Category Archives: Quantum Mechanics

An easy method to understand quantum mechanics

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.

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.

Quantum Mechanics Series -5 : Thermal Equilibrium and Fluctuations

The Thirty Year War (1900−30) – Quantum Physics Versus Classical Physics

There were three critical experiments in the pre-quantum era which could not be explained by a straightforward application of classical physics.
Each involved the interaction of radiation and matter as reported by reliable, experimental scientists.

The measurements were accurate and reproducible, yet paradoxical… the kind of situation a good theoretical physicist would die for.
We will describe each experiment step-by-step, pointing out the crisis engendered and the solution advanced by Max Planck, Albert Einstein and Niels Bohr respectively.

In putting forward their solution, these scientists made the first fundamental contributions to a new understanding of nature. Today the combined work of these three men, culminating in the Bohr model of the atom in 1913, is known as the Old Quantum Theory.

Black-Body Radiation
When an object is heated, it emits radiation consisting of electromagnetic waves, i.e. light with a broad range of frequencies.

1. Measurements made on the radiation escaping from a small hole in a closed heated oven – which in Germany we call a cavity – shows that the intensity of the radiation varies very stronger with the frequency of the radiation.

The dominant frequency shifts to a higher value as the temperature is increased, as shown in the graph drawn from measurements made in the late 19th century.

A black-body is a body that completely absorbs all the electromagnetic radiation failing on it.

Inside a cavity the radiation has nowhere to go and is continuously being absorbed and re-emitted by the walls. Thus, a small opening will give off radiation emitted by the walls, not reflected, and thus is characteristic of the black body.
When the oven is only just warm, radiation is present but we can’t see it because it does not stimulate the eye. As it gets hotter and hotter, the frequencies reach the visible range and the cavity glows red like a heating ring on an electric cooker.

This is how early potters determined the temperature inside their kilns. They would notice the color of fire where pots are heated and thr color gave them idea of temperature. Already in 1792, the famous porcelain maker Josiah Wedgwood had noted that all bodies become red at the same temperature.

In 1896, a friend of Planck’s Wilhelm Wien, and others in the Berlin Reichsanstalt (Bureau of Standards) physics department put together an expensive empty cylinder of porcelain and platinum.
At Berlin’s Technische Hochschule, another of Planck’s close associates, Heinrich Rubens, operated a different oven.
These radiation curves – one of the central problem of theoretical physics in the late 1890s – were shown to be very similar to those calculated by Maxwell for the velocity (i.e. energy) distribution of heated gas molecules in a closed container.

Paradoxical Results
Could this black-body radiation problem be studied in the same way as Maxwell’s ideal gas… electromagnetic waves (instead of gas molecules) bouncing around in equilibrium with the walls of a closed container?
Wien derived a formula, based on some dubious theoretical arguments which agreed well with published experiments, but only at the high frequency part of the spectrum.
The English classical physicist Lord Rayleigh (1842−1919) and Sir James Jeans (1877−1946) used the same theoretical assumptions as Maxwell had done with his kinetics theory of gases.
The equation of Rayleigh and Jeans agreed well at low frequencies but they got a real shock at the high frequency region. The classical theory predicted an infinite intensity for the ultraviolet region and beyond, as shown in the graph. This was dubbed the ultraviolet catastrophe.
What does this experimental result actually mean and What Went Wrong ?
The Rayleigh-Jeans result is clearly wrong, otherwise anyone who looked into the cavity would have eyeballs burned out.

This ultraviolet Catastrophe became a serious Paradox for classical physics.
If Rayleingh and Jeans were right, it would be dangerous for us even to sit in front of a fireplace.

If classical physicists had their way, the romantic glow of the embers would soon turn into life-threatening radiation. Something had to be done!

