The Two Ages of Quantum Physics: Dawn of Modernity

The Birth Of Quantum Physics:

During the entire period of 250 years, the entirety of scientific world, from Newton to Maxwell, had been oscillating between one of the 2 ends of the spectrum. Is light a particle? Or is it a wave? 

We now enter the final lap in the quest of searching the answer to the question that originated 250 years ago. The problems of thermodynamics and classical electromagnetism at the end of the 19^th century will directly lead us to  a branch of physics that will change the course of humanity forever.

By 1870s J.C. Maxwell had mathematically and experimentally demonstrated the fact that light is an Electromagnetic wave. It was a commonplace understanding that light is an Electromagnetic Wave just like Radio Waves and not like regular Sound Waves.
Given the fact that light was considered as a “wave”, it was also understood that light is continuous, just like a wave. Observe a wave and you will realize that it is not made up of some “particles” with a definite position. Rather, it is a “localized disturbance” on the medium (eg. Localized disturbance of water gives rise to water waves and of air pressure gives rise to sound waves), spread over a length/area/volume. So, the energy associated with a wave was therefore not a property of its speed (as opposed to energy associated with a particle), but rather it’s intensity (or amplitude or brightness)

Through a series of parallel developments in thermodynamics, it was mathematically calculated and physically observed that every “Black-body” (A black body is an idealized object which absorbs and emits all frequencies. It is a perfect absorber of light, i.e., it is something that is perfectly black at all possible wavelengths. It absorbs radiation of all wavelengths absolutely perfectly and re-radiates energy in form of radiation that is dependent only on the absolute temperature of the black-body) in this universe above absolute zero emitted a continuous spectrum of radiation of all wavelengths, with the spectrum peaking at a particular wavelength (i.e., every object above 0K emitted radiation of all wavelengths with varying intensities. However, only one particular wavelength was emitted with the peak intensity)

Wien had given his Wien’s displacement law which said that the black-body radiation curve for different temperatures will peak at different frequencies that are directly proportional to the object’s absolute temperature. This meant that as the temperature of a body increases, the frequency of the radiation emitted with maximum intensity also increases. Room temperature objects (about 300 K) emit radiation with a peak intensity in the far infrared region; radiation from toaster filaments and light bulb filaments (about 900 K and 2,500 K, respectively) also peak in the infrared, though their spectra extend progressively into the visible red and hence filament of bulb appears to glow or Iron becomes “red” hot; while the 6,000 K surface of the Sun emits blackbody radiation that peaks in the center of the visible range, and hence our eyes have evolved to be sensitive to light within this range.

The black line is the one derived through classical Rayleigh-Jeans method.
The Blue line is for a body at 5000K, which radiates Blue colour with max intensity. Green line is for 4000K and Red Line is for 3000K.

By late 1890s, the mathematics of black-body radiation spectrum by classical electromagnetic and thermodynamics was unable to match the experimental results. In fact, the calculations predicted the absurd result that, at any temperature, the spectral intensity increases without limit, as a function of frequency. This essentially meant that as frequency increased, the spectral intensity increases…. with no upper bound. Of course, this isn’t the case. Had it been true, then every time we’d be opening our ovens, we’d be blasted with UV radiations.

The classical models rightly predicted the graph for higher wavelengths (lower frequencies) but couldn’t explain the graph for lower wavelengths (higher frequencies). This inability of classical electromagnetic models to satisfactorily explain the Black-body spectrum became known as “The Ultraviolet Catastrophe”. (Lord Rayleigh created the model and hence this is also known as Rayleigh- Jeans catastrophe)

Rayleigh-Jeans formula for Spectral Radiance or Energy Density. Here ⍴ is Spectral Radiance or Average Energy emitted/unit time/unit volume or Energy Density. As evident, ρ is inversely proportional to fourth power of wavelength. As wavelength goes to 0, the Average Energy goes to ∞ 

Why did this happen?

