## Schrodinger’s Equation – A key equation in Quantum Mechanics

Until the beginning of the 20th century, we have developed laws and equations which govern the behaviour of relatively larger objects. In the early 20th century, we started digging deeper and deeper into the matter and found out that the laws we have developed over centuries don’t work at atomic and sub-atomic levels. This was the beginning of Quantum Physics.

Quantum Mechanics was started by Max Planck who is known as the Father of Quantum Mechanics. Quantum Mechanics tells that matter also have wave-particle duality and it is significant at the atomic scale. Thus it was thought that a wave equation might explain the properties and thus the first such wave equation was developed by Erwin Schrodinger and is named as Schrodinger’s equation.

Schrodinger’s wave equation looks like:

is the reduced Planck constant,
m is the electron mass,
is the Laplacian operator,
Ψ is the wave function,
V is the potential energy,
is the energy eigenvalue,
(r) denotes the quantities are functions of spherical polar coordinates (r, θ, φ)

As always, I am not going to bore you explaining those complex terms. Instead, let us know why is this equation so important. We all know about Newton’s laws of motion, they changed the course of physics as we could tell the destination of a moving body by knowing its initial conditions. Schrodinger’s role in Quantum mechanics is somewhat similar to that of Newton.

The Schrodinger equation is used to find the allowed energy levels of quantum mechanical systems (such as atoms, or transistors). The associated wavefunction gives the probability of finding the particle at a certain position. A particle whose position and momentum was completely unpredictable according to the uncertainty principle got at least boundaries by using Schrodinger’s equation. It really means a lot. Though there are many implications of this equation, the above one is the most important.

I hope you got a glimpse of Schrodinger’s equation.

## #5 Einstein’s Field Equation

Einstein, an absolute genius, has given a lot of groundbreaking theories and the one which completely changed our understanding was the Theory of Relativity. Both Special and the General theory of relativity being two of the most famous and fundamental scientific theories have a lot of revolutionary equations. In this article, we are going to talk about one such equation which is Einstein’s second most famous equation after E=Mc2. It is Einstein’s Field Equation

where,

$Einstein's Field Equation$

– Einstein tensor

$Einstein's Field Equation$ – Ricci curvature tensor

$Einstein's Field Equation$– Scalar curvature

$g_{\mu\nu}$– Metric Tensor

$\Lambda$– Cosmological constant

$G$– Newton’s gravitational constant

$c$ – Speed of light

$T_{\mu\nu}$– Stress-energy tensor

Did not understand the terms? This equation looks pretty clean and simple but isn’t that simple, it involves many crazy Tensor calculations, matrices, etc. So, don’t get afraid, all these terms are pretty complex and need a lot of explanation which I won’t be doing here because it will be boring (Still want to read about these terms: Read here). Instead, let us know the importance of the equation and why is it so famous.

First of all, you have to know that this is not a single equation, this equation hides and summarizes 10 partial differential equations in it. This small equation that doesn’t look too complex, (at least not as complex as Schrodinger’s wave equation :-)) completely changed our understanding of the universe. It revolutionized the concept of gravity.

According to General Relativity, Gravity is because of mass curving space-time. Einstein’s field equation explains the mass, energy, momentum, pressure distribution across space-time and how does these curve space. The left side of the equation (which has Einstein tensor) is about the curving of space and the right side (which have stress-energy tensor) is about the distribution of energy and mass, so, it relates mass and energy to curvature of space-time.

This equation has been used to predict the existence of black holes and gravitational waves. It explains how black holes warp space-time. Einstein’s equation can tell us how our universe has changed over time and offers glimpses of the earliest moments of creation. This equation has been used to understand many processes of the universe.

So, that was a glimpse into Einstein’s Field Equation, Hope you liked it.

## #4 Maxwell’s Equations – The equations behind major modern technologies

Maxwell’s set of four equations forming the basis for electromagnetism are as important as Newton’s laws in mechanics. Maxwell’s equations are applied in almost all modern technologies. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar, etc. Firstly let us see these four sweet equations one by one and then discuss them as a whole.

### 1. Gauss’ Law or Maxwell’s first equation

Electric charges produce an electric field. The electric flux across a closed surface is proportional to the charge enclosed.

### 2. Gauss’ Law for Magnetism or Maxwell’s second equation

There are no magnetic monopoles. The magnetic flux-and-faradays-law-quantitative across a closed surface is zero.

### 3. Faraday’s Law or Maxwell’s third equation

Time-varying magnetic fields produce an electric field.

### 4. Ampere’s Law or Maxwell’s fourth equation

Steady currents and time-varying electric fields (the latter due to Maxwell’s correction) produce a magnetic field.

### Maxwell’s Equations as a Whole

As a whole, what do Maxwell’s Equations mean?

Maxwell’s equations describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. One important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at a constant speed (c) in the vacuum, the “speed of light“. These electromagnetic waves have a wide variety of usage, they are used in small things like routers to big things like search for aliens using radio telescopes and all these devices involves the use of Maxwell’s equations. Maxwell understood the connection between electromagnetic waves and light with these equations in 1861, thereby unifying the theories of electromagnetism and optics.

