# Wavefunction and Max Born

One of the beautiful theories in physics, which is more than hundred years old, is Quantum Mechanics (QM).  In 1900 when Blackbody Curve was satisfactorily explained by Max Planck, Quantum Mechanics saw its birth.  Later many great scientist of 20th century like Einstein, Bohr, Hertz, Heisenberg, Dirac, Schrodinger, Born, de Broglie developed quantum mechanics the way we see it now.  Almost all the people whom I have mentioned above have received Nobel Prize for their work on developing QM.  But to me the important thing is the idea behind the each stage of development in the QM.  Some of the concept and the experimental results were unable to explain with the knowledge of Classical Mechanics which was well developed during that time.  The new ideas and concepts which came up to explain these strange results were led to the development of this field.  Here I want to focus on some ideas/concepts of QM developed by different people and more importantly by a great mathematician and physicist Max Born.

Max Planck was the first person who broke the “continuity” concept of classical mechanics and introduced the “discreteness” in the energy called ‘quanta’ (packets) of energy in order to explain the “Blackbody Curve”.  The formula which he gave was in fact a perfect fit for the observed experimental curve.  His formula for the discrete energy includes the constant h = 6.023X10^-34 Joules, where h is Planck’s constant.  He was awarded Nobel Prize for his work in the year 1918.  Later in 1905 Einstein also came up with similar idea of ‘photon’ (packets of energy) to explain the energy concept in Photoelectric Effect.  The concept of ‘discreteness’ (Einstein called ‘photon’, Planck called ‘quanta’) again fits perfectly for the Photoelectric Effect.  So, it is the idea of quantization principle (discreteness) which led to the satisfactory explanation for all these experimental results.  And Einstein received his Nobel Prize for his work on Photoelectric Effect in 1921.  Another great scientist Neils Bohr who was a student of Rutherford and he was working on the model of an atom.  Rutherford atomic model was not satisfactorily explaining all the observed experimental results.  His student Bohr applied the same ‘quantization principle’ to angular momentum and by using Plank theory he was able to explain all the observed phenomena through his new atomic model.  Again quantization principle perfectly fits the theory.  So, in quantum world every quantity is quantized.  Today we know even space and spin is quantized!  But in classical world this is not at all true, everything looks continuous!  The idea of quantization by different people has led to the development of this unknown world known as ‘quantum world’.

Image: Max Planck, W Heisenberg, E Schrodinger, Max Born.

In 1924 another great scientist de Broglie came up with a strange concept that almost all particles which has mass is associated with wave nature with a wavelength of h/mv, where v is the velocity of particle, m is mass and h is Planck’s constant.  This is called Wave-Particle duality, where all particle exhibit both the properties of wave as well as particle.  This was a big blow in the development of QM and its a strange concept when compared to Classical Mechanics.  The theory predicted that if a car (mass m) is moving at a velocity ‘v’ is actually associated with a wave of wavelength ‘h/mv’.  Classically it is impossible for a car to move like wave, but still the theory predicted wave nature of a car and gave the value of wavelength.  But in reality the effect of wave nature in a macroscopic world (mass is large) is very small and can be neglected, which means the car’s wavelength is very small and its wave nature can be neglected!  But when it comes to an electron which has very less mass, the wave nature of an electron magnifies and it can be studied under suitable condition.  The concept of QM is to study the dynamics of these tiny particles such as electrons and elementary particles.  Theory of QM is applicable to almost all particles, but for macroscopic particles the effects are very negligible and it is neglected.  This actually makes QM theory a Universal Theory to some extent.  For predicting the strange (classical sense) theory of wave-particle duality of nature de Broglie was awarded Nobel Prize in 1929.  Meanwhile in 1925 W Heisenberg explained the basic principles of QM through his papers ‘Quantum- Theoretical Mechanics based exclusively on relationships between quantities observable in principle’.  And in the very next year E Schrödinger came up with a different approach to QM through his ‘wave equation’, which is a very famous equation in physics called as ‘Schrodinger Equation’.  In classical mechanics we write the famous Newton’s law as F=ma, where ‘F’ is force applied, ‘m’ is mass of an object and ‘a’ is the acceleration of an object.  By solving this equation for a particular system (ex pendulum) we can get the equations of motion.  Which means the solutions can predict the motion of a system (ex Pendulum) after some time (t).  Newton’s law is actually the foundation to classical mechanics. And this equation holds good for almost all systems classically.  On the other hand in QM the same equation is replaced by ‘Schrodinger Equation’ which is given below.  Compare to Newton’s law, Schrödinger Equation is not so simple and difficult to solve.

