, and an induced voltage based on Faraday's Law. See
http://www.aip.org/png/html/planck.htm

I think he's referring to the historical, initial definition, which as far as I know is correct.

Planck's constant was defined as the proportionality factor relating the energy of a photon to its frequency.

 

[Fredrik]

As far I can tell (by taking a quick look at Wikipedia), I was right about the definition. The page that Bob S linked to describes how to measure Planck's constant, and suggests that this method might be used to find a better definition of a kilogram than the current one. It doesn't really say anything about the definition of Planck's constant.

 

[f95toli]

Just to clarify:
The basic idea of the Watt balance is (or at least was) to measure Planck's constant with very high precision and then use that value for a new definition of the kilogram.
It is (was) essentially an attempt to improve the SI; i.e. it was in no way an experiment in "fundamental" physics so it is more or less irrelevant for this discussion.

 

[Count Iblis]

The definition of Planck's constant can also be interpreted as a matter of defining units. I.e. allowing one to use inconsistent units (for energy, time, momentum, length etc.) and then having to introduce a conversion factor to compensate for that inconsistency.

Michael Duff has argued strongly in favor of this view, see e.g. here:


http://arxiv.org/abs/hep-th/0208093

 

[canoe]

Planck's constant is a constant of nature. Historically (circa 1900), it was empirically derived by Max Plancks's examination of the emissive power of a black body. Planck originally fit the curve of black body radiation based on data obtained from the experiments of Lummer and Pringsheim. In order to do so he had to make the assumption that E=hf. Later, Planck went back and formulated his radiation law based on thermodynamic principles. However, he could never justify his original assumption that E=hf. Planck just thought it was a useful construct. In 1905 Einstein derived a linear relationship (in his explanation of the photoelectric effect) between the frequency and the energy of a light particle "(photon)" where h turned out to be the slope of the relationship. Einstein was basically saying that E=hf was more than a useful construct - that it was real. Millikan later proved Einstein's Theory regarding the photoelectric effect experimentally and also found Planck's constant within his results.

So we can assign it (h) a precise definition based on experiment, but we don't know where it comes from. It is a constant within nature that has never been explained or derived by theory.

I had read about this just recntly and actually found an old schematic of the device used by Lummer and Pringsheim. It consisted of a cavity oven, a focusing spectrometer, a bolometer darkened with lamp black, and a bridge circuit.

 

[Bob S]

Early measurements of h, or actually h/e were done by measuring the end point (short wavelength limit) of the x-ray spectrum with a defined voltage on an x-ray tube. This is W. K. H. Panofsky's thesis in 1941:
http://etd.caltech.edu/etd/available/etd-06182004-143223/
There were also related wavelength measurements of x-ray lines using Bragg diffraction or reflection. The development of the Josepheson junction (and SQUIDS) in the late 1960s repaced the old Planck's constant measurement with a new fundamental constant, h/2e = magnetic flux quantum = 2.0578 x 10^-15 tesla-m². This changed h by about 60 parts per million. So the present value of Planck's constant does not depend on measuring the wavelength of photons.

 

[canoe]

...So the present value of Planck's constant does not depend on measuring the wavelength of photons.

Yes. Planck's constant is ubiquitous. It pops up all over the place.

 

[janakiraman]

Yeah that was my basic idea, how did it come up. But I have two important questions from the replies above

@Pengwuino

Well i don't know much about general relativity, but i had thought that based upon the mass of Earth and other related values, you could calculate the gravity of Earth to be 9.8 m/s^2. I'm surprised to know that its stilla fundamental constant like 'h'

@canoe
variation of 60 parts per million of 'h' is not a small value if I'm right. I have also read that based on qm, we were able to calculate the energy level of Hydrogen spectrum to a very very accurate value of around 1 part per million. If that was true, how did this change modify those results?

 

[Count Iblis]

So we can assign it (h) a precise definition based on experiment, but we don't know where it comes from. It is a constant within nature that has never been explained or derived by theory.

If you accept quantum mechanics as the absolute truth, then the explanation is simple, essentially given by Michael Duff in his article:

http://arxiv.org/abs/hep-th/0208093

So, it is simply a conversion factor. If the true laws of physics say that
X = Y, but for historic reasons you always measure X and Y in incompatible units (which have been assigned incompatible dimensions because we used to think that there was no way you could combine X and Y), then the equation X = Y will appear as X = r Y where r is some conversion factor that will have the dimensions of X/Y.