Using these equations only we got a relation between the speed of light (an electromagnetic wave) and permeability and permittivity of free space. So now you know why the speed of light depends on the medium it is traveling in.

So, I hope you got the essence of Maxwell’s equation. So, if you want to know about these equations in-depth, then consider reading the references.

Reference:

## # 3 Einstein’s Mass Energy Relation

Einstein, an absolute genius, has given a lot of groundbreaking theories and the one which completely changed our understanding was the Theory of Relativity. Being one of the most famous and fundamental scientific theories, it has a lot of revolutionary equations. In this article, we are going to talk about one such equation which is one of the most well-known equation in Physics. It is the Mass-Energy Relation.

Where,

• E is energy
• m is mass
• c is the speed of light

I am sure that most of you might be knowing this equation. Did you ever think that why is this equation so important and groundbreaking? Let us know.

Before Einstein giving this equation, the world used to think that energy and mass are two very different things. Newtonian mechanics told that object at rest has no energy but this equation opened our eyes and told that even an object at rest has an energy of mc2. This equation has drastically changed our understanding and told us that mass and energy are the same and are interconvertible.

So, the equation tells that you can create mass out of pure energy and you can get energy from the mass. When you add one joule of energy to a system, you are adding 1.11×10−17 kg of mass to that system. In daily life this is negligible but we see its consequences when we travel at speeds close to that of light. Einstein has given a mass velocity relation which tells the increase in mass with an increase in velocity.

Coming to the reverse process, the mass to energy conversion, 1 Kilogram of mass is equal to 9×1016 Joules which is really huge. If we could invent some method which can harness complete energy from mass then we wouldn’t need any other source for power. We know a process called Nuclear Fusion which can harness 0.7 times of this energy but sadly we don’t have the technology to do nuclear fusion on earth yet.

So, I hope that you got an essence of what this equation means and how did it change our understanding regarding mass and energy.

## #2 Einstein’s Mass Velocity Relation

Albert Einstein, an absolute genius has given groundbreaking theories and one theory among them which was a revolution was the Theory of Relativity. It took 100 years to get strong proofs for this theory.

In 1905 Einstein proved that the laws of physics are the same for all non-accelerating observers and that the speed of light in a vacuum was independent of the motion of all observers. This was the Special Theory of Relativity. Einstein then spent 10 more years to include acceleration in his theory and published his General Theory of Relativity.

These theories had many revolutionary equations and relation. Let us see one such equation which is Einstein’s mass-velocity relation.

Where,

• v is the magnitude of the velocity
• c is the speed of light
• m0 is the rest mass of the body
• m is the relativistic mass

Now, what is so special about this mass-velocity relation. Let us understand what does this equation mean. This equation tells us that if we travel with speeds approaching the speed of light then our mass will increase with speed. So, do you really gain mass? Do you become fat? No, let me tell you that we take inertial mass into consideration here. Inertial mass measures an object’s resistance to being accelerated by a force. Now, every body has rest mass which is m0 here in the formula.

If an object moves with some speed then the kinetic energy adds up to the rest mass and overall the inertial mass increases. This means that if an object approaches light speed then its inertial mass increases rapidly and accelerating it further becomes more and more difficult. If any object reaches light speed its inertial mass approaches infinite according to the above equation. Thus, Einstein stated that no object can travel faster than light speed.

## #1 Most Beautiful Equation in Mathematics – Euler’s Identity

Mathematics is a vast subject and is considered as the mother of all the science as it is a tool to solve problems in all other sciences. Mathematics is a subject with so many equations which can fill oceans.

There are many wonderful equations in mathematics which really surprises us as to how can so many different things can be connected with a single equation.

One among such equation is the Euler’s identity, and it is considered as the most beautiful equation in mathematics especially Number theory. A poll of readers conducted by The Mathematical Intelligencer named Euler’s identity as the “most beautiful theorem in mathematics in 1990.

So, Euler’s identity is

Where

1. e is the Euler’s number
2. Π is the ratio of circumference to diameter of a circle
3. i is the imaginary unit satisfying   i2 = −1

Now, let us know why is this equation so special. We know that Mathematics has Real numbers and Complex numbers which are two different things. Then Real numbers have rational numbers and irrational numbers which are two different parts of real numbers.

To understand its beauty we also have to know about transcendental numbers. These numbers are real or complex numbers which are not a solution of a non-zero polynomial equation.

Now, as we have good background let us know the specialty of this equation.

In this equation, three of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation

The equation is considered as deep mathematical beauty because it connects very different things in mathematics and those are

1. Two irrational and transcendental number that are e and Π
2. The imaginary unit i
3. Rational number 1 and 0

I hope you understand and feel the beauty of this mathematical equation.

Reference:

1. Euler’s Identity(Wikipedia)