Lecture 9: Operator Methods for the Harmonic Oscillator
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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 discusses an alternative method to solving the harmonic oscillator problem using operators.
License: Creative Commons BY-NC-SA
More information at ocw.mit.edu/terms
More courses at ocw.mit.edu
This guy's attire gets more and more laid-back as these lectures progress, not sure if I'm the only one noticing that.
you make me fall in love with physics over and over again, each time I watch this lecture series, thank you Professor! We need teachers like you!
He picked up his drink 15 times and never took a sip lmao
He does that often... See his other lectures also...😁
what is he drinling??
a a damn you really count it XD
14:09 sip confirmed
He's just flexing on everyone that's drinking green juice.
Every year I watch these lectures.
One thing I've DEFINITELY learned is that MAN knows what he's talking about .
What a cool dude 😎
absolutely loved the dimensional analysis in the beginning
Isn’t it asymptotic analysis, ie, wavefunction has to be normalizable when |ξ| -> infinity?
32:40 That giggle haha. Those subtle funny moments loosen up the mood in the lecture.
Just revisited this to refresh the reason why [H,a^+]=wa^+ leads to quantization (52:59) to understand quantization of free fields in David Tongs lectures(and notes) on QFT. Gosh Allan is such a fun and motivating lecturer! He will always be my favourite QM teacher
Subtitles: 1:02:22 I think I can tell what the "[INAUDIBLE]" is. It should be
"Any of you who are doing a UROP in a lab that has graphene or any material".
which makes perfect sense, because UROP = Undergraduate Research Opportunity Program, a popular acronym at MIT.
Thanks for your note! The caption has been updated.
🤔 awesome, inspired, and seemingly accurate given the use of in the box standard operational linear plane fundamentalism. Awesome instructor.
19:17 "is that a dagger I see before me?"
Would you (just) look at it if it dug in? ¿
He said Hamlet.. But it's actually Macbeth!
At the end around 1:16:00, he messed up an x0 with x
29:10 I was so fucking confused until he said surface terms vanished, lmao.
Yeah same. Then I couldn't tell why they would vanish until he put the limits and then it immediately clicked that f and g should vanish at (+/-) \infty.
lol
This classes are excellent to go along with Griffiths' textbook, since prof. Adams is one of the best lecturers I've seen and he clearly bases his lectures on that textbook
Which book may i ask?
@@estefanoandres "Introduction to Quantum Mechanics" by David Griffiths
@@jscruz685 Thank you!
17:18 Reaction when you beat your younger brother and he starts crying, but suddenly mom arrives.
At 54:07 , I was impressed.... Thank god I can watch this ocw lecture video...
17:43 - "and what is the quantity we wanted is x nut square and peanut squared" i love peanuts
Explanation of how the energy and ladder operators commute through to give the eigenvalues of the Schrodinger equation was great; way better than Griffiths. Of course, in Griffiths defense, he's trying to compact everything into a textbook and doesn't have the luxury of explaining everything in real time.
I just like this professor and I love the way he teaches. he makes me excited for physics. When the professor refers to a darer does he mean transpose of the operators.
Never thought teaching would be that much fun !!
Wow... this is just... beautiful
thx for posting these lectures
24:14 Prof. Adams is the slim shady of quantum mechanics... hands down
Ok, I really got a kick out of the Shakespeare reference at 19:20 or so...
In allan adams we trust.❤️
The operator method starts at 10:01
There are three methods for solving the simple harmonic oscillator. (1) The analytic method (2) the power series method and (3) the operator method. However these methods are somewhat tedious and drawn out. Whew! Furthermore one method is not considered the perfect method which will solve all these 2nd order differential equations to derive solutions to the quantum harmonic oscillator. It would be so much better if one could simply use the Hermite differential equation in place of the Schroedinger equation, find the recursion relation and then determine parity and get on with it to ascertain the eigenfunctions and eigenvalues.
MuckrakerW M. listing ways to solve one problem won't get you anywhere.
