Lecture 7: More on Energy Eigenstates
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- Опубліковано 20 сер 2024
- MIT 8.04 Quantum Physics I, Spring 2013
View the complete course: ocw.mit.edu/8-0...
Instructor: Allan Adams
In this lecture, Prof. Adams outlines how to use energy eigenfunctions to conveniently solve quantum mechanical problems involving time evolution. He then discusses various properties and features of energy eigenfunctions and their superpositions.
License: Creative Commons BY-NC-SA
More information at ocw.mit.edu/terms
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I took 3 semesters of graduate level mathematical quantum physics, so I already understand the mathematics a lot better than it is presented here, but the amount of physical insight provided in this lecture series is amazing.
Maths courses in Linear algebra, ODEs, Calc of var, fourier transforms etc can get you to quite a high level in quantum to be fair. This course seems to just be a classic introductory quantum class.
Were you one of my students?
@@thesmilegame I have no idea what your name actually is, so I have no way of knowing.
We would have all left with at least the same understanding of your comment if you had simply written, "the amount of physical insight provided in this lecture series is amazing".
But if it means that much to you - Congrats on taking math classes in grad school.
Last year, I took Quantum Physics courses at my university. Now, by watching these videos, I am understanding what they passionlessly tried to explain in those courses.
Thanks! :D
the last 15 mins of the graph about allowed region and forbidden region saved my life. my prof never analyze the formula and derive the graph. allan adams is amazing.
The quick questions part is most amazing. It shows how difficult it is to really understand the subject. Great work.
At 14:31, where the subtitles says:[INAUDIBLE], I think the guy in the audience said: "do we ever find discretization for position".
And thank you for uploading such amazing lectures!
He gets an applause in every lecture! Just amazing!
This instructor is so good, just amazing.
Man I wish I would’ve watched these videos when I took my physical chemistry coursework lol 😂 it all makes sense now! I always thought my professors were crazy trying to explain this to us.
Physical chemistry explains quantum in such a painful way... I'm a chemist, and I've had to use these MIT lectures to redo all my math and physics education, because my chem department told me a lot of baloney (or explained things in such a way that I came up with my own baloney)...
I still think there's a more concise way to teach all of it though. Chemistry provides the perfect motivation for all of the physics and math. It's so tangible, and still so fundamental to all other sciences. Except computer science.. but you could teach that by doing chemical computations! And of course, quantum computing is big fun
The ending is better than that of an episode of Breaking Bad!!
It’s refreshing to see such passion
it all makes sense now! I always thought my professors were crazy trying to explain this to us.
Can I add my own "thank you AV person" ... clear crisp audio, clear crisp video... nice.
Hey MIT, any chance of posting the rest of the lecture notes? The site provides just the first eight... Thanks anyway for the great initiative!!
They ever answer you? Still only the first 8 lectures.
@@JohnVKaravitis The rest of the lectures aren't on youtube but they have them on their website ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2013/lecture-videos/
@@JohnVKaravitis PS: Reading the video description helps ;D
@@JohnVKaravitis PPS: I'm sorry, apperently they are now on youtube too
@@Rastafa469 @John Karavitis is asking about LECTURE NOTES..
The solution to the SE near the end of the class (~1:04:00) is quite intuitive and inspiring.
A really nice set of lectures. I am up to #7 so far.
I am not sure all the lecture notes will be available though. They use three reference books for the class. Must cost the MIT students a fortune!
Wonderful content. Much appreciated.
Just a minor technicallity: (phi)" should not be read phi prime prime, but phi second [derivative]. And of course superscripts this kind may proceed with third ('''), fourth, fifth, etc., in small italic Romans, wherein prime means 'the first.' Useful for derivative functions, but also for any other numerable, sorted stuff. For instance, time divisions: hours (h), [prime] minute [hour]s ('), second [minute hour]s ("), and angular units: degrees (º), [prime] minute [degree]s ('), second [minute degree]s ("), whose extended forms have almost fallen into oblivion, and only their abbreviated ones remain in use.
I appreciate your comment. Never realized that's why they're called "seconds"
Perfect title. When I watch this I truly feel like a More on.
