# What was Planck's motivation for the frequency dependence in $E=nh\nu$?

Many accounts of the history of quantum physics explain how Planck resorted to quantizing energy in an "act of desperation" while attempting to solve blackbody radiation, only to discover by surprise that a nonzero value of $h$ in $E=nh\nu$ reproduced experimental results.

1. What was Planck's motivation behind the $\nu$ dependence in this expression?
2. Did classical physics provide any hints for this frequency dependence?

Einstein used this same relation to help explain the photoelectric effect, but that came later.

Finally, to emphasize why I have this question, consider these seemingly contradictory facts:

• Planck was treating the quantized EM waves as harmonic oscillators. However, the relation between energy and frequency for a classical harmonic oscillator has a square dependence: $E=\frac{1}{2}m\omega^2A^2=2\pi^2m\nu^2A^2$, where $A$ is the amplitude.
• In classical electromagnetic theory, the average energy density of a plane wave in vacuum has no frequency dependence: $u=\frac{1}{2}\epsilon_oE_o^2$, where $E_o$ is the amplitude of the electric field part of the wave.
• It's easy to imagine postulating $E=nh$ as a first attempt to quantize energy. The $n$ part of this expression is the quantization piece, which was a new idea that I can understand as a hopeful guess or mathematical trick—but the $\nu$ part seems a priori unmotivated, and this isn't addressed in any of the sources I looked through.
• $h$ doesn't have units of energy. For the dimensions to work, you need it to be multiplied by something with 1/time units. – probably_someone May 5 '18 at 21:16
• But this was the first appearance of $h$, so he was free to assign it whatever units the equation would have demanded. – WillG May 5 '18 at 21:24
• As a first approximation: because it worked. Planck already had an analytical expression that provided a good fit to the experimental results, and this hypothesis allowed him *some form of basis to derive it from. At that stage that was plenty. – Emilio Pisanty May 5 '18 at 21:42
• I read somewhere that Planck was trying to figure out how to fix the fact that the classical analysis of blackbody radiation leads to infinite emitted power. He found that discretizing the energy in each mode lead to non-diverging power so he ran with it. Supposedly, he saw this as an act of desperation because he had no physical reason to think the energy should be discrete. – DanielSank May 5 '18 at 23:05

In Planck's original paper (http://myweb.rz.uni-augsburg.de/~eckern/adp/history/historic-papers/1901_309_553-563.pdf), he first posits that the total internal energy of a group of oscillators might be an integer multiple of an "energy element" $\epsilon$. Using this assumption, he finds that the entropy per oscillator of a collection of oscillators should be approximately

$$S=k\left[\left(1+\frac{U}{\epsilon}\right)\log\left(1+\frac{U}{\epsilon}\right)-\frac{U}{\epsilon}\log\frac{U}{\epsilon}\right]$$

At the same time, Wien's Law, which was a well-known result at that time, predicts that the entropy per oscillator should take the following functional form:

$$S=f\left(\frac{U}{\nu}\right)$$

for some function $f$. Planck compared these two expressions and realized that the only way for statistical mechanics and Wien's Law to be consistent is if

$$\epsilon=h\nu$$

for some constant $h$.

In short, he did start with the assumption that energy was quantized. The dependence of the quantum of energy on frequency was required for consistency with Wien's Law.

• Thanks! This is very specific and helpful. My one remaining curiosity is, does this result have any classical analogy? Or is it a completely axiomatic piece of quantum theory with experimental motivation (Wien's Law) but no theoretical origin? – WillG May 6 '18 at 0:21
• @WillG What specifically are you looking for in a "classical analogy"? And the quantization of energy is not exactly axiomatic in most treatments of quantum theory. The axiom that it derives from is either the definition of time evolution by the Schrodinger equation, or the commutation relation between position and momentum. There's no agreement here on "which comes first" logically, because you are largely free to choose any axioms that explain experiment. (hsm.stackexchange.com/questions/3184/…). – probably_someone May 6 '18 at 0:29
• Even though $E=h\nu$ can be derived rigorously from the Schrödinger equation, my understanding is that $E=h\nu$ helped motivate this equation in the first place. (You can arrive at Schrödinger's equation by assuming $E=\frac{p^2}{2m}+V$ holds true for plane waves of the form $\psi=Ae^{i(kx-\omega t)}$, along with $E=\hbar \omega$ and $p=\hbar k$.) Anyway, seeing as there are now people claiming to be able to "derive" QM from purely mathematical probability axioms (arxiv.org/pdf/quant-ph/0101012.pdf), I would expect at least some form of a priori justification for $E=h\nu$. – WillG May 6 '18 at 16:48
• @WillG Are you asking for the justification we used today, or the justification that Planck used in his time? Because the latter is already in the answer, and the former is a different question entirely. – probably_someone May 6 '18 at 18:31
• Ideally I'd like to know both—Planck's best reasoning at the time (which you've given), and our current best explanation for "why" $E=h\nu$. I suppose the second part deserves a separate thread, and several disclaimers about what axioms we are starting from. – WillG May 6 '18 at 19:04