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

The mini update at the beginning of the video is that I have a Patreon page now! If you'd like to support me on there, please head over to patreon.com/parthg

The document containing questions for you to attempt (after watching this video) can be found here: [ Ссылка ]

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!

As described in my quantum tunnelling video ([ Ссылка ]) particles interacting with potential wells have a small (but NON ZERO) chance of being found in "classically forbidden" regions of space. This is like a spring undergoing simple harmonic motion and then suddenly being found stretched way further than it ever was before. It's strange to think about, and goes against common intuition. However, quantum tunnelling (quantum tunneling for my American friends) is a very well established phenomenon, with lots of experimental evidence to back up its existence.

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!
Second Channel - Parth G's Shenanigans
Instagram - parthvlogs

свернуть