How Energy is Created in Quantum Mechanics

The Creation and Annihilation Operators (collectively known as the Ladder Operators) are a very useful tool in quantum mechanics. We'll be taking a look at what they represent and how we can use them.

Before delving into the world of quantum mechanics, we'll first be looking at an important classical system - a classical harmonic oscillator. This is any system that undergoes simple harmonic motion, where the acceleration of our system is proportional to its displacement, and these quantities act in the opposite direction. A common example is a mass on a spring.

We can look at the potential energy of the system, which happens to be quadratic for a simple harmonic oscillator. Thus, we can make an analogy for our system by thinking of a ball rolling around in a quadratic potential well. The position of the ball matches the position of the real harmonic oscillator, and the height of the ball inside the well relates to the energy of our harmonic oscillator system.

This quadratic potential can also be translated to a quantum system. In other words, we can study a quantum harmonic oscillator (QHO). QHOs are difficult to find in real life, but many systems approximately behave like QHOs. The biggest difference between a QHO and a classical harmonic oscillator is that the QHO can only be found with specific energy values. The classical one can take any possible value in the range determined by the potential well.

The quantization of the QHO allows us to ask a few important questions. Firstly, what are the actual energy values of each allowed energy level? Secondly, how far apart are these energy levels? And thirdly, can our system transition between different energy levels over time?

This is where ladder operators come in. Without them, we would have to solve the Schrodinger equation (the governing equation of quantum mechanics) for every single energy level in order to find the energy values and the gaps. But it turns out our system can transition from one energy level to another if energy is supplied externally (such as by a photon) in order to go up in energy. And energy is released if the transition is downward.

Instead of solving the Schrodinger equation multiple times though, we can just solve it once, and then apply ladder operators to the found solution. The creation operator calculates the wave function of the most adjacent higher energy level, given the wave function we calculated from the Schrodinger equation. The annihilation operator goes the other way, calculating the wave function of the level just below. Thus, we can find all the wave functions by simply applying the ladder operators to one solution.

Additionally, the wave function allows us to count the number of quanta (e.g. photons) needed to get to a particular state from the lowest (ground) state. And with a restatement of the Schrodinger equation, we can also find a simple expression for the energy of a particular energy level without having to solve the Schrodinger equation!

Hence, the ladder operators are really as interesting as they sound, just not in the way we'd necessarily expect upon first hearing their names!

Timestamps:
0:00 - Creation and Annihilation Operators (Ladder Operators)
0:36 - The basics: Classical Harmonic Oscillator (Mass on a Spring)
1:46 - Potential Energy (and the Quadratic Potential Well)
3:01 - The Quantum Harmonic Oscillator, and Quantization
5:19 - A Brief Overview of Quantum Wave Functions
5:45 - Questions About the Quantum Harmonic Oscillator
6:26 - The Ladder Operators vs the Schrodinger Equation
7:01 - The Creation Operator
7:46 - The Annihilation Operator
8:01 - Why They're Called Ladder Operators
8:33 - Benefits of the Ladder Operators
9:18 - Counting the Number of Quanta in Our System
10:50 - The Energy of Each State

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[ParthGChannel]

Thank you so much for watching everyone! I always appreciate your comments requesting topics to be covered in future videos, so please keep them coming :)

[_..---]

It's pretty amazing how this dude consistently manages to explain concepts like these in such a clear way.

[tanvirfarhan]

I have learned more from this channel than I have in several year's worths of schooling. I can’t believe this is free content. Thank you so much. This was...far beyond what I expected. You could have gotten away with much, much less of an effort without any pushback. Instead, we are left with this...

An absolutely beautiful, visually pleasing, simple yet concise explanation that works for hand in hand to bring us an intuitive, entry-level walk-through of this topic

I'm honestly awe-struck. I can confidently say this is easily one of the best videos on quantum mechanics. I have yet had the privilege to enjoy it here on YouTube. (and I watch nothing but science and physics docs on YouTube etc)
What an absolutely superb masterpiece, an incredibly engaging tool that undoubtedly will benefit thousands and thousands of inquisitive minds. I am still struggling with the mathematics of quantum physics and complex particle interaction.it would be better if you suggest me some mathematics books to understand quantum field theory
and Thank you so much for this labor of love. It truly shows your passion for your field, and hoo boy what a treat the whole video was. It is insanely rare explanations, so deep insights and facts all come together so brilliant and organically organized in such a way that the end product comes together to create something much, much greater than each part on its own.

What an honor.

[lifeofphyraprun]

You cover quite a few things that many popular science YouTubers don't, or explain them more clearly and in detail than was previously thought would be healthy.I like and appreciate that.Making a somewhat deeper level of Physics open to the general audience while still not making it too complicated for them to get attracted.I wish you and your channel all the best and all the growth.Best of luck!👍

[fizyknaut]

I'd really like to be able to explain things as clearly as this one day.

[Lolwutdesu]

Physicist turned teacher here. I'm starting my own YT channel soon which aims to help kids manage through high school until university. I'll definitely be making a video at some point to talk about worthwhile YT stem channels to follow and this channel is, without a doubt, one of them. Keep up the great work!

[VictorKibalchich]

When I grow up I want to be a creation and annihilation operator!

[gregr]

Hey Parth, thanks for another great video. I just wanted to say that I really appreciate how you don't shy away from the equations in physics. I took a few physics courses in undergrad, but only took up to calc 2, and my graduate work was not related to physics. While I might not be able to solve the math of QM, I really like how you actually present the equations and break them down in a way that is understandable to someone without higher math education. It's hard to find sources that do a good job at this. You and PBS Spacetime are the main two that come to mind, and it's no coincidence that those are my 2 favorite physics channels on youtube. I'd love to see the possible video you mentioned in this episode about going into the math more. Thanks for the great content and keep up the good work

[johnpaulbounce]

As always, very interesting discussion. I also shared this video to my friends 😉

[dhanashrikulkarni]

You never run out of content don't you?
And please recommend some books for just fun physics

[kimii]

Great video, thank you very much! I have an exam in QM in a few weeks and your videos have been great in giving it more of an intuitive spin (pun intended) :D

[aidenwinter]

Small harmonic oscillator energy



Good stuff bro keep it up

[mairisberzins]

Flashbacks to my module in spectroscopy (1st-year chemistry).
The simplest model that is used to describe a bond vibration between 2 atoms.
Although, it's neat and all, it serves no practical purpose in spectroscopy due to the energy levels being spaced out evenly.
Enter anharmonic oscillator.

[alokbarik]

Hey parth.. thanks for making such nice n informative videos.. kindly make one video on 'dielectric constant'.. what it is..it's dependence on frequency..etc.. 👍👍

[tejashdasgupta]

Creation and Anhillation operators. What a badass name.

[wubwub]

Could you make videos where you solve typical quantum mechanics exercises using these methods you are showing us? That would really help us understand how these operators and other methods are really used. It would also give us a more intuitive way to understand them all

[bigphysics]

It's an amazing explanation ...but I have a simple query that if an electron is to be excited from one energy state to the next energy state say 2nd to 3rd, then we have to impart energy equal to hcut omega then how energy is taken from that electron so it can down excite say from 2nd to 1st?

[artificialharmonic]

Please make a video about the math of the operators as you suggested at 10:36. That would be really cool!

[nemuritai]

I am interested in why a†a=count, why doesnt it mean a destruction followed by an creation(no effect?). Good video!

[krishnaraoragavendran]

10:30 We would like to have a video on the detailed mathematics of applying the ladder operator on the wave function.