Key Concepts in Quantum Mechanics 

Chapter 1: Newton and the love of his life

Sir Isaac Newton enlightened us with the equation F=ma which means that the net force is equal to mass times acceleration. Suppose you throw a ball into the air, with newtons laws we can find the exact position, momentum, etc. of that ball at any given time! 

Chapter 2: All fun in games until Particle Physics

Oh well well! F=ma is too good to be true. When it comes to particle land, Newton’s laws don’t work anymore

Particles are too small and disobedient to obey newton's laws.

Chapter 3: The plot twist

 Sir Neils Bohr

Neils Bohr comes in and tells us, ”Hey, I know quantum things!”

Basically, he said that electrons stay in there particular shell because they have predefined amounts of angular momentum!

Oh well! Niels Bohr only did this experiment for Hydrogen and hence, didn’t have an explanation as to why this occurred.

Now this French physicist with a very fancy name came in and he claimed that electrons existed as waves! 

Sir Louis DeBroglie

He gave us the “DeBroglie Relationship” 

If electrons exist as waves, this must mean that they have momentum! 

DeBroglie said that Electrons are like Standing Waves!

So now, this cool dude called Heisenberg says that we can’t find both the momentum and the exact position of quantum objects.

Sir Werner Heisenberg

But, we can find other things like the energy level and the wave function. He gave us the "Heisenberg Uncertainty Principle".

The Heisenberg uncertainty principle is a bounding principle and it states that there is an inverse proportionality to our knowledge to the momentum and the position of particles. The more we know the position of a particle the less we can know about its momentum and vice versa.

Chapter 5: Lord and Savior Schrodinger

Dr. Erwin Schrödinger

 Dr. Erwin Schrödinger was a big philosophy fan, he was deeply entangled in the reality of life and studied various ancient texts like the Upanishads. 

"If the world is indeed created by our act of observation, there should be billions of such worlds, one for each of us. How come your world and my world are the same? If something happens in my world, does it happen in your world, too? What causes all these worlds to synchronize with each other?". 

- Dr. Erwin Schrodinger

In his book "What is life" he also dwells upon this topic extensively and gives us material capable to put millions in an existential crisis.

He is called the father of quantum mechanics, like all this also started with questions. 

Dr. Schrodinger asked two main questions:

>If the electron is a wave can we describe it as a mathematical function, which can give us the position, momentum and energy values?

>Can this function explain all electron orbits for all elements?

He starts out with the conservation of energy:

Perfect! Now the energy is described in the relationship of momentum and mass!

Now, lets introduce the concept of a sine wave denoted by ψ (psi).

Where:

ωt describes how the velocity changes in respect to time

Kx describes the change in frequency (which is inversely proportional to wavelength)

A describes the change in amplitude.

Now putting those two equations together we get the famous time independent Schrödinger's equation.

This is not as complicated as it looks.

This essentially means that, if we want to know how much energy an electron is allowed to have we need to add up the potential and kinetic energy.

In a mathematical sense, this equation describes a wave.

Chapter 6: Particle in a box model

Now, we're going to look at an electron inside a box of length L.

For a confined box the Schrödinger's equation transforms into:

If we look closely, most of these terms is a constant except n which denotes the energy level.

ħ denotes the constant h/2π  (h being Planck's constant)

π denotes the irrational number we all know!

m denotes the mass of an electron.

L denotes the length of the box we place our electron in.

These are some solution for even and odd n.

Now lets actually look at the particle in a box model. The important thing to remember is that the electron can NOT exist outside the box (i.e E outside the box is 0)

We can see that the energy function yields us waves for n=1,2,3...

What we just looked at were the 𝜓 functions.

If we take all the possible values of this and square them                ,

We will get the probability of the electron being present.

This is the contribution of Max Born.

The Schrodinger's equations wave isn’t just a wave in terms of 

Physicality, but it’s a probability wave.

For n=1

For n=2

For n=3

What we are seeing in the darker yellow color is the probabilistic wave function.

This generally means, if I open up the box at a random time the electron has a higher chance of existing at the places where the probabilistic wave is high.

Now, things get more interesting!

We will see the particle in a box model again, but this time in 3D. This will give us the idea of Electron Clouds.

For n=1

For n=2

For n=2, L=1

For different subshells:

This does not mean that electrons exist are flying around creating a cloud, but it means that the probability of the electron existing where the cloud is more dense is more.

For a visual explanation you may consider This Video.

As always,

Thank you so much for reading my blog!

-Ruhan