The Quantum Mechanical Harmonic Oscillator: An Algebraic Derivation
HTML-код
- Опубликовано: 2 окт 2024
- Merch :v - teespring.com/...
Help me create more free content! =)
/ mathable
Potential: • Conservation of Energy...
Separating Papa Schrödinger: • First order ODE: Solut...
Deriving the Constant A_n: • Deriving the Normaliza...
Let us solve the stationary Schrödinger Equation for a given classical potential. We are going to derive the corresponding wave function and its energy eigenstates. Featuring Daddy Griffiths mah boiiiiiis! =D
Twitter: / flammablemaths
Facebook: / flammablemaths
Visit my website! =)
mathable.me/
![When Studying Maths is UNSATISFYING... [ Math Joke Video ]](http://i.ytimg.com/vi/WaXU2vSgQSc/mqdefault.jpg)
![When Studying Maths is UNSATISFYING... [ Math Joke Video ]](/img/tr.png)
![FendiDa Rappa - Clock Dat ft. Shamar Marco [Official Music Video]](http://i.ytimg.com/vi/3NBU8KHdxYY/mqdefault.jpg)
I have no idea what this is, but since it's math and cool formulas, I'm gonna watch it anyways because why not.
It is a quantum harmonic oscillator. It is used to discribe vibration of chemical bonds in a way similar to that of a spring. We chemists use it in IR vib spectroscopy. Ofc this models has limitations
@@jeremydarcangeli7093 the most correct model is the one described using Dirac Equation.
"pls tunnel me daddy"
Name: Papa Flammy
Alter Ego: Jens Felhau
Super powers: ability to be everyone's papa without needing to pay child support, ability to dphi the system, ability to summon meteorite Eulers at will
Unconfirmed incidents: supposedly beat Papa Fibonacci's ass with linear algebra, has been witnessed to be more sooth than a continuous boi, can supposedly date the logo for Flammy lasers
Weaknesses: his ungrateful bois don't watch his differential equation videos enough
Resistances/Immunities:resistant to bad math like saying that (a+b)^2=a^2+b^2 or sin(θ)=θ for 0
This is the best week of my life.
Videos on tensors please
Yes, Quantum mechanics!!! We wait quantum field theory math explanation!!!
Sad thing is, nobody knows it properly to this date. :D Only quantum sytems with finite degrees of freedom are rigorous mathematically. Renormalisation in field theories makes mathematicians cry.
Germany lost today. Consoling myself watching you, papa Flammy!
I am a Mexican boi and i watch this videos lol
my country is crazy about this victory haha (nobody expected that)
I'm in love with this channel, greetings from Sonora, Mexico.
@@cristhianalanrojosauceda3042 alv yo tmb soy de sonora y veo los videos de este wey y están perros viejon ln jajajaja
omg this is my favourite channel. its like u r in my classroom. what a coincidence? for the last 3 quarters i have been able to watch videos on ur channel about something i learned in school. I hope we keep up!
This is by far the best and clearest derivation of the QHO. Your explanations are perfect. Thank you very much.
this video should be in youtube trends
Finished my quantum physics exam last week, nailed that shyt!
The first time I was shown this at university I was left in awe. We also started using Dirac notation from then on, which added to the effect.
Keep up the great videos! As a side note to anyone interested:
The reason the ground state energy is non-zero is because IF it were zero, THEN we would have complete certainty of the momentum. Since E = 0 => p = 0 => ∆p = 0 (since 0 ≤ p²/2m ≤ E). But since the particle is stationary, that also implies that the uncertainty in position x is zero
∆x= 0. Just as in the video,
[p,x] = -i hbar. There is a theorem which states that the product of the uncertainties ∆s∆q (standard deviations derived from the wavefunction) of two observables q and s (things which have an associated operator, ie measurable quantities) are related to the expectation (mean) of the commutator [q,s] acting on the wavefunction such that:
∆q∆s ≥ ½ ||. For p and x, we get that ∆x∆p ≥ hbar/2. So, it can't be the case that the energy is zero quantum mechanically as that would imply that ∆p∆x = 0 (as argued above) which is simply false.
David Alexander Somehow you managed not to directly mention Heisenberg's Uncertainty Principle once. Nice.
Deepto Chatterjee yeah, I didn't want to be generic ;)
You coulde argue if the non-zero groundstat is caused by the uncertainty.
but i prefert to thing the other way.
every combination of two operators which have a non-zero commutator are uncertain to each other.
simply becaus to exchange them means that they influence each other. therefor knowing one (beeing able to certainly interfere with it) means the total collaps of the wavedistribution and as the two wavesdistribution of the operators are influence means the other is at is maximum distribution and therefore absolutly not kown.
which you perfectly discribet with the forumular ∆q∆s ≥ ½ || :-)
There is one problem with your explanation, in general = 0 does not imply ∆p = 0. For example look at the ground state (or every other eigenstate of the Hamiltonian) and calculate the expectation value of p, it will turn out to be zero. But if you calculate ∆p^2 = - ^2 it is non zero. I would argue a bit different. Since ∆p is non zero for every physical state E = = /2m + = (∆p^2 + ^2) / 2m + is always greater than zero, if is always positive, which is the case for a harmonic oscillator.