So, we see that Planck's constant as no physical meaning whatsoever in this picture. One can set hbar = 1. This then means that time is given the same dimension as inverse energy and momentum the dimension of inverse length. If we set c = 1 too, then time and length have the same dimensions. Then if we set G = 1, everything becomes dimensionless.

Of course, physics is only about dimensionless numbers, so this is the way it should be.

 

[canoe]

Yeah that was my basic idea, how did it come up. But I have two important questions from the replies above

@canoe
variation of 60 parts per million of 'h' is not a small value if I'm right. I have also read that based on qm, we were able to calculate the energy level of Hydrogen spectrum to a very very accurate value of around 1 part per million. If that was true, how did this change modify those results?

I'm not sure. I was interested in Planck's derivation of his radiation law so I had delved into the Lummer and Pringsheim experiment. Although, I am certain that more recent experiments have refined h, I'm not acquainted with them and so I have no knowledge on their perceived variation. I say perceived variation (from some norm) because h is still a natural constant. It is what it is...and the only reason we get more refined values is that we are progressing in becoming less clumsy in its measure.

I am just now looking at a textbook where the ground state equation of a Hydrogen atom has been derived from the Schrodinger equation. it would seem that there would be variation in not only h-bar but the coulomb constant, and the fundamental charge that would far outweigh 1 part per million. Where did you read that? Are you certain that [they] were not referring to a variation in measure?

 

[Phrak]

There are at least two ways to read your question. The first way, is what almost everyone is answering--that energy is a different way of measuring 1/T.

For the second, what would happen if the proportionality constant between energy and frequency were to become everywhere different in the universe? Would the difference be physically measurable, or would some hypothetical outside observer simply notice that we were no longer using the same span for the length of a second anymore?

You could ask the same of c, units conversion factor for length and time, or the gravitational constant.

The second way to answer is to tell what physics would be like if h were a local variable--if it changed over spacetime regions. I don't know what the answer is, but I'm sure it's interesting.

 

[canoe]

So, we see that Planck's constant as no physical meaning whatsoever in this picture. One can set hbar = 1. This then means that time is given the same dimension as inverse energy and momentum the dimension of inverse length. If we set c = 1 too, then time and length have the same dimensions. Then if we set G = 1, everything becomes dimensionless.

Of course, physics is only about dimensionless numbers, so this is the way it should be.

You have just derived Planck units. If we set hbar=1, c=1, and G=1 then by definition length (space), time, and mass had to be unity (i.e. 1).

http://en.wikipedia.org/wiki/Planck_units

However, we still are pretty much rooted in anthropocentric units (kg, m, s) when performing experiments.

 

[Phrak]

You have just derived Planck units. If we set hbar=1, c=1, and G=1 then by definition length (space), time, and mass had to be unity (i.e. 1).

I don't know what you mean. I'm still free to pick any length as a unit length. I pick one lamp length, you pick 1 tree; they are not the same span.

 

[canoe]

There are at least two ways to read your question. The first way, is what almost everyone is answering--that energy is a different way of measuring 1/T.

For the second, what would happen if the proportionality constant between energy and frequency were to become everywhere different in the universe? Would the difference be physically measurable, or would some hypothetical outside observer simply notice that we were no longer using the same span for the length of a second anymore?

You could ask the same of c, units conversion factor for length and time, or the gravitational constant.

The second way to answer is to tell what physics would be like if h were a local variable--if it changed over spacetime regions. I don't know what the answer is, but I'm sure it's interesting.

That is an imaginative and fascinating thought...and I mean that in a very positive way. If it weren't late and I have to work in the AM, I would kick that can around for awhile. I might want to get back to you on that.

 

[canoe]

I don't know what you mean. I'm still free to pick any length as a unit length. I pick one lamp length, you pick 1 tree; they are not the same span.

Just take a quick gander at this url...right under Table 2.

http://en.wikipedia.org/wiki/Planck_units

 

[janakiraman]

@canoe

I don't remember exactly, but i definitely remember the precision of calculation was extraordinary. I'm trying to recollect it but not sure

@phrak

Its an amazing conception, you have put down things in a beautiful and a clear way about the vague things that i had thought, but it would be good if someone could think about it.

Also could someone answer my question related to gravity constant of 9.81 m/s^2. Is it not possible to derive it from general relativity? Also another interesting thing that emerges out of discussion (atleast in my understanding) is the linear relationship between the energy and frequency irrespective of the kind of vibration i.e., sinusoidal like EM radiations or the longitudinal vibrations like the lattice vibrations or phonons. its quite remarkable that this relationship also holds good for the entire frequency spectrum ranging from new hertz to order of gigahertz. It is something really fundamental and spectacular i guess.

