# Question about Planck's constant and De Broglie

## Main Question or Discussion Point

Hello everyone,
I've been studying the origins to quantum physics lately. It has been bugging me for a while that everywhere I look for information concerning the equation E=hf, does not explain where exactly the equation comes from. I know Plancks constant comes from the explanation Planck gave to the black body radiation problem. But, how exactly did the equation for the photon, E=hf, appear?

I have another question if you don't mind answering this as well: Given Einstein's equation for total energy E^2 = p^2c^2 + m^2c^4 ... if you want to get the energy for a photon, you have that m=0 so the energy is E=pc. Now, using E=hf, we get
c/f = h/p , so: L=h/p , where L is the wavelength. Then De Broglie says that this is the wavelength for ALL particles. But how can one say this if we used equations for energy that ONLY concern the photon. How can you use the same equation to describe all particles if we made m=0 in the process???

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Ken G
Gold Member
I know Plancks constant comes from the explanation Planck gave to the black body radiation problem. But, how exactly did the equation for the photon, E=hf, appear?
I think you answered your own question-- Planck treated each frequency f as if it could not be excited with any continuous amount of energy, but only energies in little bundles of size h*f. Doing that solved the "ultraviolet catastrophe" in a way that agreed with observations. So the clear interpretation if energy at f comes in bundles h*f is that we have a quantum there, and the quantum got called a photon.
Then De Broglie says that this is the wavelength for ALL particles. But how can one say this if we used equations for energy that ONLY concern the photon. How can you use the same equation to describe all particles if we made m=0 in the process???
We can only say it if it agrees with observations to do so. As I see it, de Broglie was essentially saying, "what if" the rules that govern waves of light also govern the waves that tell other kinds of particles how to behave, what would need to be the salient features of those waves? He took a guess that they should have a wavelength inverse to their momentum and in obeyance with what light does, and sure enough, this did describe particle behavior and helped motivate the Schroedinger equation. Of course, once you have the Schroedinger equation, you don't need de Broglie wavelengths any more, except as a pedagogical tool to help understand what the Schroedinger equation is telling you.

But, how exactly did the equation for the photon, E=hf, appear?
Einstein first identified it in his 1905 paper on the photoelectric effect. He found that the voltage created when light was shone on metal depended solely on the frequency of the light and not the intensity. Voltage translates to energy, and so he was able to calculate the energy produced in each atom by light of a particular frequency was E=hf.

Ken G
Gold Member
This Wiki (http://en.wikipedia.org/wiki/Planck's_law) entry might shed light on the
discrepancy between the above answers as to whether that law originates in Planck's or Einstein's work:

Planck made this quantization assumption five years before Albert Einstein hypothesized the existence of photons as a means of explaining the photoelectric effect. At the time, Planck believed that the quantization applied only to the tiny oscillators that were thought to exist in the walls of the cavity (what we now know to be atoms), and made no assumption that light itself propagates in discrete bundles or packets of energy. Moreover, Planck did not attribute any physical significance to this assumption, but rather believed that it was merely a mathematical device that enabled him to derive a single expression for the black body spectrum that matched the empirical data at all wavelengths.
Call it a collaborative effort!

It is still not quite clear to me how the equation came to be. I mean, was it all just by experimental observation? What is the mathematical process used to get to E=hf?

And concerning the de Broglie hypothesis, I did read about that. But still, just how can you be sure the wavelength is h/p? I mean maybe it is that, plus another very small term given the mass isn't zero? Not trying to say de Broglie is a lie of course lol, but I think there must be a good solid argument as to why one can explain the wavelength for all matter with the same equation as the photon... given that for all other physical characteristics, like momentum and energy, we treat the photon and other particles differently. I hope I'm being clear... E=hf is the energy for a PHOTON, as well as E=pc. The equations for the energy of other particles would be different so I would expect that in deriving equations for their momentums and their wavelenghts, would not turn out to be the same equations.

Thanks in advance to any responses.

Ken G
Gold Member
It is still not quite clear to me how the equation came to be. I mean, was it all just by experimental observation? What is the mathematical process used to get to E=hf?
Contrary to how it is often taught, all physics comes by experimental observation. The mathematics is just an effort to axiomatize the observations. It is often more convenient to teach it by starting from the axioms-- but it tends to lead to questions like yours!
Not trying to say de Broglie is a lie of course lol, but I think there must be a good solid argument as to why one can explain the wavelength for all matter with the same equation as the photon... given that for all other physical characteristics, like momentum and energy, we treat the photon and other particles differently.
The argument is that it works, experimentally.
I hope I'm being clear... E=hf is the energy for a PHOTON, as well as E=pc.
That also works for particles, if you apply a concept of "de Broglie frequency" instead of wavelength. Nonrelativistically, E=p2/2m=h2/2mL2 so if you equate that to h*f you get f = h/2mL2, or if you say k=2*pi/L and omega = 2*pi*f, then omega = hk2/4*pi*m. Now the "group velocity" of a wave is the derivative of omega with respect to k, so that's v = hk/(2*pi*m) = h/mL = p/m is the classical expression for velocity. So the expression is quite general nonrelativistically, and it works relativistically as well though relativistic quantum mechanics is a bit trickier so I'm not sure how much work it would take to show. All of this comes under the heading of the "correspondence principle", which says that quantum expressions cannot contradict the classical ones in the classical regime, even though the overall axiomatic structure of quantum mechanics is virtually impossible to use at the classical level in most cases.

It is still not quite clear to me how the equation came to be. I mean, was it all just by experimental observation? What is the mathematical process used to get to E=hf?
I haven't seen the derivation, but I imagine you measure the voltage and current in the metal and use Maxwell's Equations to find the energy per atom necessary to generate that current at that voltage.

The critical discovery was that voltage (potential energy) didn't depend on the intensity of the light, only its frequency, and so Einstein realized "one corpuscle of light per atom."

Notably, that in and of itself doesn't prove quantization of light - it only proves quantization of atoms. :)

Ken G
Gold Member
Notably, that in and of itself doesn't prove quantization of light - it only proves quantization of atoms.
Actually, I would have to say that the reason Einstein got the Nobel prize was because it did show the quantization of light. We have electrons that are in a metal here, not in atoms, and the kinetic energy they get does not depend on the intensity of light but on its frequency. If it were the electrons that were quantized, that would be more of Planck's way of looking at things, but there was no reason to think freely moving electrons in a metal should be quantized, whereas quantizing the light unified all these results.

We knew charge was quantized before 1905. Einstein opened the door for us to be able to prove that light was quantized, but at first he, like Planck, thought it was just a heuristic and didn't necessarily accept the reality of photons.

Ken G
Gold Member
The bottom line is, we can conclude that there were two clear "watershed" moments on the way to quantizing the photon-- one when Planck found that treating the radiation field as if it could only be excited in a quantized way solves the problem of thermal spectra, and another when Einstein found the photoelectric effect. The acceptance of the idea of quantized light may have taken a little time for these ideas to percolate-- after all, they preceded quantum mechanics by two decades!