Great video!
Could you make a video explaining the base of what you are doing? I was baffled, because I never had a quantum physics lecture and I guess, I'm not the only one
I think the hbar in the thumbnail might need a square, although my eyes are not that good.
I’m no quantum physics boi (have only taken high school physics) but I still was able to understand what you are doing. Great video Pappa Flammy! Also I couldn’t find the Euler meteorites :(.
This helped me understand Ladder operators and the Algebraic Method so quickly! My own professor couldn't even explain it as well as you did! I definitely know where I am going first when I'm stuck in mathematics!
Thanks Dani! :)
Ah, but you didn't show where Hermite polynomials come from! You have to do that, too :D
Could you do a video on operators and Schrodingers equation?
YES! I've been waiting for you to upload this for a while. Please do more videos on quantum mechanics and operators maybe? I would kill for a proof of the uncertainty principle using abstract non-commuting operators.
Big papah Flemmy flems , this is exactly what i needed for my Molecular Q.M class. Greetings from Mexico, papa bless little daddy
Papa Flami breathing down the neck of QFT with these ladder operators
:D
i love the physics videos a lot! Papa Flammy's random week has been awesome, seriously.
The Harmonic Oscillator, there is no escape: Classically, Statistically or Quantum Mechanically!
This is gonna be tough.
Thank you for fulfilling our otherwise empty and miserable days.
The Planck's constant in the thumbnail isn't squared tho .-.
Aha, I understand anything but I found this beautiful ! And, that's why I watch all your videos !
This concept is nicely given in Griffth Quantum mechanics book 😍🥳😀
"Calculating A_n will be your homework"
a+ and a- , fond memories of QM I took as a freshman 48 years ago.
Good Morning Flamamble math
OMG! that into is catchy specially during 00:23 -00:25
And I know nothing about quantum mechanic. I am not there yet
Think I might have to make it my ringtone......
Will pretend I understood a single thing other than basic level calculus
Is it just me, i. e. my browser and my cell phone, or do the links for the videos bout Schrödinger and the derivation of A_n not work?
We actually covered this in my Mathematical Modeling class, it was great!
N =1/2 (1 +2 +3+4 +5...+ infinity)=1/24 (Cat in Schrodingers hat) Numberphile v. Math: the truth about 1+2+3+...=-1/12 Ignore complex numbers Planck constant is very small :))) (I am joking)
Really nice! Slowly catching up on my Papa videos
Does anyone know why, when multiplying px by f(x), we get p(xf(x)) instead of (px)f(x)? Because that makes us use the chain-rule here
I realize this was posted 2 years ago, but let me try to answer anyway.
Operators are always calculated right to left in the absence of brackets, since it is really a composition of functions. Notice that composing functions f and g we get (f o g) (x) = f(g(x)), so g is applied first.
us comp chemists call it second quantization, sweet book on it by Jack Simons and Poul Jorgensen
How you did this off the top of your head is beyond me, this is the best approach I've seen to using the ladder operators for the QHO so far, bloody brilliant video mate keep up the good work :)
Nothing better than some good quantum mechanics in the morning
Papa Schrödy's proud
How about Schrodinger equation derivation? But without this crazy operators...
The Schrodinger's equation is an operator, its the Energy operator?
I don’t think the SE is derived as much as it’s declared
i'm late to the party but this was absurdly amazing!
You are awesome!
congrats on 16k!
This earns a sub, the book I have at home gives a table of harmonic oscillator solutions but does nothing in deriving them like this (pretty much just calls it maths magic and moves on). It really bugged me not being able to derive these.
Good job! Go on this way
(It was intended to skip the Hermite Polynomials? I would understand: so long and not so easy)
Just to point out: the solutions to the quantum harmonic oscillator are all the functions f of the Schwartz space expressed in terms of the Fourier expansion
f(x) = Sum_{k \in N_0} c_k u_k(x)
c_k arbitrary coefficients in \ell_2 (complex values)
u_k(x) sonc of the Real set made up with the Hermite polynomials of order k
To be clear with your notaion: (a_+)^k \Psi_0(x) is proportional to u_k(x)
Didnt understand anything in this vid but still enjoyed it xD
YASSS QUEEEEEN THE INTRO
A joy to watch, even on a flaming saturday evening. Cheers from nårwøy
Papa Flamming this was a fucking amazing video, I'm currently watching as much of your content as I can. Can't get enough of it. Mathematicians will rule the world >_
Can you explain why squaring the momentum operator correpsonds to a second order derivative ( I thought second order derivatives are different from derivative squared)
Deepto Chatterjee What you're actually doing is applying the operator to itself, hence obtaining a second order derivative
The square of an operator corresponds exactly to that operator applied twice, as definition.
Of course it is well defined iff the range of the operator is at least contained in the domain of such operator, like in this case
Ah, thanks everyone. I just started learning quantum and everything feels so weird about it.