 

[Borek]

i had thought that based upon the mass of Earth and other related values, you could calculate the gravity of Earth to be 9.8 m/s^2. I'm surprised to know that its stilla fundamental constant like 'h'

9.8 ms^-2 is not a fundamental value, but I think G - gravitational constant - is, in the same way h is. Both are proportionality constants that we can't calculate, we can only measure them.

 

[Bob S]

Let's first list some quantities that are absolute identifiable quantities, except that they depend on the system of units:

1) speed of light, c
2) charge of electron, e
3) etc.

Now let's list some quantities that are absolute, meaning unitless:

1) fine structure constant α = 2 pi e²/hc =1/137.036
2) etc.

Now what is Planck's constant?

h= 2pi 137 e²/c; or h-bar = 137 e²/c

We could select a system of units where we define e = c = 1, then h-bar = 137.

[Added edit]
add electron mass mc²
Rydberg energy = α² mc²/2 = mc²/(2 137²)
(defines energy scale for photons)

α β γ δ ε ζ η θ ι κ λ μ ν ξ ο π ρ ς σ τ υ φ χ ψ ω

 

[diazona]

<nitpick>Actually, it's
$$\alpha = \frac{e^2}{4\pi \epsilon_0 \hbar c}$$
(http://physics.nist.gov/cgi-bin/cuu/Value?eqalph|search_for=fine+structure+constant) If you select a system of units where $e = c = 1$, you would still have the freedom to choose $\hbar = 1$, but then your value of $\epsilon_0$ would be determined such that $\alpha = 1/137.036...$.</nitpick>

I made a post (which I consider interesting) about this very topic on this thread...

 

[Bob S]

Continuation of post 25
Rydberg energy E_R = α² mc²/2

So E_R/h = [α² mc²/2] [αc/(2 pi e²)]
is the frequency of a Rydberg photon (13.605691 eV)

α β γ δ ε ζ η θ ι κ λ μ ν ξ ο π ρ ς σ τ υ φ χ ψ ω

 

[Count Iblis]

You have just derived Planck units. If we set hbar=1, c=1, and G=1 then by definition length (space), time, and mass had to be unity (i.e. 1).

http://en.wikipedia.org/wiki/Planck_units

However, we still are pretty much rooted in anthropocentric units (kg, m, s) when performing experiments.

Indeed. It is also interesting to take the opposite perspective, i.e. start from a purely dimensionless setting and then ask where the incompatible diminsions and dimensional constants come from. The only way that can happen is by introducing rescaling constants, rescaled physical variables and then study how the rescaled physical quantities depend on each other in the scaling limit where the rescaling constants tend to zero or infinity.

In the scaling limit you can lose relations that do exist at the fundamental level. We then get independent physical variables which at the fundamental level are not independent. This is then what we could call the classical limit. But the real world we live in is not quite the classical limit; the rescaling constants hbar, c, G, k_b etc. are not quite zero or infinite.

 

[janakiraman]

Indeed. It is also interesting to take the opposite perspective, i.e. start from a purely dimensionless setting and then ask where the incompatible diminsions and dimensional constants come from. The only way that can happen is by introducing rescaling constants, rescaled physical variables and then study how the rescaled physical quantities depend on each other in the scaling limit where the rescaling constants tend to zero or infinity.

In the scaling limit you can lose relations that do exist at the fundamental level. We then get independent physical variables which at the fundamental level are not independent. This is then what we could call the classical limit. But the real world we live in is not quite the classical limit; the rescaling constants hbar, c, G, k_b etc. are not quite zero or infinite.

Can you explain in detail about what you have said? Maybe with a good example. I'm not getting anywhere by reading this :)

 

[Count Iblis]

Ok., let's do this in case of special relativity where we start with setting the speed of light c equal to 1. The starting point is then a setting where space and time have the same dimensions. As a consequence all speeds are dimensionless. We want to derive the classical limit from this setting.

Let's look at the expression for energy and momentum:

$$E = \gamma m$$

$$\vec{P}= \gamma m \vec{v}$$

where

$$\gamma = \frac{1}{\sqrt{1-v^2}}$$

Suppose we are interested at what happens at very low velocities, e.g. collisions between massive objects at extremely low velocities (note that v is dimensionless, so we can call velocities small in an absolute sense). We can then simply expand the above equations in powers of v and keep only the leading velocity dependent parts:


$$\vec{P}= m \vec{v}$$

$$E = m + 1/2 m v^2$$

These relations are not exact, we have ingored higher order tems in v. Now, the classical limit is not obtained by simply letting v tend to zero, as then the momentum becomes zero and the energy is equal to the mass and then nothing interesting is visible. Instead, when we approach the low velocity regime we need to zoom in by rescaling the velocity, so that when we approach the limit at which v has gone to zero, we still have a finite rescaled velocity. This means that phenomena that are infinitessimally small in the v = 0 limit remain visible.