Traditionally, a wave was defined as a “continuous” system in itself. If you look at a wave you’ll realize that it’s a continuous stream of oscillations. Classically, a wave is defined as a disturbance in a medium which travels through the medium, transporting energy from one point (source) to another, without any transport of matter. This process of transfer of energy from one point to another is a continuous process. The energy is transferred continuously and since, the electromagnetic radiations emitted by a black body are also waves; therefore the energy is continuously being emitted by the black body.

Standing Wave

Periodic Travelling Wave

This is, also, the classical explanation of emission of energy by a black body and was supposed to explain the spectral intensity vs wavelength graph of black body spectrum. As mentioned above, the above classical description broke down when Rayleigh and Jeans tried to come up with a mathematical model using the knowledge of classical electromagnetism and classical thermodynamics.

The answer? Max Planck and Quantum Physics.



1^st Era Of Quantum Physics (Black-Body Radiation and Photoelectric Effect):

Max Planck was a fairly unassuming German physicist whose primary topic of interest and research was thermodynamics. In a now famous and path breaking lecture given by him in Deutsche Physikalische Gesellschaft, DPG (German Physical Society), on 14^th December, 1900, Planck announced that the theoretical observations of Rayleigh-Jeans model could be rectified and the Ultraviolet Catastrophe can be avoided if one postulates that energy can only exist in the form of certain discrete packets or bundles called “Quanta”, and that the transfer (either emission or absorption) of this energy, via Electromagnetic radiations, also takes place in discrete bundles of energy packets (then called “quanta” and hence giving the birth to “Quantum Theory” ) known as “Photons”.

According to many contemporary writers, Planck had been tirelessly working on the Rayleigh-Jeans problem for years. He decided to focus on the experimental results and compare these results with the Rayleigh-Jeans curve. As said earlier, the Rayleigh-Jeans curve predicts the spectral intensity values at higher wavelength but approaches infinity at lower wavelengths. From the graph, he observed that the average energy of each wave for limits of higher wavelength must be equal to a constant value that depended only on the Temperature, since the graph changed only on changing the temperature of the black body (and hence depended on the temperature of the black body only). This value was calculated to be = kT where k = 1.6*10^-23 J/K and is called as the Boltzmann Constant while T = Absolute Temperature of the body.
Furthermore, it was quite obvious from the experimental data that for lower wavelengths, the value of average energy of each wave must be = 0.

On these observations he made another observation and proposed that the average energy of the wave must then be dependent on the frequency of the wave. Planck’s biggest contribution came when he realized that he can obtain the required cutoff in energy if he rectifies one of the basic tenets of classical thermodynamics- Boltzmann distribution's calculation of average energy. Instead of treating the physical quantity “energy” as a continuous variable of Classical Physics, he treated it as if it was a discrete variable. In particular, he imagined that energy could take only certain discrete values and that the total energy radiated by the object must therefore be an integral multiple of a basic fundamental amount of energy.

The results were profound. Planck’s small but monumental modification led to the development of mathematical model/equation of the spectral intensity emission from a black body at absolute temperature T which matched almost identically with the graph of experimental data. Also, since the graph obeyed Rayleigh-Jeans law at higher wavelengths, therefore, the equation of the graph obtained after Planck’s modification also satisfied Rayleigh-Jeans Law at higher wavelengths. The value of average energy tends to kT as the frequency tends to 0 (obtained from experimental data) while the value of average energy tends to 0 as the frequency of radiation tends to ∞.

Formula for Average Energy of an Electromagnetic Wave (as derived by Planck on the basis of his Quantum Theory) and it's relationship with frequency and energy. Here ε is the energy of one light quanta.

Planck’s modification took him on a journey which led to the establishment of an equation that explained the spectral intensity vs wavelength graph. According to the equation obtained, the value of average energy tends to kT as the frequency tends to 0 (obtained from experimental data) while the value of average energy tends to 0 as the frequency of radiation tends to ∞.

The startling and sparking correlation and similarities between both the Planck’s modified “Quantum” model and the graph plotted on experimental data meant that there has to be some connection between the Energy of the wave and its frequency. In fact, the energy of the wave is directly proportional to the frequency of the wave.

