Quantum Mechanics Series – 6 Planck’s Theory

Max Planck Quantum Theory
Plank’s story begins in the physics department of the Kaiser Wilhelm Institute in Berlin, just before the turn of the century.

Plank was repeatedly being confronted with reliable experimental data on Black-Body Radiation. He was trying to explain black body radiation, but could not explain with available theoretical tools at that time.

Planck was a very conservative member of the Prussian Academy, steeped in traditional methods of classical physics and a passionate advocate of thermodynamics. In fact, from his PhD thesis days in 1879 (the year Einstein was born) to his professorship at Berlin twenty years later, he had worked almost exclusively on problems related to the laws of thermodynamics. He believed that the Second Law, concerning entropy, went deeper and said more than was generally accepted.

Planck was attracted by the absolute and universal aspects of the black-body problem. Plausible arguments showed that at equilibrium, the curve of radiation intensity versus frequency should not depend on the size or shape of the cavity or on the materials of its walls. The formula should contain only the temperature, the radiation frequency and one or more universal constants which would be the same for all cavities and colours.
Finding this formula would mean discovering a relationship of quite fundamental theoretical interest.
1. This radiation law, whenever it is found, will be independent of special bodies and substances and will retain its importance for all times and cultures… even for non-terrestrial and non-human ones.

History has proved Planck’s insight to be more profound

than even he thought. In 1990, scientists using the COBE satellite measured the background radiation at the edge of the universe (i.e left over from the Big Bang), and found a perfect fit to his Black-body radiation Law.

Pre-Atomic Model of Matter
Planck knew the measurements by his friends Heinrich Rubens and Ferdinand Kurlbaum were extremely reliable.
Planck’s oscillators in the walls of the cavity


Planck started by introduced the idea of a collection of electric oscillators in the walls of the cavity, vibrating back and forth under thermal agitation.
(*Note! Nothing was known about atoms.)
Planck assumed that all possible frequencies would be

present. He also expected the average frequency to increase at higher temperatures as heating the walls caused the oscillators to vibrate faster and faster until thermal equilibrium was reached.

The electromagnetic theory could tell everything about the emission, absorption and propagation of the radiation, but nothing about the energy distribution at equilibrium. This was a thermodynamics problem.
Planck made certain assumptions, relating the average energy of the oscillators to their entropy, thereby obtaining a formula for the intensity of the radiation which he hoped would agree with the experimental results.

Planck tried to alter his expression for the entropy of the radiation by generalizing it, and eventually arrived at a new formula for the radiation intensity over the entire frequency range.

The constants C1 and C2 are numbers chosen by Planck to make the equation fit the experiments.
Among those present at the historic seminar was

Heinrich Rubens. He went home immediately to compare his measurements with Planck’s formula. Working through the night, he found perfect agreement and told Plank early next morning.

Planck had found correct formula for the radiation law. Fine. But could he now use the formula to discover the underlying physics ?

Planck’s Predicament
1. …..From the very day I formulated the radiation law, I began to devote myself to the task of investing it with true physical meaning.
2. After trying every possible approach using traditional classical applications of the laws of thermodynamics, I was desperate.
3. I was forced to consider the relation between entropy and probability according to Boltzmann’s ideas. After some of the most intense weeks of my life, the light began to appear to be.

Boltzmann’s statistical version of the Second law based on probabilities seemed Planck’s only alternative. But he rejected the underlying assumption of Boltzmann’s approach which allows the second law to be violated momentarily during fluctuations.

S = k log W

(Boltzmann’s version of the second law of thermodynamics.)
Not once in any of the forty or so papers that Planck wrote prior to 1900 did he use, or even refer to, Boltzmann’s statistical formulation of the second Law!

Chopping Up the Energy
So, Planck applied three to Boltzmann’s ideas about entropy.
1. His statistical equation to calculate the entropy.
2. His condition that the entropy must be a maximum (i.e. totally disordered) at equilibrium.
3. His counting technique to determine the probability W in the entropy equation.
To calculate the probability of the various possible arrangements, Planck followed Boltzmann’s method of dividing the energy of the oscillators into arbitrarily small but finite chunks. So the total energy was written as E = N e where N is an integer and e an arbitrarily small amount of energy. e would eventually become infinitesimally small as the chunks became infinite in number, consistent with the mathematical procedure.

A Quantum of Energy
1. I found that I Had To Choose Energy units proportional to the oscillator frequency, namely e = h f, In Order To obtain the Correct form for the total energy. F is the Frequency and h is a constant which would eventually decrease to zero.
2. BUT THEN A REMARKABLE THING HAPPENED. IF I ALLOWED THE ENERGY CHUNKS TO GO TO ZERO AS THE PROCEDURE DEMANDED, THE GENERAL VALIDITY OF THE DERIVED EQUATION WAS DESTROYED. HOWEVER…
3. I NOTICED THAT IF A DID NOT REQUIRE THAT ENERGY OR h GO TO ZERO, I OBTAINED MY OWN EXACT RADIATION FORMULA….WHICH I KNEW WAS CORRECT.

Eureka! Planck had stumbled across a mathematical method which at last gave some theoretical basis for this experiment radiation law – but only if the energy is discontinuous.
Even though he had no reason whatsoever to propose such a notion, he accepted it provisionally, for had nothing better. He was thus forced to postulate that the quantity e = h f must be a finite amount and h is not zero.
Thus, it this is correct, it must be concluded that it is not possible for an oscillator to absorb and emit energy in a continuous range. It must gain and lose energy discontinuously, in small indivisible units of e = h f, which Planck called “energy quanta”.


Now you can see why the classical theory failed in the high frequency region of the Black-Body Curve. In this region the quanta are so large (e = h f) that only a Few vibration modes are excited.
With a decreasing number of modes to excite, the oscillators are suppresses and the radiation frequency end. The ultraviolet catastrophe does not occur.

Planck’s quantum relation thus inhibits the equipartition of energy and not all modes have the same total energy. This is why we don’t get sunburn from a cup of coffee. (Think about it!)

The classical approach of Rayleigh-Jeans works fine at low frequencies, where all the available vibrational modes can be excited. At high frequencies, even though plenty of modes of vibration are possible (recall it’s easier of stuff short waves into a box). Not many are excited because it costs too much energy to make a quantum at a high frequency since e = h f.

During his early morning walk on 14 December 1900. Planck told his son that he may have produced a work as important as that of Newton. Later that same day. He presented his result to the Berlin Physical Society signaling the birth of quantum physics.

It had taken him less than two months to find an explanation for his own black-body radiation formula. Ironically, the discovery was accidental, caused by an incomplete mathematical procedure. An ignominious start to one of the greatest revolutions in the history of physics !
From this start would come an understanding of why statistical rules must be used for atoms, why atoms don’t glow all the time and why atomic electrons don’t spiral into the nucleus.

In early 1901, the constant h – today called Planck’s constant – appeared in print for the first time. The number is small –
h = 0.000 000 000 000 000 000 000 000 006 626
-but it is not zero! If it were, we would never be able to sit in front of a fire. In fact, the whole universe would be different. Be thankful for the things in life.
Surprisingly, in spite of the important and revolutionary aspects of the black-body formula, it did not draw much attention in the early years of the 20th century. Even more surprisingly, Planck himself was not convinced of its validity.

I was so sceptical of the universality of boltzmann’s entropy law that I spent years trying to explain my results in a less revolutionary way.
Now of the second experiment which could not be explained by classical physics. It is more simple, yet inspired a more profound explanation.