The Ultraviolet Catastrophe
Everyone agreed that Rayleigh and Jeans’ method was sound, so it is instructive to examine what they actually did and why it didn’t work.
1. We applied the statistical physics method to the waves by Analogy with Maxwell’s gas particles using the equipartition of energy, i.e. we assumed that the total energy of radiation is distributed equally among all possible vibration frequencies.
2. But there is one big difference in the case of waves. There is no limit on the number of modes of vibration that can be excited…
3. …Because It’s easy to fit more and more waves into the container at higher and higher frequencies (i.e. the wavelengths get smaller and smaller).
4. Consequently, the amount of radiation predicated by the theory is unlimited and should keep getting stronger and stronger as the temperature is raised and the frequency increases.
5. No wonder it was known as the ultraviolet catastrophe.

Quantum Mechanics Series – 4 : The Existence of Atoms

The Existence of Atoms

India Philosopher “Kanad” before 600 B.C. said that each matter consists of small particles which are called “Kan”. He gave those small particle his own name.

A Greek philosopher named Democritus (c. 460−370 B.C.) also first proposed the concept of atoms (means “indivisible” in Greek).

The idea was questioned by Aristotle and debated for hundreds of years before the English chemist John Dalton (1766 – 1844) used the atomic concept to predict the chemical properties of elements and compounds in 1806.

But it was not until a century later that a theoretical calculation by Einstein and experiments by the Frenchman Jean Perrin (1870−1942) persuaded the sceptics to accept the existence of atoms as a fact.

However, during the 19th century, even without physical proof of atoms, many theorists used the concept.


Averaging Diatomic Molecules

The Scottish physicist J.C. Maxwell, a confirmed atomist, developed his kinetic theory of gases in 1859.


This was qualitatively consistent with physical properties of gases, if we accept the notion that heating causes the molecules to move faster and bang into the container walls more frequently.


Maxwell’s theory was based on statistical averages to see if the macroscopic properties (that is, those properties that can be measured in a laboratory) could be predicted from a microscopic model for a collection model for a collection of gas molecules.


Maxwell made for assumptions :

Maxwell : gave distribution of velocity for gas particles








This last assumption was the most unusual and revolutionary showing a great deal of physical insight by Maxwell.

It would be impossible by try to compute the individual motions of many particles. But Maxwell’s analysis, based on Newton’s mechanics, showed that temperature is a measure of the microscopic mean squared velocity of the molecules. 

The real importance of Maxwell’s theory is the prediction of the probable velocity distribution of the molecules, based on his model. In other words, this gives the range of velocities…how the whole collection deviates from the average.

Postulates of Maxwell Theory helps to calculate probability that a molecule chosen at random would have a particular velocity.

Maxwell velocity distribution curve:

This is the well known curve which physicists today call the Maxwell Distribution. It gives useful information about the billions and billions of molecules, even though the motion of an individual molecule can never be calculated. This is the use of probabilities when an exact calculation is impossible in practices.





Ludwig Boltzmann and Statistical Mechanics

In the 1870s, Ludwig Boltzmann (1844−1906) – inspired by Maxwell’s kinetic theory – made a theoretical pronouncement.

  • He presented a general probability distribution law called the canonical or orthodox distribution which could be applied to any collection of entities which have freedom of movement, are independent of each other and interact randomly.
  • He formalized the theorem of the equipartition of energy.

This means that the energy will be shared equally among all degree of freedom if the system reaches thermal equilibrium.

  • He gave a new interpretation of the Second Law.


When energy in a system is degraded (as Clausius said in 1850), the atoms in the system become more disordered and the entropy increases. But a measure of the disorder can be made. It is the probability of the particular system – defined as the number if ways it can be assembled from its collection of atoms.

More precisely, the entropy is given by :

          S = k Log W −−−−

Where k is a constant (now called Boltzamann’s constant) and W is the probability that a particular arrangement of atoms will occur. This work made Boltzmann the creator of statistical mechanics, a method in behavior of their constituent microscopic parts.

Quantum Mechanics Series- 3 : What is Thermodynamics

Quantum Mechanics 3

What is Thermodynamics ?

The word means the movement of heat, which always flows from a body of higher temperature to a body of lower temperature, until the temperatures of the two bodies are the same. This is called thermal equilibrium.