This is like watching brain ssurgery, because in Principle, you know to dig out a splinter before infection gets out of control. But surgery can become a technique in isolation if the fundamental elements of biology are marked by years of successful practice? Ie recognition of real-time calculation of logarithmic singularity location techniques is simply termed "Mathematics" instead of temporal mechanics.
Or, why Beautiful Mathematics is so satisfying.., because it shows hidden possibilities in a meaningful way. Great lectures.
0:35 Just realized he was 100% making a pun when he said "I went to a math conference... it was pretty surreal"
His remark that "we are always looking at physics in the room" around 31:00 is very unfortunate. That is exactly how we DO NOT look at quantum mechanics in the general case. It's only in non-relativistic quantum mechanics that we are looking at normalizable wave functions. In quantum field theory we are looking at the exact opposite: plane waves from infinity and to infinity.
This lecture is itself a raising operator with the energy gaps decreasing upwards. Everytime you see this lecture, you can apply an a* on QM understanding but the gap decreases.
I LOVE THIS TEACHER
Does this mean that a sinusoidal contained within a wavelets, is a solution 4:40 since wavelets can have exponential growth then decay container envelopes? Is the complex conjugate of an operator, a reflection? 20:10 (reference Hermetian Adjoint 22:51)
Oh thank you! that integration by part thing was never explained to me
Can we get the lecture notes from lecture 9???
OCW has only uploaded upto lecture 8...
Thankyou MIT
@mitocw Yes the full set of lecture notes for the course would be very helpful and much appreciated!
It seems that around 1:04:12 he uses the general form of Parseval's relation
int f(x)^* g(x)dx = 1/(2 pi) int f(w)^* g(w)dw,
without mentioning its name: int f(x)^* g(x)dx = 1/(2 pi) int f(w)^* g(w)dw, and without (1/2pi).
In physics, most of the texts define Fourier transform and inverse Fourier transform with a factor of 1/sqrt(2 pi) unlike engineering texts where inverse is defined with a factor of 1/(2 pi) so the Parseval's relation is without the factor of 1/(2 pi)
At 1:04:36 the caption shows "doing the 4a transform for the momentum term and not doing the 4a inaudible" but Professor means "doing the Fourier transform for the momentum term and not doing the Fourier transform"
Thanks for your note! The caption has been updated.
Will all of the lecture notes will be uploaded?
Awesome explanation
日本の学部ではこのような生成消滅演算子の導入は紹介されないからとても参考になりました👍
Make sure to check out 8.04 (currently running) on Edx: www.edx.org/course/quantum-mechanics-a-first-course
8.05 & 8.06 coming next!
大学にもよるかも知れません。
i find it quite poetic that this video's "most replayed" wave looks the most like an actual wave that I've ever seen..... because people are trying really hard to understand waves
Ingenious!
Great professor
Allan Adams' lecturing style reminds me of Feynman. I actually hope he doesn't see this comment. :)
the answer is security 'maybe'
Se In un intorno di intervallo la tangente che passa da seno e coseno e continua ,si ha un MOVIMENTO si dice oscillatorio dell' andamento della curva,se No ,si dice discono e alternativo.
This is helpful ❤️🤍
Ah, I love mathematic equations on the blackboards. Try running a Powerpoint presentation with the same contents!
Fantastic leassons! Real quality here, but it really bothers me when he forgets to put hats on operators :)
Thanks 🤍❤️
Does anyone know where can I get the lecture notes from this chapter onwards??
Sorry, that's all the material we were given to publish. You might find more if a student decided to publish their own notes.
Why does operator 'a' acting on wave function give us other state of the particle? Isn't 'a' created just to factor the energy operator?
that's the motivation for it, but it actually serves to provide the next state up or down in the tower of states that are allowed by this potential. It's not an observable as Prof Adams said, so it's not nicely thought of in the same way the other operators so far have been.
i want to drink what he is drinking
If comprehension of QM were to be reduced to resolving a natural confusion between words collected from conventional usage, choosing a math-philosophy reasoning nomenclature, then the near-absolute m-p definition of duality in singularity, ..that is Planck's constant for physics, and the basic element of QM existence for spacetime chemistry and all coordinated science, ..is a convergence of constants in a common (experiential) meaning.