Meanwhile in 1927 Heisenberg came up with another law known as ‘Uncertainty Law’.  He said it is impossible to measure the position and momentum (mass x velocity) simultaneously for microscopic particles.  It means we cannot say the position and momentum of particle (ex. electron) at the same time.  This kind of simultaneous measurement is impossible in QM.  But in classical mechanics we can easily say the position of car and its momentum at any point of time simultaneously.  In macroscopic world everything looks simple and obvious, but the same is not true in microscopic world.  This restriction has reveled striking feature of microscopic world (nature) that we cannot pin point where is the electron in a system at a given point of time.  So, the traditional way of writing atomic model (fig a) breaks down and electron never revolve around the nucleus in a circular way as we have learnt in school.  Instead the circular line becomes a disk of width ‘a’ (two dimensional representation) (fig b).  Electron is present in the region of disk, but where in the region is unknown even today. There is a debate that may be we lack the technology of doing experiment and our instruments are incapable of doing such sensitive measurement.  But this is ruled out by scientist!  It is nothing to do with our technology or the sensitivity of instruments, Uncertainty Law is a law in nature! Neither our future technology nor our highly sensitive instruments in future can go beyond this level.  It is just there in the nature, and it magnifies in microscopic level.

As I said the Schrödinger equation actually tells the dynamics of the particles (say electron).  It should describe how an electron moves in a given condition (ex. Infinite Square Well Potential).  While expressing this equation Schrodinger assumed that all the required information about the particle (electron) is hidden in the quantity called wavefunction (Ψ). By applying the given condition (given potential) to the equation one should get the value of wavefunction which is normally in complex form.  After obtaining the wavefunction in its form, the required information should be extracted from the wavefunction.  And this wavefuntion is a not a localized function but a spread function!  Uncertainty Law which is present in nature is actually a hidden property of Schrödinger equation and it shows up in the wavefunction as a spread function (not localized like classical mechanics).  But the question is how do we extract the required information about a particle from wavefunction?  Here comes Max Born a mathematician and physicist with his revolutionary approach to QM through Statistical Mechanics in the year 1926 said ‘modulus psi square(IΨ(x,t)I^2) is actually gives the probability of finding the particle at a point x and time t. Which means squaring of modulus of a wavefunction is actually a real quantity not a complex number! (But remember wavefunction is a complex quantity)  And Max Born was the first person to identify this quantity as the ‘probability’ of finding electron at x at time t.  This is called Born’s Statistical Interpritation.    I feel this concept is a revolution in QM, because without this concept we cannot interpret the wavefunction.  The normalization concept which is the direct product of this idea is a very handy tool in handling wavefunction.  I feel the statistical approach is the most significant step in the development of QM.  All credits go to Max Born who identified ‘modulus psi square’ is actually a probability.  Heisenberg and Schrodinger were awarded Nobel Prize in 1932 and 1933 respectively.  Max Born was awarded Nobel Prize on Dec 11, 1954 for his statistical approach to QM, after 29 years of his most important concept.  Coincidentally his Nobel Prize ceremony date was his 72nd birth anniversary (Born: 11 Dec 1882, Died: 5 Jan 1970)! In his Nobel Lectures he began his talk like this ” The work, for which I have had honour to be awarded the Nobel Prize for 1954, contains no discovery of fresh natural phenomenon, but rather the basis for a new mode of thought in regard to natural phenomena.”  What matters more is the way we think on some observed results/phenomenon.  All great scientist have just did that.  If you ask me what is QM, in simple words I can say Quantum Mechanics is just solving Schrodinger Equation.
(Max Born visited Bangalore in the year 1935.  He worked with Sir C V Raman at Indian Institute of Science for six months.)

On an application level QM has wide verity of application.  In a real world application it is applied in Laser, MRI, Quantum Cryptography, Transistor and many more.  It is also applied in the study of Atoms and Molecules, Nuclear Physics, Astrophysics, Solid State Physics and many more.  At some point of time I said QM is a Universal Theory, but still it fails to explain some important concepts.  Even today there is a debate on the fundamental concepts and principles of QM.  It is still not a complete theory even after more than hundred years.  Never the less it is one of the famous theories in Physics.

In the developing stages, the unique ideas by the scientist played a vital role in shaping QM in a proper way.  For student like us the important thing is the ideas behind each successful stage of development in QM or in general Science.  The ideas/concepts which are different from normal thinking can actually change the world totally.  This has actually happened in the history.  When I met my friend Suraj at Bangalore Planetarium, we were discussing about the great scientist and their concepts.  He said “look Einstein is famous for E=mc^2, Stephen Hawking is famous for his chair, Feynman is famous for his teachings, but there are other great scientist with their ideas/concepts, revolutionized the world, but they are not famous among us (to some extant this is true even among science students and teachers).  Everyone tells Faraday, Einstein, Edison, Hawking but nobody remembers Maxwell, Tesla, Born, Dirac, Pauli and many more who are equally important like others.  The reason is we never tell the stories of these great peoples among students.   I feel it is the stories of these great people and their ideas are important to students rather than the confined textbook chapters.

[If you are still interested in quantum mechanics and its development I strongly recommend you to read the preliminary chapters and Nobel Lectures in the book “Quantum Mechanics: Theory and Applications” by A.K. Ghatak and S. Lokanathan.     (S. Lokanathan was my Classical Mechanics teacher in REAP Course at Bangalore Planetarium)].

[I have just shown only four scientist pictures in this article, but all of them are equally important.]

Viswa Keerthy S
21/10/2013