The really blew my mind at the end, I was like woah dude.
I watched this after a touch of dinner liqueur and I finally understand quantum physics! I better enjoy it while it lasts.
By the way, I got "C" as my answer!
HEART BEAT FROM WHERE RESUSCIATION IS POSSIBLE
Beauty of physics to discribes unknown things .
♥️♥️♥️
This vid series is a wonderful supplement to my textbook (Griffiths). While Griffiths lays out proofs relatively clearly he seems to side step a lot of the physical meaning of the equations. And though the appendix gives a refresher on linear algebra, I find that it too fails to attach the math to physical meaning. Its really easy to get very deep into methods for solving systems while forgetting the physical interpretation of the methods and solutions. This vid series corrects that.
Griffith's is a great introductory text, but like any textbook it needs to be supplemented with quality lectures.
Inspiring :) Thanx for sharing knowledge and make it accessible for everyone around the world :)
ЕО
в
2:00 !!!!!!
7:00 degenerate eigenfunctions + linear combinations of them
25:30
33:00 they’re incompatible & do not commute.
43:00
53:00
Lebesgue integral takes care of both discrete sums (with discrete measure) and "continuous sums".
Hongbo Li Yeah, but seeing how even linear algebra isn't required for this course, I don't think the students are able to grasp measure theory...
thanks MIT ....this dude is awesome......free mind candy 4 all....1 way 2 help fix the future 4 sure ! .....only hundreds of views wtf !?.....prob shoulda had piano kitty teaching instead...jeez
He may as well be teaching this in pandamento for all the sense it makes to me. Its not through lack of trying though. I just don't get any of this.
BuzzTeddyHead You should of course not START your education with this stuff :) No one without a proper background would understand.
lol
@BuzzTedyHead I'd start with Calculus & Linear Algebra, then Classical Mechanics & Statistics. Also learn how to code with a library/language that supports complex data structures (Python, Mathematica, Matlab, C++, etc.) However, you'd be surprise how little of a background in any of these things you need to star comprehending what's going on. If you solved some 50 Integral/derivative/vector space problems and get the gist of what does it mean (in general) to integrate/differentiate something, or what makes some set a vector space, then these lectures will start to make sense to you. Whenever you have a doubt pause the video and google the mathematical definitions of whatever he's talking about. You can learn anything on your own; don't give up and good luck.
In Allan Adams we trust.
This course is a million times easier to understand after taking graduate linear algebra
You are supposed to have had a class on linear algebra and one on theoretical classical mechanics before you take this one. It's also a good idea to take a class on atomic physics in parallel or at least shortly thereafter.
watching that final analysis was wild
At 43:05 where the subtitles say, "[INAUDIBLE]", I think he is saying, "Frequencies are like energies *modulo on* h-bar".
Thanks for your note! The captions have been updated. :)
Pretty good teacher
The site provides just the first eight... Thanks anyway for the great initiative!!
Just wanna be a MIT student...
Why would you pay a lot of money for something you can find for free...
The reasoning at the end seemingly leading to nothing and then all of a sudden the aha that's why discrete Energies was beautiful lol
There are no discrete energies in quantum mechanics, either. That's just a by-product of an oversimplification of the actual physics.
The statement at 1:10:23 doesn't seem to be true. The amplitude is captured at the constant A and B in the solution, and they are not k dependent, i.e. do not change with the difference between U and E
Incredibly insightful
... It would be cool if we had informations about analycity on wave function. He autorizes nonzero functions with a finite support in this video (twice), but on a previous one he explains that translations are exponential of differentiation, admitting the analicity of wave functions... (sorry for my english :3)
I find it so hard to accept that electrons are binary like that . It has to be the scale like day and night isnt binary but at a relative scale it would sure look at way . But anyway ,why do blood cells stick together exposed to non ionizing and conversely separate when grounded . They are acting like their missing electrons and when in contact with the earth they return. Is there an energetic state that you can force an electron into , not by observation and wave collapse. But with emf , non ionizing excitation of valance electrons ?