Heat is correctly described as a form vibration…


The First Law of Thermodynamics


Steam Engines

James Watt (1736 – 1819), A Scot , who had built a working steam engine in 19th century.  





Soon after, the son of a Manchester brewer, James Prescott Joule (1818−19), showed that a quantity of heat can be equated to a certain amount of mechanical work.

Then somebody said…. “since heat can be converted into work, it must be a form of energy” (the Greek word energy means “containing work”) But it wasn’t until 1847 that a respectable academic scientist, Hermann von Helmholtz (1821-94), stated…..









This is called the law of the conservation of energy.  It remains a foundation of modern physics, unaffected by modern theories.


Rudolf Clausius: Two Laws

In 1850, the German physicist Rudolf Clausius (1822-88) published a paper in which he called the energy conservation law The First Law of Thermodynamics. At the same time, he argued that there was a second principle of thermodynamics in which there is always some degradation of the total energy in the system, some non-useful heat in a thermodynamic process.

Clausius introduced a new concept called entropy – defined in terms of the heat transferred from one body to another.

Entropy is measurement of disorderness of any system. The entropy of an isolated system always increases, reaching a maximum at thermal equilibrium, i.e. when all bodies in the system are at the same temperature.

Quantum Mechanics Series – 2: Solvay Conference 1927 – Formulation of Quantum Theory

The Solvay Conference 1927 – Formulation of Quantum Theory

A few years before the outbreak of World War I, the Belgain industrialist Ernest Solvay (1838-1922) sponsored the first of a series of international physics meetings in Brussels. Attendance at these meeting was by special invitation, and participants – usually limited to about 30 – were asked to concentrate on a pre-arranged topic.

The first five meeting held between 1911 and 1927 chronicled in a most remarkable way the development of 20th century physics. The 1927 gathering was devoted to quantum theory and attended by no less than nine theoretical physicists who had made fundamental contributions to the theory. Each of the nine would eventually be awarded a Nobel Prize for this contribution.

This photograph of the 1927 Solvay Conference is a good starting point for introducing the principal players in the development of the most modern of all physical theories. Future generations will marvel at the compressed time scale and geographical proximity which brought these giants of quantum physics together in 1927.


There is hardly and period in the history of science in which so much has been clarified by so few in so short a time.

Look at the sad-eyed Max Planck (1858−1947) in the front row next to Marie Curie (1867−1934). With his hat and cigar, Planck appears drained of vitality, exhausted after years of trying to refute his own revolutionary ideas about matter and radiation.


A few year later in 1905, a young patent clerk in Switzerland named Albert Einstein (1879−1955) generalized Planck’s notion.


That’s Einstein in the front row centre, sitting stiffly in his formal attire. He had been brooding for over twenty years about the quantum problem without any real insight since his early 1905 paper. All the while, he continued to contribute to the theory’s development and endorsed original ideas of others with uncanny confidence. His greatest work – the General Theory of Relativity – which had made him an international celebrity, was already a decade behind him.


In Brussels, Einstein had debated the bizarre conclusions of the quantum theory with its most respected and determined proponent, the “great Dane” Niels Bohr (1885-1962). Bohr – more than anyone else – would become associated with the struggle to interpret and understand the theory. At the far right of the photo, in the middle row, he is relaxed and confident – the 42 year old professor at the peak of this powers.

In the back row behind Einstein, Erwin Schrodinger (1887−1961) looks conspicuously casual in his sports jacket and how tie. To his left but one are the “young Turks”, Wolfgang Pauli (1900−58) and Werner Heisenberg (1901−76) – still in their twenties – and in front of them, Paul Dirac (1902−84), Louis de Broglie (1892−1987), Max Born (1882−1970) and Bohr. These men are today immortalized by their associate with the fundamental properties of the microscopic world: the Schrodinger wave equation; the Pauli exclusion principle ; the Heisenberg uncertainty relation, the Bohr Atom…. and so forth.

They were all there – from Planck, the oldest at 69 years, who started it all in 1900 – to Dirac, the youngest at 25 years, who completed the theory in 1928.