So, let's introduce a rescaled velocity $\vec{v}_{r}$ by putting:

$$\vec{v}= \frac{\vec{v}_{r}}{c}$$

where c is an arbitrary rescaling constant. We can now approach the limit of v to zero by keeping $v_{r}$ fixed and let c go to infinity, so the velocity does not become invisible. In terms of $\vec{v}_{r}$, the equations become:

$$\vec{P}= m \frac{\vec{v}_{r}}{c}$$

$$E = m + 1/2 m \frac{v_{r}^2}{c^2}$$

We see that in the limit c to infinity the momentum becomes zero, so it makes sense to define a rescaled momentum:

$$\vec{P}_{r} = \vec{P} c$$

Then, in terms of rescaled variables, we have:

$$\vec{P}_{r}= m \vec{v}_{r}$$

We can then take the limit of c to infinity and still have a relation between finite quantities. In this limit, the higher order tems we have neglected are exactly zero.

If we look at the energy equation, we see that the energy becomes the mass in the limit c to infinity. Since energy is conserved, this means that the total mass is conserved in the classical limit (one thus assumes that when we let c go to infinity the rescaled velocities stay finite). The kinetic energy is given by:

$$E_{k} = E - m = 1/2 m \frac{v_{r}^2}{c^2}$$

This goes to zero in the limit c to infinity, so it makes sense to rescale the kinetic energy by multiplying it by c^2:

$$E_{k,r} = E_{k} c^2 = 1/2 m v_{r}^2$$

Then we can call the rescaled kinetic energy simply "the energy", and the original energy can be called the mass. Conservation of energy thus implies conservation of mass and of kinetic energy separately.

Note that in the original theory mass was simply the rest energy and not an independent physical quantity. But in the rescaled theory when we let the rescaling parameter tend to infinity mass has become independent of energy.

 

[Phrak]

That is an imaginative and fascinating thought...and I mean that in a very positive way. If it weren't late and I have to work in the AM, I would kick that can around for awhile. I might want to get back to you on that.

@canon and janakiraman

It's one answer to the question "What would happen if h were different, or if c were different?", where usually the question is pu-poohed, and the questioner is left unsatisfied?

Those who know the least physics have the best questions, in my opinion. Why?

If you have the background, http://en.wikipedia.org/wiki/Gauge_theory should be an interesting place to start, under Classical Gauge Theory.

 

[canoe]

@canon and janakiraman

It's one answer to the question "What would happen if h were different, or if c were different?", where usually the question is pu-poohed, and the questioner is left unsatisfied?

Those who know the least physics have the best questions, in my opinion. Why?

@Phrak

Here it is late again, and I have about 3 minutes...but if h changed as conjectured than uncertainty could likley be causal.

 

[Phrak]

...Here it is late again, and I have about 3 minutes...but if h changed as conjectured than uncertainty could likley be causal.

I'm not sure what you mean, but if h were not everywhere constant, then it would give rise to a field. What sort of field? I don't know. This is the basis of quantum field theory. The mathematic basis qft originated with the connection coefficients found in general relativity.

 

[diazona]

9.8 ms^-2 is not a fundamental value, but I think G - gravitational constant - is, in the same way h is. Both are proportionality constants that we can't calculate, we can only measure them.

Yeah... and on that note (for completeness, since nobody's posted this yet): the gravitational acceleration $g = 9.8 \mathrm{m}/\mathrm{s}^2$ comes from Newton's universal law of gravitation (and second law of motion),

$$F = G \frac{Mm}{R^2} = mg$$

with M as the mass of the Earth and R its radius.

$$g = G\frac{M}{R^2} = \left(6.67\times 10^{-11}\frac{\mathrm{m}^2}{\mathrm{kg}\cdot\mathrm{s}^2}\right)\frac{5.9736\times 10^{24}\mathrm{kg}}{(6371\mathrm{km})^2} = 9.8\frac{\mathrm{m}}{\mathrm{s}^2}$$

In general relativity, the equation is slightly different (I don't remember exactly what the higher-order corrections are) but the procedure is basically the same.