Ok so its a wave that goes across the entire universe and an electron is like a tiny bit of that wave getting sucked in like a tractor beam to matter , its wave is everywhere and completely interconnected with itself in a 4d matrix , like tapping into gods brainwave and being like . Why you so weird
Why does he write the dx before the integrand? Sorry if this is a stupid question - I'm a high schooler that only knows basic calculus
Cypress It's just notation. It's easier to read for some people when you have multiple integrals since the traditional Leibniz way can be a bit confusing when you have more than one integral.
Physicists are more prone to this while mathematicians usually like to put the dx at the end
Physicists prefer this notation. It's more common in the physics literature compared to Mathematics
Incredible lecture!!! THANK YOU!!!!!
At 45:33 i think that the [INAUDIBLE] part says:" frequency but then you have that group velocity is still time dependent".
Thanks for your feedback! We'll update the captions. :)
The question at 37.00 - shouldn't have answer C since we just know the coefficients & not the eigenfunctions, isn't it?
The square of the eigenvalue also gives the probability of an eigenstate. But I think the answer should be -1/2 rather than just 1/2.
probability is the norm of eigen function, probability distribution is a strictly positive function
Adams must have known his position since he kept speaking so fast.
Great lecture
6:29 who is Shri Kulkarni and why is “phi star” in honour of him? (Sorry I’m sure it’s a hilarious joke that explanation detracts from!)
Did you find the answer?
Rupesh R K not yet :( I googled him and he is an astronomer though
Also, I'm taking DE right now and we're talking about eigenvalues and eigenvectors with matricies. What does "eigen" mean? It seems that the name is all over the place
eigen comes from eigenwert which is german for intrinsic value
"eigen" means "own" in German. So any equation where the result has some simmetry with the initial thing, physicists like to call the element which provides the simmetry a "eigen-soemthing".
@1:10:38 Why does the wave function not equal to 0 at the turning point? And why it must be continue at boundary of the allowed region? What does he want to say at all at last 5mins lecture?
When he says chat with your neighbor but you have no neighbor 😔
thank u
amazing lectures
I wish I was there in that class......
At about 42 minutes, he forgot to take integral when computing the probability. Second, the answer should be "Yes, always" because we are talking about eigenstates and eigenstates are always orthogonal on each other.
Dan Fulea they may be orthogonal but this fact is only important when ure integrating them which u will get zero. But for the probability distribution function the interference terms are still there, so there can still possibly be a t dependence for the probability function. I hope Ive made my point clear!
If you are talking about probability density, yes you are right. In this particular case we are talking about eigenstates, and things which are orthogonal they remain orthogonal at any t.
Dan Fulea Yeah I'm talking about probability density because the characteristics of orthogonality will only be seen when u integrate psi*_n and psi_m over the whole space. Over a finite region it is definitely time dependent, u can plot them on mathematica or some other software to check if it's true.
at 17:06 he says solutions of the shrondinger equations add to become solutions of the schrondinger equation what does it mean?
If there are two solutions to a linear differential equation then the linear combination of the two solutions will be the solution of the same equation as well
是不是由于能量是一个标量，并且这个标量不随时间而变化。因此波函数的能量形式才能有包含时间的简单形式。（17：34）
At 56:34 there shouldn't be a square root for (2/a). Anyone please confirm this.
@23:12 why at position L "the fact that this goes to 0"??
@25:20 when n=10000, then there're 10000 point in 0~L where particle couldn't appear because the probability of these point = 0. Is that possible??
What are u trying to ask? Do u mind rephrasing ur qn?
In case of electron around hydrogen nuclei values of momentum are also discrete. Why don't we use the sum symbol for k, like we use it for energy?
momentum is not discrete for the H atom. I think your thinking of angular momentum.
first... also what did he mean by "morally equivalent" at 2:05???
EpicName meaning that (f|g), to have some intuition for it, should be thought of the same way you think of the dot product of two vectors. In general, "morally" in maths and physics means that the stated fact is not quite true/exact, but it is helpful to think of it that way. By the way, later you will see that the reason it behaves like a dot product is that it IS a dot product, in the (vector) space where these complex-valued functions are the vectors.
curious about what professor drinks using that bottle, coffee?
