UNDERSTANDING THE SCHRODINGER WAVE EQUATION AND ITS IMPORTANCE.

The quantum theory that was established by Max Planck, Albert Einstein and Niels Bohr enjoyed experimental success for nearly two decades. But by the year 1920, this theory was soon referred as the ‘old quantum theory’. From here on, a ‘new quantum theory’ would be developed by a radical group of scientists comprising Schrodinger, de-Broglie, Dirac, Heisenberg, Ehrenfest, Born and Jordan.

The old quantum theory showed that the radiation or the light has particle like nature associated with it. Scientists then questioned the converse of this: Can we associate a wave with all the material objects? The answer was Yes! The de-Broglie hypothesis, that every particle has a wave nature associated with it was confirmed by the famous Davisson-Germer experiment.

Quantum mechanics now needed a sound mathematical platform. With time, two mathematical formulations of quantum theory emerged: Wave mechanics by Schrodinger and Matrix mechanics by Heisenberg. Both the formulations had different approach towards the same problems and yet both provided the correct results. Today, let us talk about the flagship equation of quantum mechanics, the analogue to Newton’s second law in classical physics: The Schrodinger Wave Equation.

The mathematical form of the Schrodinger Wave Equation

This equation tells something you all know. You have studied the meaning in your high school. The meaning of the equation is:

The total energy of a particle is the sum of its kinetic and potential energy.

Is that all it means? Yes! If you look closely, this is what the Schrodinger wave equation means. The left part of the equation adds kinetic energy(h^2/2m ∇^2) and potential energy (V) and sums it equal to the total energy E. SWE is the most fundamental equation of quantum mechanics. All the information about a quantum system is already embedded in this equation just like everything about a mass m is contained in Newton’s second law. The difference is that in classical mechanics, the fundamental quantity is position and in quantum mechanics, it is wavefunction.

Consider throwing a particle in a box, you can only tell where the particle will probably be. You can visualize it using an analogy. Suppose your friend is in his bedroom. Knowing the nature of your friend, you can tell that there is 80% probability that he is playing video games, 15% probability that he is sleeping and 5% probability that he is studying. So there is a spread of probability (probability distribution) but when you open the door, he is, say, sleeping. The distribution collapses into one value.

This is what this ψ(x) tells us. ψ(x) is the wave function of the particle. Since by being probabilistic, a particle exhibits a wave like nature in the box, ψ(x) is hence known as the wave function. Itself it doesn’t represent anything physical but its square represents the probability of finding a particle in the particular state.

The image above shows a typical wavefunction of a particle in a box. One feature of the wavefunction is that it must be zero at the boundary of the box. This means the probability of finding the particle at the boundary is 0. Look carefully, this is exactly what every graph shows.

Another important implication of SWE, after energy conservation is quantization of energy. Consider the same problem of particle in a box as taken above. When you solve SWE for this particle, the energy levels that we get for a particle are quantized. This means the particle can only take certain energy values, unlike in classical mechanics. See Particle in a box.

The reason why this equation is the analogue of Newton’s second law from classical mechanics is that in the latter, we solve for the position of the mass. Once we know the position, we get everything: velocity, acceleration, momentum, energy etc. Similarly in quantum mechanics, the SWE solves for the wavefunction of the particle, from which we can learn about the entire quantum system. So the Shrodinger wave equation tells the evolution of the wavefunction of a particle. If you know the wavefunction, you can derive all other quantities of the system.

THE ENTRY OF RUTHERFORD AND BOHR IN QUANTUM PHYSICS.

Today is the second day of our Quantum Week. In the first article (click here to read), we discussed 3 major events that gave a blow to the old classical physics and showed that a new theory is required to explain what was happening at the tiniest level. Today we will learn the work of two eminent scientists, who had a major contribution in explaining the structure of the atom: Ernst Rutherford and Niels Bohr.

In 1896, J.J. Thomson discovered the electron. It was a turning point in the history of mankind as it was the first elementary particle to be discovered. Based on his discovery, Thomson tried to explain the structure of the atom. So using best of his knowledge, he developed the Plum Pudding model of the atom. In this model, the atom is considered to be a positively charged sphere with electrons embedded in it, just like plums on a pudding.

This model of the atom had some unseen flaws in it. They were only noticed after a major experiment by his own student, Ernst Rutherford in collaboration with Geiger and Marsden. The name of the experiment was the Gieger-Marsden Experiment or the Rutherford’s alpha scattering experiment. In this, alpha particles were bombarded on a gold foil kept at some distance. It was observed that the alpha particles were deflected by the gold foil in different directions. A very few alpha particles were deflected at an angle close to 180 degrees (this is opposite to the incident direction). These few alpha particles startled Rutherford.

