Why Quantum Mechanics Uses the Physics of SPRINGS - Quantum Harmonic Oscillators EXPLAINED
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- Опубліковано 14 лип 2024
- 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: drive.google.com/drive/folder...
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 ( • Quantum Tunnelling: Wh... ) 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!
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Hi friends, thank you for watching this video! You can find the worksheet containing 4 questions to have a go at here: drive.google.com/drive/u/2/folders/1SKtjN7GQTxEjlUUXAkuIMEg9hWwBWEUS
Also, feel free to check out my Patreon page (where I'll be uploading video solutions to each worksheet question) here: www.patreon.com/parthg
i know i am late but time is relative
Hi Parth I love your Videos so much !
Sounds like fun. Hope you try to duplicate my spring model. I think it shows what going on with fields. The area under is an energy packet in the field. The compressing members represent other field. The fit curve is exponential. But the real curve is sawtooth. A lot can be learned from it. I think it even shows some quantum effects are really classical.
Bro next video on Fourier and Taylor series please
Parth G you’re killing it bro thanks you get a sub
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.
I know right, and all because we interpret the square modulus of the wave function as a probability distribution :D
@@ParthGChannel I definitely would not have come up with that. If it were up to me to discover this stuff then energies would be continuous, intuitive, and wrong.
@@AndrewDotsonvideos Hahaha mate if it were up to me, then nothing would have been discovered at all :D
It's just about math: a self-adjoint Hamiltonian operator H with negative eigenvalues (i.e., the particle is bounded by the potential, E < 0) has a discrete spectral decomposition. The particular mathematical theory is not easy. On the other hand, we can't expect that the (very difficult!) variational equation + = E, that has \psi(x) as a variable, can have a solution for every possible constant energy E. Equivalently, not every real number E is an eigenvalue of H \psi = E \psi.
Hi Andrew ! go physics !
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.
Short and intuitive UA-cam 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.
Hey thank you so much! :)
I've been teaching myself the whole stack thanks to YT videos like these. Parth is very clear, consise and to the point. Thus far with a MOOC undergraduate linear algebra im following along as best I can.. enough to dare dream a few ideas.. and appreciate your work. And yes working through the math is something I must do to accompany the lectures.
Absolutely agree. For all the horrendous downsides of the internet, the fantastic thing it has done is platform the great natural explainers like Parth, 3B1B. Eigenchris, Khan academy and all the many, many others; all of them light-years ahead of the average classroom drudge teacher.
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!
Thank you so much for uploading a video about that topic! I truly needed a clear explanation as yours :)))
YAY! New video! Cool! Always best quality :)
First and bro you are just amazing
Thank you so much :)
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! : )
Hey thanks so much for the kind words :)
love your clear and compact explanation in every video😁😊
Parth you are a genius and a pleasure to watch / listen to.
I start my physics degree this month. Super excited and these videos really help. Thanks a lot
Oww, another interesting content. Thanks, man.
Love the way you represent the concept.
Animation 🔥❤️
Cleared some of the doubts I had on this, thanks!
Great video! :)
The questions in the document was pretty funny to do! Looking forward to do more of them.
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.
Thnx bro for this easy and short time explaination. It was awesome
Thank you so much sir. 🙏 This video really helped me a lot in clearing my doubts. ❤️
Cool stuff!
You are a blessing to all Physics Majors!
Your videos help me understand quantum mechanics sooo much better than my professor!!! Thank youu so muchhh !!
Youre helping me out a whole lot more than my physics prof rn Parth
Excellent work keeps it up... 🌟
Fantastic explanation
Amazing explanation and cool video
Awesome video.
Left everything just to come here! You're sooooooo good!
Thanks so much for watching! :)
This was a fantastic explanation. Thanks from a current modern physics student.
Hey why did you remove your Cambridge vlogs and videos?
Love ur videos brother❤️❤️❤️
Well done sir.. signed a new subscriber
Hey Parth !
I love your Videos !
But Parth, I have a request !
Please..... 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
sir u r just awsm in explaining physics to the easiest way i'd heard a big thanks to u fr that........plz make a video on Young's Double Slit exp. too
Amazing video!! And the questions was also pretty helpful..... Which book will good to read such a cool stuff of quantum physics?
Hey parth great video again man, I'm so glad I came across you on UA-cam. I'm sorry to ask for something so tangential to the topic of this video but I was wondering what you (and the followers of this channel.. fellow engineers and physicists) thought about Liquid Fluoride Thorium Reactors as a potential energy solution? I can see plenty of advantages to this system but a problem I can see (though it's an assumption) is that in order to achieve the Uranium 233 necessary for the fission of Thorium, won't we need Uranium 235? Which is pretty damn rare? I want to know everyone's thoughts on this. I'm an engineer but I'm not a nuclear engineer so I'd love to get some more clarity on this subject. The person who got me really interested in this is Kirk Sorensen. Definitely worth a look.