Pretty sure it's some form of amphetamins. This prof is on high voltage...
36:27 Why B please?
I'm also confused.Did you get the answer?
@@piyushgalav6483 Nope :/ Learn as fact
In the definition of (f|g) ( look at 0:50 ) we take complex conjugate of f ie f*.
So when we have (cf|g) it has the complex conjugate of cf ie (cf)* = c* times f* , so when we take c out of integral we have to take that into account and write c* times (f|g).
Hope that clears your doubt.
@@CoolDesaster thanks for helping
Since it's in the first half we take the complex conjugate (notated with *) of C instead of C itself
How do I show equation 1 by explicit calculation?
why energy is not be in a integral form of superposition ?
i asked this ques at 11: 00 so i am not aware that someone ask this ques also at 11: 54 sorry for asking
Of course you can have a superposition of states with different energy.
"Everyone cool with that?" Errrr..Sure.
Is there an infinite state of infinite states ??
Yes. That's what you will have to learn about in quantum field theory.
Yeah but for me its like super symmetry.... cool so were there already
And three disliked this? :/
This one is a little bit hard
Professor Adams is just Awesome Sauce! 🤣💦
great great great
Can someone help me out about the question at 49:25 ish, i thought when width of well goes large in number wave function became something has more waves which mean higher frequency and it leads to higher energy, but i know k equal 1 over L and k and energy are proportional but i dont know why i had the feeling of what i wrote first.
The wave number is k_n = (n+1)pi/L where n is from the given energy state E_n.
The question specifies the particle is in the ground state: E_0, therefore, k_0 = pi/L.
pi is half a wavelength, and so pi/L means half a wavelength is stretched out over the length L.
For the sine function the first half-wavelength is a convex curve beginning and ending at zero.
I realise I didn't answer your question! xD
I am afraid I will say something incorrect so I will keep it vague. Remember that these wave functions describe probability distributions of properties of a single particle. So, if you stretch out the space around the particle, it's wave function has to stretch out to fill the space, losing amplitude.
If you make your well wider you would certainly increase your wavelength (to reach at L) thereby decreasing k (k=2π/wavelength)and thus decreasing energy (E is proportional to sq of k)
@2:41 why does he thank the AV person?
Beginning of the lecture he was talking about something not working (projector or something). I guess he was thanking him for "coming to the rescue".
It through me off as well. Then I remembered
5:25
What???!
[INAUDIBLE]
Lol I am just here to mind fuck myself. This is insane!!!!!
👍🙏
add
cool?
Anybody from India who called them Baingan values?
Smartest people in the world can't get the chalkboard to work...wait for the janitor. Doubt if any one in the room could change their own oil.
I could always make chalkboard work. You simply use a window cleaning squeegee. One wipe with the sponge, one with the rubber. Done.
Mind-tools: Correlate the Intuitive Equivalence of the Black-body chalkboard and Temporal Singularity projection-drawing positioning in Eternity-now, as hyper-hypo temporal vector fluidity-> Superspin Polar-Cartesian coordination Calculus.., wave-package localization as density-intensity Vector-values, ..and the lecture has to "make sense", or no matter how long we look at it, we do not see and perceive the circumstances. (Speaking from experience)
Something in Nothing is probability ONE Origin of omnidirectional-dimensional logarithmic interference probabilities in self-defining potential possibilities, of real-time significance.., ie Vector spaces.
Eg Anti de Sitter space is the same context/concept of instantaneous real-time e-Pi axial-tangential sync-duration in i-reflection=> Containment (Eigenvalues) in the Universal standing wave-packaging of mass-energy-momentum, or Superspin-spiral embedded Modulation proportioning-positioning.
This takes practice, all-ways starting with AM-FM Communication in real-time pictures of Temporal Superposition-point Positioning Singularity Conception. Childhood development experience is critical to Intuitive capabilities, that is nature, and mnemonic memory associations/techniques are nurture, often self managed.