I play Pokémon Go everyday...
Thank you for the videos papa flammy. They save me from the voices in my head, especially this wavy boi
Papa Flammy just became Daddy
i want a version of one of your videos where every time you tap the blackboard with your knuckles the screen goes red, it zooms in to your finger, and it's bass boosted.
o.o
Daddy Griffiths is a lifesaver.
I was literally lost. Your video saved me and your way of explaining made understanding these equations much easier! Thank you very much!!
Oh my god I love the way you say Schrödinger 😍 thanks for another great video papa 😊
What's your favourite topic in maths(besides calculus)?
Videos in abstract algebra would be a real treat Flammy ;)
Did you mention Papa Griffiths? ;-)
There's a typo on the thumbnail, the momentum operator is p-hat = i h-bar and kinetic energy is p squared over 2m. Which means it should be -h-bar squared upon 2m ...
ℏ is pronounced "h-bar", and I don't know if that is a language barrier that you didn't know, but figured it would be respectful to bring it up :^)
OIAAAAA
PS I found all the euler meteors
Btw could you do a video on solving the heat equation??
29:12 i thought that the wave function cannot be real, is this because it is for some fixed state Psi 0?
derive schrodinger eqn please
Schrodinger equation can't be derived. Just like F=ma, the schrodinger equation is a fundamental law of mechanics
Can you help me apply this equation. Suppose that my wave equation (Psi) is composed of oscillators tuned at F1, F2, F3...FN orthogonal frequencies.
24:40 Sir, why you equated the a_(psi_0) to the 0? What is the idea behind that? thank you
EXCUSE ME, did you just say the naturals, including 0, as if 0 wasn't included in the naturals already?? Shameful, I would have thought papa Peano would have taught you better. Otherwise, that was an awesome video, haven't looked at quantum mechanics before and until now I thought it was totally inaccessible. You made it make so much sense though!
Gamma Knife There are two separate conventions that are widely used, and one of them does not have 0 included as a natural number. In particular, including 0 in the natural numbers is very exclusive to some specific fields in mathematics.
Do you know an argument why the set of solutions you found is exhaustive, i.e. why there are not more solutions? You showed that there is a "ladder" of solutions, starting with psi_0 (Gaussian) and energy hbar*omega/2 and that is not bounded from above. But there might be a second (or infinitely many) ladders. I think I was presented a mathematical argument in a lecture on Lie algebras but that was quite mathematical and I cannot remember.
Noob Question: there's obviously a reason why you can't cancel the wavefunctions. Why? Is it something to do with their complex branches? EDIT: I see now the things. The purpose of the shrodinger equation is not to find the energies, but the wavefunctions.
Happy Father's Day, Papa Flammy
How about some functional analysis for quantum theory video?
Please make some more videos about physics, maybe electromagnetism, retarded sphere (circle) and derivation of feynman's electromagnetic force of accelerated charge? Best regards
Do you care about football? I don't but admittedly seeing Germany lose made me thrilled-im from England, we're losers😂
5:53 I did...
Pass experimental physics 1 first time. Oof.
a challenge! can you remember the intro at the end of the video :D???
Where do you make your thumbnails?
en argentina hoy es el dia del papá(padre(pappa))
Another derivation: (Quite a bit shorter at the cost of being a tiny bit too profane)
ruclips.net/video/b8AX7QN68bw/видео.html
20:18 same
29:47 so the integral is=1?
Sympathetic excursion. Thanks.
Sup buddy, new subscriber here. I'm the Number 24 000 yayyyyy...
If someone actually has a quantum computer- what worth has this to him?
Thumbnail is wrong, misses an h bar. :D But great video.
I challenge you to solve this:
solve for x in x= tan^-1(i)
0:47 WTF is the classical harmonic oscillator? I only know that it is the spring potential, damn I am such a noob !!!
great one today Papa Flammy!
Just when you thought you could just straight up FOIL the shit out of that, turns out you can’t.
How i know about plank constant about any building its strength is known by us with instruments without any big torsion
Hättest du auch mal ein Jahr früher uploaden können ...
:'(
Where does that momentum operator come from? I've never seen it before but it's intriguing
cool.... mathematical expresion I ever seen..
Me discovering this video after getting a B+ on my QM course:
FFFFFFFFFFFFFFFFFF
F
F
FFFFFFFFF
F
F
F
F
> this is the difference of two squares
bro it already was earlier lol
Dont u just love this guy..
Lol ...do a fourier series for psi...thats madd, and use it for the function for defiinite momentum space.....
I really love this guyyy
Thanks for the explanation of ladder operators
hey bad boi, the Planck's constant in your thumbnail picture is not s- to tha -quared, just sayin' ;-). Apart from that keep it up, u got a good sense of humour and an unhealthy appetite for doing nasty integrals :-)
FYI for those confused on the momemtum operator, the momentum operator is positive i*h dx so when it is squared the square of an i value makes it negative. whereas in this video a -i value that is squared leaves us with a positive value and thus the wrong p^2 value of h^2