Why Quantum Mechanics Uses the Physics of SPRINGS - Quantum Harmonic Oscillators EXPLAINED

A spring is a great example of a Classical Harmonic Oscillator. The physics behind it is insightful and interesting... but it becomes even more amazing when applied to the world of Quantum Physics!

Hey guys, I'm back with a video discussing Simple Harmonic Motion - something you may have studied at school already - and how it can be extended to the world of Quantum Mechanics! I find this topic really interesting, so I hope you enjoy this video. Let's start with some timestamps:

0:00 - Intro (incl. mini UPDATE!)
0:43 - Simple Harmonic Motion (Classical Harmonic Oscillator)
2:14 - The Potential Energy Stored in a Spring (it's Quadratic!)
4:46 - Let's Go QUANTUM (Quantum Tunnelling)
5:24 - The Schrodinger Equation and the Quadratic Potential Well
6:01 - The Quantum Harmonic Oscillator (QHO) and Its Properties
6:43 - Zero Point Energy
7:36 - Wave Functions of the QHO
8:20 - The Forbidden Regions
9:07 - Why Do We Care About This? (Applications of the QHO)
10:31 - My Links and Socials

Anyway, let's start with a quick discussion about Simple Harmonic Motion. Springs display this kind of motion when connected to a mass since as soon as the mass is displaced, the spring exerts a force on it that is proportional to the displacement. Also, the force is in the opposite direction to the displacement - it tries to bring the mass back to its equilibrium position.

Importantly, we can plot the potential energy stored in the spring as a function of the spring's extension. This is a quadratic function, and we find that the system can have basically any amount of total energy (within reason of course), and it will never oscillate more than the initial displacement we gave it.

We should keep the above points in mind, because we will compare these properties to those of the quantum harmonic oscillator. We can imagine a system where a quantum particle interacts with a similar (quadratic) potential well. To do this, we need to use the Schrodinger equation (the big boi of quantum physics). We substitute in the quadratic potential well into this equation, and the results are very interesting.

We find that the quantum harmonic oscillator has "quantised" (quantized for my American friends) energy levels. This means that particles interacting with this potential well can only have specific total energies. This contrasts with what we saw with the classical harmonic oscillator, which could have basically any energy value, depending on how far we pulled the spring initially. The energy levels in the quantum harmonic oscillator are all evenly spaced, which makes further study of this system extremely enjoyable (mathematically speaking). This is not true for all quantum systems, only for the quadratic potential well. Another interesting feature of the system is that each particle MUST have a minimum amount of energy that is NOT zero. This is known as the zero point energy of the system!

Now all of this is well and good. But why do we care about a quadratic potential being analysed using quantum mechanics? Well, because particles in real life have to interact with often complicated and messy potentials. And most of these potentials are smooth rather than jagged or discontinuous. Which means many parts of these smooth potentials can be approximated as quadratic - at least near stable turning points. In other words, without doing a lot of complicated mathematics, we can work out approximately how a quantum system will behave by just approximating various regions as quantum harmonic oscillators. This is done a LOT, including in the study of solids where atoms generate periodic potentials.

Anyway, with all of that being said, thanks so much for watching! If you enjoyed the video, please hit the thumbs up button and subscribe to my channel for more fun physics content. Hit the bell button if you'd like to be notified every time I upload. I also have a Patreon page now, so please support me on there if that's something that would interest you!
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[AndrewDotsonvideos]

Use to really confuse me why the energies had to be quantized since the uniqueness and existence theorem guarantees that there formally be a solution to the S.E. for any value E. Interesting that all the blame for the discrete spectra just comes from additionally insisting that the wave function be square integrable.

[AnujShahshahmanuj]

Short and intuitive YouTube videos like this represent the future of physics education!
Of course, you also require solving numericals and practice problems in order to really grasp the concepts.

[xjuhox]

The harmonic potential is one of the rare situations that allow an analytical solution to the Schrödinger equation. Thus, we may solve the Schrödinger equation for a harmonic potential and then calculate the expectation values of potential and kinetic energy for every solution. At the ground state, the potential and kinetic energy are non-zero and equal (that is, the Virial theorem) and that corresponds to the maximum entropy state of the least internal energy. Higher energy states require some "squeezing" of the wave function and that loads energy into it. Every quantum system seeks to the lowest possible energy state by dissipating its internal energy.

[apooravsinghdeo]

It took me months to understand Quantum Harmonic Oscillator and exactly why it is used in Quantum Mechanics. Wish I had this video then. I love the way you explain this concept in sort of entertaining way. Great work, Keep it up! : )

[michaeldamolsen]

Your videos are such a pleasure to watch. You strike an excellent balance between technical detail and a casual presentation style. Thanks for all the time and efforts you are putting into these!

[mobilephil]

Parth you are a genius and a pleasure to watch / listen to.

[psydfx]

You are a blessing to all Physics Majors!

[kanerylander]

This was a fantastic explanation. Thanks from a current modern physics student.

[lanaalabbasi]

Thank you so much for uploading a video about that topic! I truly needed a clear explanation as yours :)))

[alexianakoutsi]

Your videos help me understand quantum mechanics sooo much better than my professor!!! Thank youu so muchhh !!

[satyamverma]

Hey why did you remove your Cambridge vlogs and videos?

[andreas]

The questions in the document was pretty funny to do! Looking forward to do more of them.

[snakez]

I start my physics degree this month. Super excited and these videos really help. Thanks a lot

[joshuasorell]

I appreciate how at the end you explain the applications of this. Lot's of times in my QM 1 class it feels just like a math class, and I wonder where the math we are doing is every actually used.

[horace.e]

Youre helping me out a whole lot more than my physics prof rn Parth

[chinmaydubey]

Thnx bro for this easy and short time explaination. It was awesome

[nabhoneelchatterjee]

love your clear and compact explanation in every video😁😊

[themartian]

Hey Parth !
I love your Videos !
But Parth, I have a request !
Make a video on Vedic Physics and Discuss Vedic Rashmi theory which describes the creation of the universe by a structureless and beginning less particle

[TheLethalDomain]

This explanation literally directly explains why a "true vacuum" doesn't really seem to exist, because with the ground state being non-zero, and with Heisenberg's Uncertainty Principle guiding along, there MUST be something going on, and thus the closest to "nothing" we ever really achieve will be at the limit of the ground state, which is certainly something.

[vishakhasasuke]

Thank you so much sir. 🙏 This video really helped me a lot in clearing my doubts. ❤️