A direct relationship between energy of light quanta and its frequency was established by Max Planck. h was the proportionality constant now known as Planck's Constant.

E=hf. 

Where  E= Energy of the wave
         f= Frequency of the wave.
         h= Planck’s Constant.

After doing some complex calculations and applying the modifications of his newly developed theory of Quanta, Planck came up with a relation of spectral density as a function of wavelength and when plotted, the graph matched almost identically with the observed experimental data. Planck’s equation of spectral intensity had the added bonus of reducing to Rayleigh-Jeans equation at higher wavelengths, which further proved that his Quantum Physical model was correct and reduced to the Classical Electromagnetic and Thermodynamical model at higher wavelengths as it should.

Planck's formula for Spectral Radiance/ Energy Density. Here ⍴ is Spectral Radiance or Average Energy emitted/unit time/unit volume or Energy Density. According to this formula, as wavelength goes to 0, the Average Energy goes to kT 

Quantum Theory established that energy in Light (and in any Electromagnetic Radiation) can only take some discrete values and not every and any continuous value

In 1916, detailed experimental work done by William Coblentz provided wonderful confirmation of Planck’s spectral intensity of radiation vs wavelength plot. Coblentz used his curve to calculate a value for Planck’s constant which came out to be pretty close to modern more accurate values.

The dots on the graph are the values predicted by Planck's equation

Well, here in our story enters a relatively unknown German clerk who went by the name of Albert Einstein.

While Planck had solved the ultraviolet catastrophe by using atoms and a quantized electromagnetic field, most contemporary physicists agreed that Planck's "light quanta" represented only flaws in his model. A more-complete derivation of black-body radiation would yield a fully continuous and "wave-like" electromagnetic field with no quantization. Planck’s model was under intense scrutiny and in order to attain complete credibility in the scientific circles, it needed to explain another outstanding problem of the era. That outstanding problem turned out to be- The Photoelectric Effect.

In 1905, Albert Einstein provided an explanation of the photoelectric effect, an experiment that the wave theory of light failed to explain. He did so by postulating the existence of photons, quanta of light energy with particulate qualities.

In the photoelectric effect, it was observed that shining a light on certain metals would lead to an electric current in a circuit. Presumably, the light was knocking electrons out of the metal, causing current to flow. However, using the case of potassium as an example, it was also observed that while a dim blue light was enough to cause a current, even the strongest, brightest red light available with the technology of the time caused no current at all. According to the classical theory of light and matter, the energy of a wave was inn proportion to the strength or amplitude of a light wave, which was in proportion to its brightness: a bright light should have been easily strong enough to create a large current. Yet, oddly, this was not so.

Einstein explained this predicament by postulating that the electrons can receive energy from electromagnetic field only in discrete units (quanta or photons): an amount of energy E that was related to the frequency f of the light by

E = hf

where h is Planck's constant (6.626 × 10⁻³⁴ Js). Only photons of a high enough frequency (above a certain threshold value) could knock electrons free. For example, photons of blue light had sufficient energy to free an electron from the metal, but photons of red light did not. One photon of light above the threshold frequency could release only one electron; the higher the frequency of a photon, the higher the kinetic energy of the emitted electron, but no amount of light below the threshold frequency could release an electron.
The photoelectric effect is the scientific principle that makes modern solar power possible.

No electron emission if frequency of incident light is less than the threshold value of the material. 

Electron emission if frequency of incident light is greater than the threshold value of the material. 

Experimental demonstration of The Photoelectric Effect.

Max Planck was awarded the Nobel Prize in Physics in 1918 for his discovery of the law of the quantization of light.

Albert Einstein was awarded the Nobel Prize in Physics in 1921 for his discovery of the law of the photoelectric effect.