-----
The story goes that anyone who defends themselves in a Court of Law, (law-based on pure-math empirical laws of shape shifting temporal Calculus), "has a fool for a Client", which might be why Feynman said the easiest person to fool is yourself when deriving physical theories.
The laws of Mathematical relative-timing ratio-rates in terms of AM-FM e-Pi-i sync-duration resonances are derived from logarithmic Inflation, Condensation FM raised to natural log/antilog powers into the Eternity-now density-intensity numberness Interval Conception of reference-framing Holographic Principle Imagery of perceived Reality. This is the inside-outside Superposition-point Singularity Superspin Modulation Mechanism by which we make Theoretical Judgements about abstract reasoning components of perceived Reality.
When Reality teaches us, this is Actual Intelligence, when we project accumulated Intuitions such a Euler's e-Pi-i Mathematical Conjectures, (confined to 2-ness duality.., i-reflection picture-plane containment, theoretically => 3D-T), we are compelled to assume the continuous creation connection cause-effect context of ONE-INFINITY Singularity.., the Universal pivotal concept of vanishing point-zero-infinity Black-body containment.
Under the circumstances, BBT is non-sense because of the implied discontinuity of a zero Time beginning, when the actual Centre of Time Duration Timing, or here-now-forever, is self-defining zero-infinity sync-duration, e-Pi-i connectivity. Disproof Methodology is always apparent in the Observable Eternity-now Actuality Reality, logarithmic quantization Interval.
Review, reorientate and reiterate from/by First Principle Observation.., relies on a beginng teaching observable manifestation of Actual Intelligence inherent in Actuality.
Eg Ancient Greek (revision) for Geometry, Conics and point-line-circle drawn alignments from observed Celestial time-timing phenomena. Followed by Euler's e-Pi-i Mathematical assembly of what constitutes a Disproof Methodology.., elimination of disconnect to reveal universal Centre of Time Duration Timing e-Inflation potential +/-, Pi-bifurcation radiance and i-reflection containment in vanishing point-zero-infinity Black-body Singularity reflection of/by No-thing-defined eternally, equivalent to instantaneous trancendental expansion-condensation .dt omnidirectional-dimensional cause-effect self-defining Origin.
.
Why did physicist insist so much on bracket notation? So annoying from a math person perspective.
Because it's less work than writing out integrals and infinite sums. To a physicist the deeper results of functional analysis don't matter. We treat the Hilbert space as a simple vector space and the brackets are our notation for a vector product. It would probably be more useful to adopt a tensor product notation, though, because in the end we are in need of fully symmetric and fully antisymmetric functions. I would dare to make the prediction that in about a hundred years physicists will employ a completely different notation that is much closer to the geometric structure of these quantities.
Consciousness is all there is. See my paper "How Self-Reference Builds the World", author Cosmin Visan*
Nonsense.
@@schmetterling4477 Care to write more than 1 word to explain your view ? Or are you a kindergarten kid ?
@@ROForeverMan Physics PhD. Nonsense was all that needed to be said. :-)
@@schmetterling4477 Ok. Good luck in graduating from kindergarten!
@@ROForeverMan Nonsense. I graduated from university decades ago. How is your fast food service job going? :-)
morally equivalent to a dot product? 🤣🤣🤣🤣🤣
I know a 16 year old shouldn't be watching this playlist but I just wanna say, he talks way too fast, especially when i don't understand the topic or equation.
+Eric Herbert 15 year old here.
Hit me up @ Quantum4lif3@gmail.com
+Eric Herbert I wouldn't say he talks too fast, I watch the lectures at 1.5x speed otherwise it's just boring so... personal preference
Robert Lyness
I kinda studied this in advance and came here. It's interesting but the math is tedious and sometimes difficult.
you have to take a linear algebra and an analysis course to probably understand it. I couldn't get it at your age either properly, but after my first two semesters of physics i'm now able to watch these lectures with 3x speed and solve the problem sets without any problems.
He is so slow, I watch it on x2 speed and its still not fast enough I want x10 speed
Sadler A i am very smart
I understand u
But… they don’t
We r at war
Your not speaking english
cool?