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A friend asked me to explain "why" a particle's energy is proportional to it's frequency, i.e: $$E=h\nu$$

The reason this result is so un-intuitive, is that in the macroscopic world, A wave's energy (e.g electric power dissipated through a resistor) is not frequency dependent but rather amplitude dependent. Of course a "pure" photon doesn't really have an amplitude due to it's description as a plane wave, but that's beside the point.

So in short: Is there an easy explanation of "why" the energy is only dependent on frequency? can this be "derived" from other principles (say, the particle-wave duality)?

N.B. I'm aware there are no real "reasons" for any law of physics other then the fact that they agree with experiment, but laying those laws on a smaller number of basic principles is always nice when possible.

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Why the -1 ? It seems on topic to me. – Frédéric Grosshans Jun 6 '12 at 10:38
up vote 5 down vote accepted

As other have said, the answer to "why ?" is basically "because !". It is the definition of quantum physics.

However, one could have an answer linked to the anthropic principle, which is basically :

If the particle energy would not be increasing with its frequency, the blackbody radiation would have an infinite power, because of the ultraviolet catastrophe.

Let's suppose the energy of a photon to be $\varepsilon(\nu)$. In quantum physics, $\varepsilon(\nu)=h\nu$, but we are precisely exploring other possibilities.

If $\varepsilon$ were constant or decreasing, when $k_BT$ is at least of the order of $\varepsilon$, each mode of the electromagnetic field would carry an energy of the order of $k_BT$. Since $\nu$ is a priori unbounded, there are infinitely many modes, and the sum is infinite. This problem, is the ultraviolet catastrophe and was solved in 1900 by Max Planck when he established Planck's law. (Edit: The previous sentence is historically inaccurate: Planck did not want to solve this problem, which was only stated in in 1905.) If $\varepsilon$ increases with $\nu$, then the average population of the high frequency modes decrease exponentially, allowing to sum over all the frequencies and have a finite integral.

More quantitatively, the population of a mode with frequency $\nu$ is $P(\nu)=\frac1{e^{{\varepsilon}/{k_BT}}-1}$ and Planck's law is $$ I(\nu,T) =\frac{ 2 \varepsilon\nu^{2}}{c^2}\frac{1}{ e^{{\varepsilon}/{k_BT}}-1}.$$ If one wants the total radiated power $\int_0^{+\infty}d\nu I(\nu,T)$, one needs $\varepsilon(\nu)$ to increase fast enough ($\varepsilon\propto \nu^l$ seems enough $\forall l>0$, but $\varepsilon\propto\log \nu$ is not fast enough.)

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Thank you - nicely stated food for thought. – nbubis Jun 6 '12 at 13:50

If you already know that why-questions are always a little problematic and you have an understanding regarding amplitude assumption involving a quantization of the field into photons, then a short argument why the frequency-energy relation pops up in quantum mechanics is that the axiom

$$\left(i\hbar\frac{\partial}{\partial t}\right)\Psi=H\Psi,$$

already contains the relation, the Hamiltonian giving the energy. Check the units - it basically says that energy is one over time. Consider a decomposition of the wave into plane waves

$$\psi\sim \text e^{-i\frac{t}{\hbar} \omega}\ \ \ \Longrightarrow\ \ \ \left(i\hbar\frac{\partial}{\partial t}\right)\psi=\omega\ \psi$$

then the energy is a decomposition into frequencies $\omega$.

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Yes, that's true, but Schrodinger's equation was written precisely to retrieve the de Broglie / Planck relations, so that's a bit of a tautology. – nbubis Jun 6 '12 at 10:01
@nbubis: Historically, yes. But I think that if you reject the Schödinger equation, or if you want a justification, then you'll end up with a discussion about spacetime symmetries together with continous probablity distributions. Basically, if you have waves and intergrals of motion, you have some relation in that vain. The concept of energy is a major tool and the frequence is a quantity distinctive for the object you want to consider. If you don't want to go over the axioms of the underlying theory, then I guess you'll have to take the route of discussing fundamental constants of nature. – NikolajK Jun 6 '12 at 10:28
Dear nbubis, apologies, I just want to admit that I downvoted your question because the comment above has led me to believe that despite the ambiguous wording, it wasn't meant to be a constructive question. Schrödinger's equation (or Heisenberg equations which would produce the simple energy-frequency relationship in a similar way) is a fundamental and universal law of physics. The energy operator is linked to the change of phase of the amplitudes (wave function) per unit time. There isn't any more fundamental explanation. – Luboš Motl Jun 6 '12 at 10:29
Your question could and should have been asked in 1924 or so and the right answer would have been exactly what Nick gave you: it's the more fundamental explanation of why de Broglie's guess worked. In fact, in some sense, this answer was given around 1925 by the fathers of quantum mechanics, just a year after de Broglie, so quantum mechanics could develop. The relationship depends on Planck's constant which is a constant measuring the strength of quantum effects so it's simply not possible to separate the relationship from discussions about quantum mechanics. – Luboš Motl Jun 6 '12 at 10:31

I suspect this argument just exchanges one non-intuitive bit of physics for another, but you could try it on your friend and see.

The key point is that photons aren't (just) a convenient way of chopping up a light beam into bits. The photon is the unit of interaction between light and matter i.e. when matter interacts with light it does so by absorbing or emitting a photon.

Also, light doesn't interact with all matter, it specificially interacts with electric dipoles. Absorbing light makes a dipole oscillate and an oscillating dipole emits light.

Let's take the quantum version of the oscillating dipole to be a simple harmonic oscillator. I think the argument is simplest if you consider the oscillator emitting light (absorbing light is the same process but time reversed). If you solve the Schrodinger equation for a simple harmonic oscillator you find that if the frequency is $\nu$ the energy levels have a spacing $h\nu$. The oscillator emits a photon when it drops down an energy level, and the frequency of the emitted photon will be the same as the frequency of the oscillator. Since the energy of the photon must match the energy spacing in the oscillator, the energy of a photon of frequency $\nu$ is $h\nu$.

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But it again, we end up with a using the Schrodinger equation axiomatically (without it the energy transferred to the dipole would depend only on amplitude). I think I'll just have to accept that the energy frequency relation is just one of the axioms of quantum theory that was discovered experimentally, without any classical counterpart. – nbubis Jul 18 '12 at 15:45

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