Thanks again. Keep doing what you're doing. I recommend you all the time because you're able to take complex subjects and explain them in a way anyone understands (personally, I dont have that patience!)
thanks bunches
Nice video sir. I have a question please make a video on it that why electrons have negative charge and where it comes from to be negative .
Thanks a lot 😊 sit
Sir
Thanks
Nice video .. Please a video on Copenhagen interpretation
calling it an interpretation is almost misleading, since it doesn’t have much of an answer to many of the questions it’s being asked
Hey Parth, I like your videos so much. But I'm a beginner in physics and I'm in Alevel now!
Can you suggest me some beginner tips? And also I need some mathematical tips too!
Love ur Explanation brother ❤️❤️❤️
Thanks for watching!
🙂🙂🙂🙂
@@ParthGChannel 😀😀
Dead Famous Albert Einstein his Inflatable Universe Book..
I want this book ..
Wanna buy but can't find anywhere..
Any1 have pdf or e-book of it..
Pls share..
I've been searching for it since 4 yrs😭😭😭
Hi parth am big fan of your physic video. Pls make video about quantum computing, atomic fusion power . Thank u...👍👍👍
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.
I would say that Schrödinger Equation is likely the most useful equation contributing to our technological advances in past 100 years.
Hello Parth, hope you are doing well. I attempted the questions and really liked them. Can you post some more questions but please increase the difficulty level a bit. If you can, please post some questions on SR and GR as well.
Hi Harshit, thanks for giving them a shot! I will definitely try and make some more complicated next time :)
@@ParthGChannel I had some confusion regarding the difference between the courses- ' mathematical physics ' and ' physics(natural sciences)'. So, can you please clear this out?
How do we explain the physical significance of zero-point energy to someone? Nice video :)
If you want you can do a video about electromagnetic radiation and antennas
I hope you live forever sir
I hope that you teach physics at some high school, because you could inspire future generations of physicists.
Single handedly carrying me in my physics class
I have a question about planck's explanation about black body radiation
Planck imagined some oscillators in the blackbody which oscillates and radiation comes out of it...
My question is what kind of oscillators he assumed ?
Exactly what was his thought about oscillations of oscillators in the black body ?
Wow
Hii Parth, why cant I find the old how to cambridge videos anymore ??
Uh hi....I'm new to the channel but I have a question
Springs work with classical physics but in quantum physics, we have the Heisenberg uncertainty principle...now how do they relate to each other?
There's something else I'd like to add...you've already made a video on the quantum wave equation, can you also make one on the quantum wave function??
Hey, welcome! Springs only relate to quantum mechanics in the sense that quantum particles experience potential energies that look similar to the potential energy of a spring (i.e. quadratic potential energy wells can commonly approximate regions of potential energy experienced by real quantum particles). So they are only mathematically similar, but some aspects (e.g. expectation values of certain variables) of some quantum systems can also ping back and forth over time, just like a mass on a spring can :)
And yes, I definitely want to make a video discussing the wave function as used in quantum mechanics!
I think so that has already been discussed in his schrodingee equation video.
Isn't the the ball analogy exactly correct in a uniform gravitational field with no friction, since potential energy is mgh in a uniform field?
3:35 What happens in the intersection of those graphs?
I am baffled and bedazzled
I have a question about Zero-point energy. As you mentioned, quantum mechanical systems in a quadratic potential well have a minimum energy state, which is the ground state. According to the Heisenberg uncertainty principle, particles fluctuate in their ZPE, and as I understood ZPE is lower than the ground state, so how can particles fluctuate in an energy state that is lower than their minimum/lowest state (ground state)?
the uncertainty principle in this case only refers to non-stationary states, meaning states whose potential energy varies with time (or are not conservative). In the case of the quantum harmonic oscillator, this is a stationary state, so the uncertainty principle need not apply.
@@bobross5716 Thanks a lot
Hay Parth , can you make a video on quantum computer.....☺️
Ha please..
What is legendre polynomials ?
Your explanations are very interesting
Diffraction means bending of light. But we have diffraction gratings which have normal slit I don't understand can you please explain
Questions were great, although you could have included some more challenging (optional/extra) ones for viewers with a deeper mathematical background... :)
Thanks for providing the wonderful content you make!
The biggest reason why a quantum mechanic In quantum Springs are so big right now It's because in theory we Can use them in Generating power Because at the fact that they're wiggling We can use that as a parasource. Since nothing can be at zero It has to be either negative energy or positive energy. You can use that as a quantum alternator / generator.