                                    

If Thomson was right, then there would have been no deflection of the alpha particles. This is because the electrons that are embedded in the positive sphere of atoms are 8000 times lighter than the alpha particles. This was so unexpected that Rutherford later said, “It was the most amazing moment of my life. It was as if I fired a 15 inch shell on a tissue paper and it came back and hit me.“

Using his observations, Rutherford developed his new model of the atom.According to his interpretation, an atom was mostly empty space, with most of its mass being concentrated at its centre in the form of positive nucleus. The electrons revolved around the positively charged nucleus in certain predictable paths known as orbits at very high speeds, just like the planets revolving around the sun. Due to this similarity, his model was also named as the planetary model of atom.

No doubt, Rutherford’s model of atom was based on experimental observations, still it failed to explain certain things. The first thing it could not explain was the stability of an atom. According to Maxwell, any charge that accelerates in the presence of an electric field will lose its energy, consequently emiting radiations. Since, in Rutherford’s model of atom, electrons are accelerated due to electric field produced by the nucleus, so they should also lose energy, and should collapse into the nucleus tracing a spiral path. However, the atom was observed to be quiet stable, which means that Rutherford’s model didn’t go well with Maxwell’s theory. This was its first limitation. The second thing that Rutherford’s model could not explain was the arrangement of electrons in their specific orbits and didn’t tell anything about their energies. Moreover, electrons were supposed to be continuously losing energy, if this was true, then the atom should have been observed to emit radiations of continuous frequencies, but this was not the case. On the contrary, the atoms were only observed to emit radiations of certain discrete frequencies.

                                           

                                                                 Rutherford-Bohr Model of an Atom

Due to these shortcomings, again there was a need for a modified model of atom and this is when Niels Bohr came to rescue. Bohr proposed his model of atom in 1915. Since the Bohr model is a modification of Rutherford’s model of atom, so it is also known as the Rutherford-Bohr model.

Bohr proposed that the electrons are moving in fixed orbits around the nucleus, having a fixed set of energies.This explained the stability of an atom. The energy of an orbit is related to its size, ie. the nearest orbit to nucleus being the smallest one has the minimum energy, with the farthest one having the maximum energy. Bohr stated that the angular momentum of electrons revolving around the nucleus is quantized. Moreover, according to his model of atom, the electrons jumped from orbits of lower energy to those of higher energy by absorbing energy of certain frequencies and they emited radiations of certain frequencies while jumping from higher to lower energy states. This explained the fact that atoms only emited EM radiations of certain fixed frequencies. Bohr’s model was also helpful in explaining the spectral emission lines of atomic hydrogen. 

Even though, not all the early atomic models were accurate and some discrepancies existed, still all of them formed the basis of the quantum mechanics and played a key role in the development of quantum mechanics in one way or the other. Time and again, scientists are developing new and improved theories on what the atom looks like, still the most commonly used model of the atom till date is the Rutherford-Bohr model, thus making Rutherford and Bohr, two of the most important pillars of quantum theory!

Rutherford (Left) Niells Bohr ( Right)

Reference: www.secretsofuniverse.in

                                                                       

Simple Meaning of Time Dilation

WHAT DOES ‘TIME DILATION’ REALLY MEAN?

Equation of Time Dilation

 Meaning of Equation

The faster you travel through space, the slower you travel through time

To understand this, let us go back in the year 1905 and meet a patent clerk at Bern, Switzerland who is trying to find a solution to a question no one was asking. So, Einstein had this question: Suppose I have a master clock in a town that sends signals to synchronize all the other clocks in the town, at the speed of light. The clocks will synchronize. No problem so far. But what happens if we try to send the signal to a clock on a moving train? Light will have to speed up or slow down to catch up with the train. But according to Maxwell’s equations of electrodynamics, light always travels at the same speed. So either Newton is right  (in saying time is absolute, same for everyone) or Maxwell is. It can’t be both. Einstein wanted to know who was right between the two.

Einstein then came up with another brilliant thought experiment. Imagine yourself standing on a platform of a station. Two lighting bolts strike in front of you as shown above. The bolts were 300 m apart. For you, on the platform, they were simultaneous. But if a person views the 2 bolts by standing in the middle of a 300 m long moving train, they would not be simultaneous because he would be moving towards one bolt and away from the other. So then who is right? The person on the platform who says the bolts struck at the same time or the person on the train who says the bolts were not simultaneous? It turns out that- both are correct. Absolute simultaneity is not possible. Thus, time is not absolute. It means Newton is the one who gets it between the eyes.

Since light traveled equal distance for both the observers, they’ll always say that speed of light is constant but argue on the fact of simultaneity. Thus, time is not absolute. It is different for people in different reference frames.

