The quantum harmonic oscillator
Вставка
- Опубліковано 2 бер 2021
- 📝 Problems+solutions:
- Quantum harmonic oscillator I: professorm.learnworlds.com/co...
- Quantum harmonic oscillator II: professorm.learnworlds.com/co...
💻 Book a 1:1 session: docs.google.com/forms/d/e/1FA...
📚 The harmonic potential is key in understanding many classical physics problems, from the vibrations of strings to the behavior of electronic circuits. In quantum mechanics, the harmonic oscillator plays a similarly important role, and in this video we explore the reasons why.
⏮️ BACKGROUND
Operators: • Operators in quantum m...
Position and momentum: • Position and momentum ...
⏭️ WHAT NEXT?
Ladder and number operators: • Ladder and number oper...
Eigenvalues of the quantum harmonic oscillator: • Eigenvalues of the qua...
Eigenstates of the quantum harmonic oscillator: • Eigenstates of the qua...
~
Director and writer: BM
Producer and designer: MC
I love how each one of you has his signature move before starting the whiteboard (you pointing at the camera, and the other guy doing the "lets go"). I find it very wholesome and encouraging.
Your videos are all great and very helpful. The quality is honestly amazing. This is what top-tier educational content should look like.
Much love from germany!
Thanks for watching, and glad you like our little details like the signature move :)
I have been preparing for my Phd interviews for condensed matter physics. These lectures are very helpful.
Thank you so much❤️
This is great to hear, where are you applying? And good luck with the interviews!
@@ProfessorMdoesScience I am from India and I am applying in Indian institute of technology.
Thank you very much! I have an engineering background, and your lectures helped me pass my Physics qualifier exam. Thank you for making Physics journey a little smoother. I am very grateful to both of you!
Great to hear we were helpful! May we ask where you study?
“Harmonic oscillator is very essential in qm”
Then rip my qm grade
We hope the videos help!
don't worry, it's (insha'Allah) easier than you imagine!
Simplicity and precision.. Well done
Glad you like it! :)
I like that you in the beginning give a reason to learn the adressed subject
Glad you find this useful! :)
I’m so envious of anyone that’s good at math… and THIS?? This is a language I swear. What I’d give for my mind to understand any little bit of this. 😩
Maths is the language of science, and although it may appear challenging at first, it is worth the time investment :)
Thank you for this clear video!
Glad you like it!
Thank you. It’s very nice presentation, which softwares and tools have you used for creating this?
Thanks! We use Explain Everything to prepare the videos.
Thank you -- very clear!
Glad it was helpful!
Very Accurately Displayed!
Glad you like it!
What a great video!
Glad you enjoyed it! :)
this is a key teaser video for further interest
It is an extremely important topic, so hopefully it will spark interest! :)
thank you
Can't wait for the rest of this series! Just as a quick enquiry, I was wondering if you guys have any plans to cover perturbation theory, specifically of the time-dependent kind, at some stage? Also, super job on these videos in general! They're wonderfully clear and really entertaining.
Thanks for your kind words! And yes, we are planning on covering multiple approximation methods, including perturbation theory (and its time dependent version). Not sure when exactly we'll get there, but we'll do our best.
@@ProfessorMdoesScience ❤
Fantastic series of videos. I wonder whether you would upload a video on perturbation theory as applied to the solution of the time-independent Schrödinger equation. Please do. Thanks.
Thanks! We do hope to do a series on perturbation theory, hopefully soon!
thank you for the videos. is there any difference between the eigenfunctions and eigenstates for the quantum harmonic oscillators?
The eigenfunctions are the position representation of the eigenstates, so they do describe the same thing. You can check out our videos on wave functions to explore the relationship between wave functions and more abstract states here:
ua-cam.com/video/2lr3aA4vaBs/v-deo.html
I hope this helps!
I was hoping you would talk about transition amplitudes vis a vis the quadrature - but guess I'm not accurate there. thanks
We hope to talk about transition amplitudes in the future, but this "quantum harmonic oscillator" series is looking at the basics for now.
Thank you for your awesome tutorials. Since I am new to quantum mechanics I was thinking that if the first derivative of V(x) is zero at x0, then why the second derivative is not equal to zero at x0? Because mathematically I think it should do so. Am I interpreting something wrong? Cause I can not get that. I would appreciate it if you enlighten me. Thanks again for your fabulous series.