Maybe that’s why my model is worth studying.
Thanks for this interesting video 😊😊
Please make a video on quantum computers
I love your videos
I am also an Indian
I have watched all your videos
I am in 10 th grade what guide can you give me
Like what should I take in engineering if I like quantum mechanics and relativity very very much
Actually I like everything in physics and maths
7:43 i found a mistake? On the y-axis the energy states (n) are all named n=2. Shouldn't they be called n=1, n=2 and n=3?
Amazing content!!! Please what does this symbol lY》stand for?
farside.ph.utexas.edu/teaching/qm/lectures/node9.html depends what Y is but it's in a ket vector, so it's an operator of some sort... depends on the system
That was very intuitive 😅so mentally palatable
how only 1.1k likes??
Can quantum particles have "harmonic particles" or "ghost particles", generated by the quantum particle fluctuation?
Can someone solve a Schrodinger equation with potential function v(x)=e^x
Please
Theory
I thought the questions that weren't classical physics were a bit too hard. Just giving feedback that you asked for Parth.
Thanks Zack, my next couple of videos should address how to tackle them :)
Did you listen to John Petrucci's new solo album? And if so what did you think about it?
Comments are good for the UA-cam algorithm
I read that string theory
Lol
Well what is a 'spring' but a wiggly string?
Have a question, How can the first questions (a) part can have an answer of zero.?
I think. 3.1 (a) part and (b) part are incorrect.
Question 2.1 a) is O joules of energy because it said the system was at rest meaning the spring was in it's natural position and therefore had not been extended or compacted at all leading to no stored energy hence 0 joules. For question 2.1 b) you needed to account for the fact that the compression of the spring was measured in centermeters, but the equation E = 1/2kx^2 uses meters so therefore what would have been 4 ((2cm)^2) will be 0.0004 ((0.02m)^2).
Ok. But in (b) part the length is 3 cm ( and ya I converted it)
@@internationalfatherinlaw5585 the spring compressed by 2 cm (5cm - 3cm) though.
Ok so that was the problem. Hmmm
Thanks for correcting me.
Hey we share the same hair style
I love how you say fun physics when physics is a nightmare 🤪
Spring or string? Where is it correct?
spring
The Grand theory of everything turns out to be: spring theory
More accurately string theory
Yea, that's the joke 😊
I am first... Please bro give a heart
Thanks for watching! :)
Everything is a Harmonic Oscillator if you're brave enough.
It’s something about his man-bun
Jk, I actually love your vids, keep it up my man
I know that iam in the 3rd on the list ...But i must ask your opinion about future of coumtum computer
dude i wish you could reply to this comment
i always have the anxiety that i will not understand the concept before studying the concept itself
the really problem i have is that i always think that i will never reach an expert level understanding of physics i actually have reasonable intelligence but still i have anxiety can you help me with this
Hey! I think everyone suffers from this a bit, but the key is to realise that the end goal is not to achieve an expert level understanding. Nobody becomes an expert before they become a beginner, so do whatever works for you to take the pressure away from learning. Do it for fun, with zero expectations of an outcome. This is what I find works for me! Experts only become experts because they enjoy what they learn and so they can do it over and over for years and years until they get good at what they do... but they also started as beginners!
@@ParthGChannel *THANK YOU*
SUSAH URUS SAING
SUSAH NGITUNG GOSIP
SUSAH NGITUNG SAING
Hi
Are you parsi indian origin 🤔🙏🤲?
Yeah this hairdo is the best . IMO.. just saying
Is anyone else getting triggered by the obvious linear (definitely not harmonic) movement of the pink weight? I find it extremely distracting.
SUSAH GALAK
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SUSAH GOSIP
Are you Indian 🇮🇳?
SUSAH BAHAS
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SUSAH TIDAK PUNYA SAHABAT
Are you Indian
You look Indian, Bhai, why the accent?
Some things oscillate, it's true. And some of the things that oscillate do so harmonically.
Most events in nature, however, happen at random intervals and their collections tend toward power-law distributions, not normal curves.
We've been studying trigonometry for three or five thousand years now, so its "laws" are as attractive as the lamp-post to the drunk -- nice easy places to rest.
Fractal mathematics has only been around for about three or four generations -- since Victorian times, or since the Civil War for Americans. But it's much more likely than good ol' Greek (and Egyptian, and ancient Indian) stuff to describe what's going on out there.
Avoid the easy, slack, assumptions: look for accurate descriptions of observable, empirical fact, OK? It's about time we got his "science" thingie moving, and got ourselves out from under all the damn metaphysics!