The day after this photograph was taken – 30 October 1927 – with the historic exchanges between Bohr and Einstein still buzzing in their minds, the conferees boarded trains at the Brussels Central Station to return to Berlin, Paris, Cambridge, Gottingen, Copenhagen, Vienna and Zurich.

They were taking with them the most bizarre set of ideas ever concocted by scientists. Secretly, most of them probably agreed with Einstein that this madness called the quantum theory was just a step along the way to a more complete theory and would be overthrown for something better, something more consistent with common sense.


But how did the quantum theory come about? What experiments compelled these most careful of men to ignore the tenets of classical physics and propose ideas about nature that violated common sense ?

Before we study these experimental paradoxes, we need some background in thermodynamics and statistics which are fundamental to the development of quantum theory.


Quantum Mechanics Series – 1: introduction

Quantum Mechanics series Chapter 1

Quantum Mechanics is one of difficult and interesting topics for students at higher studies level.  

Introducing quantum Theory…..

Just before the turn of the century, physicists were so absolutely certain of their ideas about the nature of matter and radiation that any new concept which contradicted their classical picture would be given little consideration. Scientist were considering that they know almost everything and not much is left to understand. 

     Isaac Newton

      James Clerk Maxwell








Not only was the mathematical formalism of Isaac Newton (1642-1727) and James Clerk Maxwell (1831-79) impeccable, but predictions based on their theories had been confirmed by careful detailed experiments for many years.  The age of Reason had become the age of certainty !


Classical Physicists

What is the definition of classical” ?

By classical is meant those late 19th century physicists nourished on an academic diet of Newton’s mechanics and Maxwell’s electromagnetism – the two most successful syntheses of physical phenomena in the history of thought.

Testing theories by observation had been the hallmark of good physics since Galileo (1564-1642). He showed how to devise experiments, make measurements and compare the results with the compare the results with the predictions of mathematical laws.

The interplay of theory and experiment is still the best way to proceed in the world of acceptable science. 

It’s All Proven (and Classical)…

During the 18th and 19th centuries. Newton’s laws of motion had been scrutinized and confirmed by reliable tests.


“Fill in the Sixth Decimal Place”

A classical physicist from Glasgow University, the influential Lord Kelvin (1824-1907), spoke of only two dark clouds on the Newtonian horizon.

In June 1894, the American Nobal Laureate, Albert Michelson (1852-1931), though he was paraphrasing Kelvin in a remark which he regretted for the rest of his life.


The Fundamental Assumptions of Classical Physics

Classical physicists had built up a whole series of assumptions which focused their thinking and made the acceptance of new ideas very difficult. Here’s a list of what they were sure of about the material world.

(1)     The universe was like a giant machine set in a framework of absolute time and space. Complicated movement could be understand as a simple movement of the machine’s inner parts, even if these parts can’t be visualized.

(2)     The Newtonian synthesis implied that all motion had a cause. If a body exhibited motion, one could always figure out what was producing the motion. This is simply cause and effect, which nobody really questioned.

(3)     If the state of motion was known at one point – say the present – it could be determined at any other point in the future or even the past. Nothing was uncertain, only a consequence of some earlier cause. This was determinism.

(4)     The properties of light are completely described by Maxwell’s electromagnetic wave theory and confirmed by the interference patterns observed in a simple double-slit experiment by Thomas Young in 1802.

(5)     There are two physical models to represent energy in motion : one a particle, represented by an impenetrable sphere like a billiard ball, and the other a wave, like that which rides towards the shore on the surface of the ocean. They are mutually exclusive, i.e. energy must be either one or the other.

(6)     It was possible to measure to any degree of accuracy the properties of a system, like its temperature or speed. Simply reduce the intensity of the observer’s probing or correct for it with a theoretical adjustment. Atomic systems were thought to be no exception.


Classical physicists believed all these statements to be absolutely true. But all six assumptions would eventually prove to be in doubt. The first to know this were the group of physicists who met at the Metropole Hotel in Brussels on 24 October 1927.