Glad you like the videos! Mathematically, there is no reason why a function with a vanishing first derivative at some point should also have a vanishing second derivative at that point. Mathematically, the function x^2 at x=0 is a good example of this. The function itself vanishes at x=0. The first derivative is 2x, which also vanishes at x=0. But the second derivative, which is 2, does not vanish at x=0. From a more conceptual point of view, a vanishing first derivative at some value x0 indicates we have a stationary point, in the case of x^2 a minimum at x0=0. But the second derivative is a measure of the curvature of the function, and x^2 has a non-vanishing curvature at x0=0, which should hopefully be clear from the plot of x^2. I hope this helps!
@@ProfessorMdoesScience Ohhh my bad...! yeah thanks a lot. The first derivative is zero at a specific point and the whole function is not constant! I have to watch the videos deeper 😀
can you explain how traformation of x to new set u ...changes the potential becomes decoupled
We will cover this in more detail in later videos, but for now reading the Wikipedia page on quadratic forms may help: en.wikipedia.org/wiki/Quadratic_form
Can you please give some hint on changing from x1, x2.. xn to u1, u2,... un coordinate?
We hope to publish videos on mathematical methods, but briefly, what we are doing is trying to find a new basis for describing the N-dimensional space in which the function we are looking at becomes a simple sum over N independent functions. Mathematically, the second derivative terms in x can be collected together into a matrix called a Hessian, and its eigenvectors are then related to these normal coordinates. I hope this helps!
Thank you for your reply . I am waiting for your videos on mathematical methods.
Your video is clear.Cute aesthetic handwriting but could be less curving for clarity❣️thank you for the video
Glad you like the video, and thanks for the feedback!
How do I know when this video was made and when it was published?
I believe UA-cam features the publication date below the video, although it may depend on the device/browser.
Thank you for this very nice video ! I am stuck with one problem, I cannot get the classical energy of the harmonic oscillator when n is very large. D
The relation between the classical and the quantum harmonic oscillators is really interesting, and leads to the concept of coherent states. We have a playlist on these coherent states (which we are still building, more videos coming up in the next few weeks): ua-cam.com/play/PL8W2boV7eVfmrgKnhZEzdYY9EfriRz4Tb.html
A particularly relevant one for your question is the second video on the playlist, which compares the quantum oscillator with the classical one: ua-cam.com/video/0ef1rLO6DTU/v-deo.html
The next two upcoming videos (first one tomorrow) will look at the wave functions, which will also help make this connection clearer.
I hope this helps!
@@ProfessorMdoesScience Great feedback ! Thank you, I will leave a comment on the other video. Already subscribed ! Hope your channel keeps growing )
Thank you so much... Love from india 🇮🇳🇮🇳
Glad you like it!
How about a discussion of the entropy of coherent harmonic oscillator states?
Thanks for the suggestion! We do hope to publish videos on statistical mechanics in the future, but in the meantime, we have a series on coherent states of the harmonic oscillator, you can find it in this playlist: ua-cam.com/play/PL8W2boV7eVfmrgKnhZEzdYY9EfriRz4Tb.html
@@ProfessorMdoesScience Many thanks. Have you done anything on entanglement or decoherence? When I first learned QM 45 years ago, these topics were not taught. The way you teach the subject is excellent -- very insightful -- rivaling Susskind's Theoretical Minimum. Keep up the good work.
@@markdenny3374 We haven't done anything on entanglement and decoherence, other than a very brief mention of entanglement in our video on tensor product states: ua-cam.com/video/kz3206S2B6Q/v-deo.html
Hopefully we'll get there at some point! And glad you like our teaching approach :)
@ 3:55 THE USUAL ERROR (repeated over and over again by others too): the particle is only moving back and forth along the x-axis! It's a 1D-problem. The curve is just an energy-curve, not the trajectory of a particle.
Thanks for the feedback! You are absolutely correct that this is a 1D problem, and the system should move on a line and not on the potential curve. However, I wouldn't call this a usual "error", I would instead call it a usual "liberty taken in the depiction". The aim of a schematic diagram like this is to highlight how the potential affects the motion, I think everyone still understands that this is really a 1D motion. Moving forward, we'll try to explain more clearly when we take a license such as this for describing a certain idea.
@@ProfessorMdoesScience I understand, but your presentation of the physical situation is wrong. The particle is really moving like a mass (on a frictionless, horizontal line) attached to a spring.
Can you post a video on Decoherence!??
Thanks for the suggestion, we'll add it to our list!
All we have to do to move to the quantum harmonic oscillator is promote x and p to operators in the energy function. But we never even _used_ the full energy in the description of the classical case. We used F=ma and the potential to get a function of position over time.
In the quantum case we use the Schrödinger Equation to get a function of probability amplitude over position _and_ time.