Reflection = Time is the inclusive word-element form-ulation for all information in probability superposition. (Where Reflection is the "spin"-rate range, zero-infinity, of temporal superposition, ..Eternity-now; ie spacetime in timed/inflation by duration, space, in a universal wave/echo Big Bang theoretical format)
So the "Mathematical Aside" required to interrelate e, Pi, i etc, in temporal equivalence is expected to be?..
Because 1-0 = 0-1 probability in which "=" , "i" , +/- is a dynamic zero are all elemental simultaneous-superposition, "cyclical-echo" equivalent states of the natural logarithm, self-measured unit-area, and probability one, differentiated-integral.
It's the dimensional structure of geometry at the one-zero origin of +/-infinity reciprocal-infinity conjugation.
(This is a hypothetical math-physics-Philosophy comment about the existential requirements for a combined nomenclature of a quantum logic)
Nested Turtle eggs(?).
Is it possible to access the lecture notes from Lecture 9 and above?
Sorry, that is all we were given by the instructor to publish!
Is there any way to get the lecture notes??? :(
Selected lecture notes are available on MIT OpenCourseWare at: ocw.mit.edu/8-04S13. Best wishes on your studies!
@@mitocw notes only till 8th lecture are available...Plz upload rest of the lectures' notes too.
Why does he put the dx of the integral in front?
Maybe he wants to "highlight" the integrand
That’s a standard convention in physics
I have never felt so stupid
there are no lecture notes available for this one. :(
Can someone elaborate on the surface terms of the integration by parts vanishing at infinity, please.
+Jarrod Mccarthy Dunno if you googled this already, but basically it's because the values at infinity of the functions f and g given that they're probability distributions are zero. Therefore, evaluating them or their products from negative infinity to infinity also gives you zero.
40:20 Why are all operators corresponding to observables hermitian? What about dx?
when you measure the velocity of an object you measure the position and use that to calculate the velocity.
How do I take my test over UA-cam? I feel as prepared as these students
You can check out the problem sets on the course website of OCW
For "Harmonic Oscillator" read dualistic shaping, resonant density-intensity sync-duration, of the e-Pi-i standing wave-packaging, in formation.
WYSIWYG here-now-forever Hologram.
52:59
@38:11 what is on that student's head?
Traditional jewish cap
It's called kippah (Hebrew) or yarmulke (Yiddish). You can see the different types associated with different Jewish sects here: en.wikipedia.org/wiki/Kippah#Types_and_variation
[a, a†]=1, but [a, a†]Φ_E=0 (if you raise and lower, or lower and then raise, you get the same thing). Could someone explain please?
This question messed me up. You asked this two months ago so you probably have already figured it out, but I believe the reason is that applying the raising and lowering op. messes up the normalization your wavefunc, ensuring that [a, a†]Φ_E=Φ_E.
@@davidfenoll2332 Yeah that's right, the a and a† acting on Φ_E multiply it by some square-rooty factor depending on n, such that [a, a†]Φ_E=Φ_E.
He forgot the "bar" in h-bar @1:00:17
Anyone else cheering when momentum turns out to be Hermitian?
When he says E and sounds like Kermit
Make a donation to the brightest students in the world
Is an Energy Eigenstate Wave function zero if its Eigenvalue is 0?
The point of that a(ground state)=0 business was not to show that the energies are zero. It is exploiting the fact that a wave function cannot be identically zero everywhere (since that corresponds to zero probability density everywhere and breaks normalization). When we act on an energy eigenstate with a we go down a level on the ladder. If there is some state that, if we try to go one level down (act with a) again, returns zero, then it means that acting with a on this state is forbidden, since doing so would result in a wavefunction that's identically zero. Therefore there exists this state where we cannot go down any further, hence the "ground" state.
Haochen Wang Woahhh thank you!!!❤️
Got it!!🦾
@@sriramvs2140 Eigenvectors are non-zero by definition. If we allowed eigenvectors to be zero, the eigenvalue could then be anything and hence not unique (per eigenvector).