Time dilation is an outcome of the fact that light travels at the same speed in all reference frames.  The flow of time in that frame adjusts itself to keep the speed of light constant. Hence, moving clocks run slower than the stationary clocks and this is exactly what the equation tells us. If v, the speed of moving clock gets larger, Δt’ gets larger, which means the time duration between two events gets longer and hence time slows down. So moving clocks run slower than the stationary clocks and this is exactly what the equation tells us.

Begening Of Quantum Physics.

HOW WAS QUANTUM PHYSICS BORN?

Quantum physics is more than 100 years old. Unlike relativity, the development of this theory was slow and required many minds. However, it is one of the most successful theory that the mankind has in its hands. Quantum mechanics has passed numerous tests through time. Today, on the first day of the Quantum Week, we’ll learn about the birth of quantum mechanics. What went wrong with classical physics and how the new theory saved the entire physics.

Quantum mechanics is the theory of matter. It answers the questions: What is matter made of? How the constituents of matter behave? At the end of the 19th century, matter was clearly divided into two main forms: particles and waves. Particles were the localized lumps of stuff that flew about like little bullets. The best investigated of the fundamental particles was the electron. The other form was wavelike matter. The one well-investigated form was light or, more generally, electromagnetic waves.

We had pretty much figured out how things work around us: The ripples on a pond, the motion of planets, the projectile etc. But in science, whenever we think ‘We Got It All’, nature gives new problems to solve. The clear distinction between the two forms of matter was soon going to get a blow in a series of events. We will discuss 3 major events that lead to the development of this beautiful theory.

1. Max Planck: The Blackbody Radiation

The typical plot of blackbody radiation

At the turn of the 20th century, a few engineers approached Max Planck and asked him, “How to make light bulbs more efficient?” This was the question that began quantum mechanics. Physicists were unable to explain the nature of the blackbody radiation. A blackbody is a perfect absorber and emitter of radiation. Sun, stars and light bulbs are good approximations of a blackbody. The curve of radiation from a blackbody has a particular shape as shown below.

Several attempts were made to explain blackbody radiation curve. The theories by Rayleigh-Jeans and Wien, however, provided partial success. This is when Max Planck comes across this problem. He makes an assumption to which he had no logic. He assumed that the light is made up of discrete, rather than continuous, packets of energy. Each packet has energy equal to hf, where h is Planck’s constant and f is the frequency of light. This solved the problem mathematically but Planck wasn’t convinced if his idea had any practical implication or logic. He said, “quanta (energy packet) are just a mathematical concept.” In December 1900, Planck announced his results and this was the beginning of quantum theory.

2. Albert Einstein: The Photoelectric Effect

In 1905, when scientists like Marie Curie, Max Planck, Phillip Lenard and Wilhelm Roentgen were at the peak of their career, there was a young, rebellious and ambitious man working as a third class patent clerk at Bern, Switzerland, trying to get noticed by these giants. That man was Albert Einstein, someone who was going to change the course of history in the coming decade. Einstein recognized the beauty and importance of Planck’s work, something that Planck himself could not.

When light is shed on a metal surface, electrons are emitted from the surface. It seems convincing that if we increase the intensity of light, more energetic electrons will be emitted and vice verse. Light intensity can easily be changed by moving the source of light towards or away from the metal. The experiments on photoelectric effect had a different story to tell. It was seen that the intensity of light has no effect on the energy of electrons that were emitted from the metal. Intensity did not matter to the ability of light to produce photoelectrons. All that mattered was the frequency of the light. If light was of low frequency, it could not generate photoelectrons, even if the light were very intense. If the light had a high frequency it could produce photoelectrons, even if the light was of very low intensity.

This, Einstein observed triumphantly and linked it with Planck’s quanta. If the frequency (energy) of the quanta that is shed on the metal is less, it will not have enough energy to knock out the electron. Increasing the intensity of the light did nothing more than increasing the number of light quanta showering on the cathode, all them too weak in energy to liberate a photoelectron.

Einstein won the 1921 Nobel Prize in Physics for his work on the photoelectric effect. His role in building the quantum theory is significant as the phototelectric effect actually shed more light on the particle nature of light.

3. Niels Bohr: The Atomic Spectrum

The most significant blow to the old classical physics came in the form of the atomic spectrum. When scientists passed an electric current through a tube containing gas (as shown below) and analysed the emitted light through the spectrometer (prism), they were shocked. They observed very fine spectral lines. The lines were sharp. Classical physics could not explain the sharpness of the spectral lines.

n 1913, Niels Bohr, a Danish Physicist, gave an explanation to this strange phenomenon. He attributed these spectral lines to the transitions made by the electrons from one orbit to the other. Bohr’s theory, however, had many flaws in it which he himself could not explain. His theory was the first leap to explain the structure of the atom, and thus an important milestone towards the development of quantum mechanics.

Different glow of gases when electric current is passed

The fine spectral lines of hydrogen that classical physics could not explain.

Reference:www.secretsofuniverse.in