I'm a bit confused how these correspond to each other.
If we use a hamiltonian approach to describe the classical case to we get a function of "occupation" over position and time?
Thanks for the comment! For the classical case, I would say we do very often use energy conservation to understand the problem (e.g. at maximum amplitude all energy is potential energy, and at the center of the oscillation all energy is kinetic energy). But more generally, the connection between the classical and the quantum oscillator is a very interesting one, and we cover it in detail in our series on coherent states:
ua-cam.com/play/PL8W2boV7eVfmrgKnhZEzdYY9EfriRz4Tb.html
I hope this helps!
@@ProfessorMdoesScience I'll check those out. Thank you!
@@ProfessorMdoesScience my point about the energy/Hamiltonian was that it wasn't used in this video. So the motivation for the operators is very unclear to me.
@@ProfessorMdoesScience There should be a way to use the Hemiltonian method to solve the classical harmonic oscillator. It should not be forbidden to consider energy, x and p as operators. I would assume that with these operators defined, the Eigenvalue problem would be identical to the differential equation and the solution results in a function of x and t that represents the function of motion by f(x,t) = 1 only it x and t comply the function of motion, otherwise 0.
Something like this would perfectly illustrate the transition to the quantum case.
@@michaelschnell5633 Absolutely, the Hamiltonian approach can be used to solve the classical problem :)
thank you madem this is really helpfull
Thanks for watching!
@@ProfessorMdoesScience madem can you make a plalist of explaination of quntum spin plzzzz🥺
@@nthumara6288 Spin is the next series we are hoping to start, likely after the summer :)
While changing to x1, x2,... xn you have done first derivative with respect to xi and second derivative with respect to xj, why? Why not with xi two times?
We are doing a Taylor expansion of a multi-variable function, and the second order terms of such an expansion include all possible combinations of pairs of variables. Note that both i and j sums run over all variables 1 to N, so that the "diagonal" terms with two xi are also present. I hope this helps!
thank you for your prompt reply.
Would you like to explain the damped harmonic oscillator quantum mechanically please .
Describing a damped system requires the inclusion of an bath or similar that can take in the energy that is being removed from the harmonic oscillator. The description of this process requires somewhat more advanced ideas, involving the density matrix. We do hope to get there eventually!
As always I am amaze with your videos , do you think you may be able to do some videos about perturbation ?
We do have perturbation theory on our to-do list, hopefully soon!
What do you think about my spring. It does not obeys Hooks law but it’s Looks more like a quantum system with energy levels and bifurcations you can see the video on my page and it’s graph 7:30
I didn't think Captain Hooke had any Law.....
Hahaha, good one! ;)
huh?
What if "dark matter" is just harmonics from the normal matter, bending the fabric of space-time or the higgs field, giving the impression of "extra matter"? The same for "dark energy".
Do you know the definitions of any of the words you used in your sentence?
@@angelmendez-rivera351 I am an RF telecom professional, specializing in transmission/reception and analog RF signal pre-processing.
Waves is my profession.
Harmonics is my fetish.
@@gt4654 Okay, so you know the definition of one of the words you used in your sentence. Do you know the definitions of _any of the other_ words?
@@angelmendez-rivera351 are you going to ask for every word in my sentence? Do you have something to share, or you just ball bustin'?
@@gt4654 You asked a question, and it is not clear you understand what the words you used to construct the question actually mean, and indeed, if you do not understand those words, then it will be very difficult to answer the question in a way that will satisfy your curiousity.
Ok.. sister..😂😂
I wish I was as cool as Ankit Pandey. What he says and the emotes he uses are so cool. All of the people that read his comments think that he is very smart, since he spends his obvious wealth of free time (instead of honing a skill) on watching videos that he does not begin to understand. The gender inference he makes in his comments immediately make everyone look up to him, wish that they could be him, all alone, wasting his own time on earth, discounting people's attempts at sharing knowledge, apparently to the point that he laughs so hard he cries.
If only everyone could follow his example, we would all be SO COOL. But at that point in time, he would still be an uneducated, jealous, biased, lonely, time wasting dolt, that nobody except for me gives any concern for. What a pitiable existence, sitting in front of his phone, smugly satisfied with the mediocrity of his life.
@@owlredshift look sir ..I don't had any intention to do such comments..
This comment is done by one of my friends..by his computer in which my you tube account is logged in.
I apologise for that..
Mam will teach really well..and I ashmed on such kind of cheap comment..
Sorry to everyone..
And I am not illetrate..
Appology sir and mam.. 🙏🙏
and I have no reference I just received a thought "quantum harmonics" wow 🤍