Despite Planck’s work being grounded in thermal radiation properties of hot objects, his path-breaking insight into the nuances of the basic structure and functioning of this universe ushered in an Age of Enlightenment for modern science, and laid the foundation to an era of physics to come that will revolutionize science and humankind forever. But before that, if one is observant enough, he/she will realize that Planck’s works had directly led to the revivalism of the corpuscular theory of light. The Photon became the particle which Newton was talking about, albeit with slight modifications. Demons of the past re-emerged. Was Newton right all along or are we fighting old battles again? Electromagnetism and its maths say that light is a wave while Thermodynamics and the maths behind thermal radiation points towards Light as stream of particles or Quanta.

So we are right back to where we started. What is Light?

2^nd Era Of Quantum Physics (Wave-Particle Duality and The Birth of Quantum “Mechanics”): 

In 1923, Louis de Broglie, a French physicist, came up with a wildest of ideas. He initially thought that if light, an electromagnetic “wave”, can exhibit particulate characteristics and phenomena, is it possible that classical “particles” exhibit “wave-like” characteristics and phenomena? Since, light shows a “dual” nature, is it possible that every matter particle also exhibits a “dual” nature? Through a series of simple and straightforward substitutions of Einstein’s most famous equation of mass-energy equivalence, E= mc² , classical definition of momentum, and Planck’s relation between energy and frequency, he came up with an ingenious idea of “matter wave” and a mathematical derivation of it.

De Broglie derived his equation using well established theories through the following series of substitutions:

De Broglie first used Einstein's famous equation relating matter and energy: E=mc²  

E= energy, m = mass, c = speed of light

Then he used Planck's theory which states every quantum of a wave has a discrete amount of energy given by Planck's equation: E=hf

E = energy, h = Plank's constant(6.62607 x 10^-34 J s), f = frequency

Since de Broglie believes particles and wave have the same traits, the two energies would be the same: mc²=hf

Because real particles do not travel at the speed of light, De Broglie substituted v, velocity of the particle, for c, the speed of light: mv²=hf

Given that the given particle also acts like a wave and that the speed of that particle, v, must satisfy the relation v= λ$\mathrm{f, }$de Broglie substituted f as v/λ$\mathrm{ <font\; face="Arial\; Narrow,\; sans-serif">and\; the\; above\; equation\; becomes: </font>}$mv²=hv/λ

Hence: 1/λ$ = mv2/hv $

or

$$

λ$\mathrm{=\; h/mv<span\; class="mn"><span\; face=""arial\; narrow",\; sans-serif"><o:p></o:p></span></span>}$

Although De Broglie was credited for his hypothesis, he had no actual experimental evidence for his conjecture. For a physicist, to take such a huge leap of faith on mere assumptions and theoretical evidence was brave but the reasoning behind his thinking was right. There was no need to judge light as a matter or a wave. Why can’t it be both? And if we extend that similar line on thought to other “particle like matter substances” as well, can all matter exhibit wave-particle duality?

It was De Broglie’s scientific bravery that ushered in the era of Quantum Mechanics, for 4 years later after he gave his controversial and yet ground breaking hypothesis with no experimental backing, in 1927, Clinton J. Davisson and Lester H. Germer shot electron particles (which classically, were defined as “matter”) onto a nickel crystal in a now incredibly famous and criminally underrated Davisson-Germer experiment. What they saw was a diffraction pattern, a very conclusive “wave” phenomenon, and in the process, demonstrating an experimental proof of De Broglie’s brave leap of faith.

According to de Broglie, matter particles like electrons also exhibited wave like behaviour, with a wavelength depending on it's momentum. Since, an electron's momentum depends on its orbit, therefore the wavelength of "electron wave" also depended on its orbit. In fact, the wavelength of electrons was such that number of "electron waves" formed on the orbit turned out to be = the orbit number.

This dual nature exhibited by all matter, laid the foundation of modern Quantum Physics There were still a few loopholes in De Broglie’s hypothesis; but all the ingredients were in place now for redefining science and physics as we knew then as well as development of an entirely new and potentially revolutionary branch of physics which we now know as Quantum Mechanics. 

The Golden Age of Modern Science had just begun.