Understanding the Derivation of E=hf and its Significance in Quantum Physics

[holtto]

What is the derivation for E=hf and why did experimental observation of black bodies show that quantization of light was necessary?

 

[tom.stoer]

E=hf was not 'derived' in the strict sense. In back body radiation one could assume that energy of a certain mode with frequency f is not related to its frequency but to the electromagnetic field strength (squared). That would mean that a 'photon' of frequency f could carry an arbitrary energy E (like a classical electromagnetic wave where the energy has nothing to do with the frequency). Doing state counting that way one finds that the total energy emitted by a black body (of temeperature T) is infinite - which is nonsense. So Planck introduced the idea that the memission process of radiation from the atoms of a black body is quantized as E=hf, i.e. that a photon of frequency f must always carry a fixed amount of energy E=hf (he did not say that these photons must always come in these quanta, he only assumed this for the emission). In that way Planck was able to guess an energy density (per frequency) which interpolated between two known formulas, one for low and one for high frequency, and to obtain a finite result for the total energy emitted by the black body.

http://en.wikipedia.org/wiki/Black-body_radiation
http://en.wikipedia.org/wiki/Planck's_law

It was Einstein who assumed (a few years later) for the interpretation of the photoelectric effect that photons of frequency f always carry energy E=hf (and that this is not only due to a specific emission process). He was then able to explain the observed energy spectrum of the photo electrons which only depends on the frequency (not on the intensity of light, what would have been guess based on classical elecromagnetism).

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

 

[mraptor]

I was about to ask exactly this questions..
I found derivatoin of E =mc^2, and de Broigle eq p = h/lambda, but not E = hf

 

[holtto]

the de broglie formula comes from E=pc=hf

 

[Jay_]

E=hf was not 'derived' in the strict sense. In back body radiation one could assume that energy of a certain mode with frequency f is not related to its frequency but to the electromagnetic field strength (squared). That would mean that a 'photon' of frequency f could carry an arbitrary energy E (like a classical electromagnetic wave where the energy has nothing to do with the frequency). Doing state counting that way one finds that the total energy emitted by a black body (of temeperature T) is infinite - which is nonsense.

I was just reading through the forum.

Can you explain "state counting" in simple words? Let's say a photon does have energy proportional to its EMfield squared. How does that make its energy infinite?

 

[jtbell]

Do a Google search for "Rayleigh-Jeans law".

 

[tom.stoer]

Can you explain "state counting" in simple words? Let's say a photon does have energy proportional to its EMfield squared. How does that make its energy infinite?

You have to use two inputs
- density of states
- equipartion theorem

Suppose you black body is an empty cube; you have to calculate the number of electromagnetic waves fitting into the cube i.e. satisfying the boundary conditions. In three dimensions the density of states for a given freqency interval reads

N(\nu)\,d\nu \sim v^2\,d\nu

The classical treatment goes as follows
- calculate the average energy contained in each mode with a given frequency
- classical statistical physics says that for a system in thermal equilibrium the equipartition theorem applies
- therefore the average kinetic energy per d.o.f. is constant, i.e. ~ kT

So

\bar\epsilon_T(\nu) = kT

Together the average energy density for a given freqency interval reads

u_T(\nu)\,d\nu \sim \bar\epsilon_T(\nu)\, N(\nu)\,d\nu \sim kT\,v^2\,d\nu

The total power (= radiation energy per time) is calculated by integrating

P(T) = \int_0^\infty u_T(\nu)\,d\nu = \infty

Therefore the above mentioned assumpttions are wrong b/c a black body of finite temperature T and finite internal energy cannot produce infinite radiation power.

It was Planck who corrected the assumptions and "guessed" a different energy density. And it was Einstein how derived this energy density using the assumption of equilibrium of absorption + spontaneous emission + induced emission.

 

[Jay_]

I don't understand the equations completely. What is the tilde '~' stand for? But thanks for your explanation :)

 

[tom.stoer]

~ means proportionality i.e. neglecting irrelevant constants b/c in the end the difference between ∞ and const. times ∞ doesn't matter ;-)