The Most Beautiful Equations in Physics
Little mathematical entities that describe the universe
One of the most fascinating things I find about learning physics is that we can describe almost every physical phenomenon in terms of mathematical equations and it gives me a great kick to understand such equations and comprehend their importance and their role in our understanding of the universe. These little mathematical equivalence entities enable us to open new doors that lead us to new and new discoveries. From Einstein’s mass-energy relation, which shows the intrinsic property of matter to convert itself into energy, to the Dirac equation that tries to include gravity in the frame of quantum mechanics and predicts the existence of anti-particles, these mathematical relations play a significant role in our understanding of nature. In this story, I shall talk about five important equations that have helped us shape our understanding of reality.
Schrödinger’s Equation
Every matter exhibits a duality nature and behaves as a wave. However, the wave nature of the matter was inversely proportional to the mass. While bigger particles could have negligible wave nature, smaller particles like electron’s wave nature were significant and could not be ignored. Looking at the energy level in an atom, electrons form a circular standing wave with an integral number of wavelengths. This was the reason why certain orbits had specific energy levels. To understand the possible trajectory an electron could have, in the 1920s, Schrödinger started to analyze this problem starting with Hamilton Jacob's theory as a reference.
Though this was an equation for classical physics, it helped a great deal in furthering the study at the quantum level. He came up with the equation in 1925 and told the whole world that just as sound is a mechanical wave and light is an electromagnetic wave, the electron can be considered as the cloud of probability density which simply means that the exact trajectory of the electron could not be determined, but the probability of it moving at certain vector field could be determined. The probabilistic character of nature didn’t sit well with some physicists then, and now.
Einstein’s mass-energy equation
This is the equation that almost everyone instantly recognizes and yet might not know the physical implications of it. This equation was formulated by Einstein in 1905. He didn’t actually come up with the relation E=mc², but throughout the years of his workings on light, he formed this universal law. He wanted to study the relativistic nature of light, and before his approach, Maxwell had already given the constant velocity of light c. But this bothered Einstein’s mind. He procrastinated on what velocity means because, in order to have speed, it must be relative to something. But then he arrived at the conclusion that the velocity of light is the absolute speed, that no matter how fast the observer is moving, he will always see the light at a constant velocity of c. In his initial conclusions, papers, he stated that if a body gives off energy in the form of radiation, the mass of the object diminishes by L/c²(L being the symbol for energy used by Einstein), which later gave birth to the famous mass-energy relation.
Newton’s gravitational equation
From a falling apple to the visualization of a falling moon, Newton came up with the universal laws in classical physics which are still prevalent to this day. Newton is best known for laying the foundation of classical mechanics also known as Newtonian physics along with calculus. When Newton first visualized the falling of the Moon, he was in awe thinking why doesn’t it directly fall to the earth? Then he realized that the moon is in a constant state of freefall, and forming an orbit around earth. The same principles now apply to satellites. Furthermore, in 1687, he gave the relation that the gravitational force between two objects separated by a certain distance is,
where G is the universal gravitational constant and its value is given as 6.67408 × 10^-11 Nm^2/kg^2. Though this equation doesn’t hold its ground on the quantum level, for the larger objects that we encounter daily and for the convenience, it holds true for them with errors that could be neglected. This law has been used for launching Rockets to find the position of Geo-stationary satellites.
Euler- Lagrange equation
This could somewhat be understood as an analogy to the famous F=ma. It means, that if you have a Lagrangian and we substitute it into the second order differential equation called the Euler-Lagrange equation for a certain dynamical system, then we will eventually come up with the relation, F=ma.
This was developed by the Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange in the 1750s, which gives it its name. For a simple force system, relation F=ma could be used. But this equation shines when there are forces whose vectors are significantly complicated. This was developed in connection with their studies on the tautochrone problem. When Langrage solved this problem in 1755, he sent it to Euler, and both further developed the equation. It was found that it had a great significance in Mechanics, which led to the formulation of Lagrangian Mechanics as well.
Dirac equation
British Physicist Paul Dirac came up with this equation in 1928. Until the time of discovery, various attempts had been made to correlate the quantum theory of the atom and make it compatible with the theory of relativity but had failed. As the Schrodinger equation is the nonrelativistic theory of electrons and Klein Gordon equation is the relativistic theory of scalar particles, we now arrive at the relativistic theory of electrons which is given by the Dirac equation. This sort provided a platform for the physicists to talk about electrons in a sense that is consistent with relativity. This could be considered a foundation for Quantum field theory. It introduced us to the concept of spinners which are invaluable in Quantum field theory. Also, it gives the theory of Fermions (a particle such as an electron, proton, or neutron whose spin quantum number is an odd multiple of 1/2). Dirac’s original intentions were fulfilled but he had no idea about the doors it opened to studies on various fields that were not just limited to electrons.
Contributed by Rishab Karki and curated by the author.
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