You should consider taking MIT's 6.04-6.06 sequence on Edx:
www.edx.org/course/quantum-mechanics-a-first-course
www.edx.org/course/mastering-quantum-mechanics
www.edx.org/course/applications-of-quantum-mechanics
6.04 has already started. 6.05 & 6.06 will start next year. Together they cover about 80% of the quantum mechanics taught in MIT's Physics PhD program.
Also, to see why the lowering operator has to annihilate the ground state (as opposed to turning it into an arbitrary function), first note that the lowest possible energy of the quantum harmonic oscillator is strictly positive (unlike that of the classical harmonic oscillator). Indeed, E = ħω(n + 1/2). Then, by factorizing the Hamiltonian, H = ħω(a^†*a + 1/2), we see that the lowest possible energy, ħω/2, is attained if and only if aψ = 0, a first-order differential equation, whose solution is the Gaussian.
Barton explains it all here: ua-cam.com/video/vnyxYtj0mfE/v-deo.htmlm54s
Note that Barton factors the Hamiltonian first as H = (1/2) mω^2 V^†*V + (1/2) ħω and then as H = ħω(a^†*a + 1/2) by defining a and a^† in terms of V and V^†, while Allan jumps to the latter directly:
H/E_0 = x^2 / x_0^2 + p^2 / p_0^2 = a^†*a + 1/2
where
E_0 = ħω,
x_0 = √(2ħ/(mω))
p_0 = √(2ħmω)
a = x/x_0 + i*p/p_0
a^† = x/x_0 - i*p/p_0
E_0, x_0, and p_0 are the natural energy, distance, and momentum scales of the system obtained by dimensional analysis.
In context to the question the student asked in the starting you didn't answer it.For even solution a_0,a_2...... etc the general solution behaves like e^(u^2) for large j. But what about odd solutions ? How do they behave for large j? Also, you answered something else.
a_1 must be zero, so that you get a normalizable wavefunction. All the odd coefficients are zero.
❤️
If I drink mysterious looking green glug will it make me smarter?
2024 👇
Amazing professor, why is he only in socks tho? xD
Please just zoom out sir and leave the camera as is Great lecture though
It should be illegal to teach QM without dirac notation
Comment using what
Grpoing arrows work
Regular expressions?
Nowater has to be at the crown at 1 on the 31st December to pick up the correction
Barton Zwiebach is way better than Alan Adams. The quality of a lecture is not determined by how often the lecturer says "cool", but by how structured and organized the presentation is.
When I watched the lectures four years ago, I had the same opinion! But now that I have a basic understanding of mathematical concepts such as Fourier transforms, linear algebra, etc., I can't help but notice how well-structured and pleasing Prof. Allen's lectures are. All his statements are well restricted. He manages to cover the mathematical treatment as well as the qualitative, physical descriptions in a manner that really aids my understanding.
Maybe you too can try going through some of the math he uses before hand? I think it will really help.
I love how messy this guy is, gives me hope that i might be smart after all :)
I suspect Allan got caned. There is no notes for lecture 9.
Nope, he was my professor last semester teaching 8.05. Still going strong!
Austin Garrett
Can you tell him to upload his lecture notes?
do you have his notes after 6th lectures? also of the 8.05 ones? if yes please help
I any have notes plz upload.
hermeeshen
Aaaaaaa
?
Regular expressions nod what
NAON SIH
Tim kruk
physicists > mathematicians
When you get to the professors level, It becomes highly interdisciplinary. So, Physicists= Mathematicians=Computer Scientists= Chemists....
1 brangroo
Prof. Allen Adams: Are you dislexic? (No offense meant, I am one myself.) So it's true what they say: dislexics are the smartest people after all :)
This comment wasn't to be taken too seriously. Just my way of sympathizing. Although I see an extraordinary amount of lefthanded people in my engineering school. But then again: correlation doesn't mean causation and it's just anecdotal as well
interesting, but why you thought of that? He clearly knows all the symbols and loves to read
great lectures, just cringey commentary/"jokes"