Ch 13: Where does the Schrödinger equation come from? | Maths of Quantum Mechanics
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- Опубликовано: 2 фев 2023
- Hello!
This is the thirteenth chapter in my series "Maths of Quantum Mechanics." In this episode, we'll finally understand where the Schrödinger equation comes from. We'll first derive that time evolution must be unitary, then use that to get to the form of the equation. We'll then use the fact that energy is the generator of time evolution to take it all the way home (after fixing the units).
If you have any questions or comments, shoot me an email at:
quantumsensechannel@gmail.com
Thanks!
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Hi everyone! A quick note:
In rewatching the video, I don’t think I give enough credit to what Planck’s constant is doing in the Schrodinger equation. It’s true, it really does settle the units, but this is way more important than you might realize.
The Schrodinger equation has energy and time in it, but how big/small are these time energies? Mega Joule seconds? Pico Joule seconds? On its own, we don’t know the energy-time scales that show up in the Schrodinger equation. So Planck’s constant does exactly this. It tells us that the energy-time scale is ~10^-34 Joule seconds, ie really really really small!!! So Planck’s constant sets the scale of Schrodinger’s equation (and hence quantum physics) to be on incredibly small energy and time scales!!! So the hbar being there is the reason we don’t regularly see our dog in a superposition of two states: you need to be in tiny energy and time scales to see the effects of quantum mechanics.
Hopefully this gave Planck’s constant the respect it deserves in the equation! Units are important because they set the scales of physics!
-QuantumSense
Where is the 14th vedio?
Am I not allowed to watch it?!!
Schrodinger's... dog?
Hmmmmm ... could this imply that there are no real continuums but the least contiguums exist to the order of 10^(-34) ?
I hope so - just to see what effect it has on the square root of 2 🙂
@@Alan-zf2tt yes study noncommutativity of Alain Connes.
@@voidisyinyangvoidisyinyang885 Downloaded the pdf - will study it. I must admit to being drawn to algebraic/differential geometries. They seem so close to reality and realism.
Plus, any measure in physical space more-or-less assumes that empty space is well defined otherwise measures do not work?
The best series on quantum mechanics I've seen. If mathematics and physics textbooks were written using your approach, more students would understand and enjoy those subjects. Thank you!!! Please continue your work. With luck, you could become an inspiration for others in how to teach.
Bro this shit blew my mind. I was easily able to use schrodinger equation but the connection to classical mechanics is enlightened my perception of the formula.
That connection is, unfortunately, false. Try to connect superconductivity to classical mechanics and see where that gets you. ;-)
I think you left out the definig characterisitc of one parameter unitary groups U(t), namely that U(t+s)=U(t)U(s), which also makes a lot of sense intuitively. With this the proof that each U(t) is indeed unitary can be shortened quite a bit, also it automatically implies the existence of the inverse U(t)^-1=U(-t).
On another note, due to Stones theorem, each unitary group can be written as U(t)=exp(itA) for some unique self-adjoint operator A. This is what i find fascinating. Each unitary group U(t) determines an observable A, and each observable A determines a unitary group via U(t)=exp(itA). It's similar to Noethers theorem in classical mechanics, that each symmetry of a system corresponds to some conserved quantity
I saw "U(t+s)=U(t)U(s)" and thought "that sure looks like something exponents/logs would do. I wonder how they're related?" And then you delivered
@@cogwheel42 You can watch a similar discussion here by Prof Leonard Susskind: ruclips.net/video/8mi0PoPvLvs/видео.html
I copied the link so that it starts at 29:38. You'll see the derivation which includes writing the time evolution operator explicitly as an exponential. Though he doesn't do it in a mathematically rigorous way, it's still nice.
The best comments are always in the comments
Hey Brandon, your Maths of Quantum Mechanics series is amazing! Keep up the great work, and thanks for creating such beautiful, quality content.
Absolutely amazing. I've never thought of proving time evolution operator have to be unitary, that's so pretty and neat derivation, love it!
OMG worth every second. And probably then most beautiful seconds of my life. The Beauty of quantum physics can't be explained better than this.
Thank you for making physics as fun and comprehensive as it should be, Huge respect for you Dr Brandon and I hope you continue doing more to humanity!
Is there going to be a continuation of the series? it is incredible!
As someone who’s just done an intro to quantum course at uni this series is really brilliant. We didn’t really approach quantum mechanics from the same kind of mathematical perspective of this series. We just started from the schrodinger equation and worked through increasingly complex systems up to hydrogen and helium and the periodic table. But we never really went into the underlying linear algebra with which you do quantum mechanics - we didn’t even learn Dirac notation.. It’s just kinda annoying that all this maths was sort of hidden from view so that we could just trudge through all the usual quantum systems that every undergrad has to learn.
I feel like this series is going to save me a LOT of headache later in my uni course, much like what 3B1B did for me in the past.
this is what i needed which griffiths failed to provide! thank you! please do some more of quantums please. We need a lot of intuitions and perspective in this
I have really been struggling while learning linear algebra and feeling like I have any appreciation for how all these different concepts I'm learning work together to do something meaningful. This video series has not only been incredibly helpful to me in crystalizing many linear algebra concepts through a fascinating application, but has provided me with fresh motivation to move forward. Really fantastic work. Best of luck in your doctorate work and I can't wait to see more!
This is the best video series on RUclips! Thank you!
Thank you so very much for this amazing video series!
My Quantum Mechanics I professor literally butchered the foundation of quantum for me with his dry "shut up and calculate approach".
He barley gave us any context, motivation or intuition for any of the derivation he thought us, so when i finished that course i could say i somewhat _knew_ the basics of quantum mechanics but i certainly didn't understand any of it.. but this video series really helped me fill in the gaps!
Please keep up the good work! you are really helping physics students everywhere understand this wonderful and confusing subject!
Your series has been helping me so much in my undergrad QM course. Thank you for all the time you put into making these videos, they are both aesthetically pleasing and easy to understand. Would you consider expanding this series to discuss more topics in QM such as expectation values, transmission probabilities, and different potential profiles (ie. Delta Potential, SHO, QMW, etc.)? I hope that your graduate studies are going well so far, and keep up the excellent work!
I was really puzzled about a lot of things in my qm classes last autumn. This series is a fantastic refresher and has helped me to finally understand many concepts!
I currently following quantum physics 2 and this helped my a lot in understanding the basics! I am looking forward to the rest of the series.
Waiting for the next set of videos!! Really couldn't wait to be honest. Thank you for these wonderful explanations!
You rightly reignited my passion to learn QM, which was lost in the last decade or so. Thank you very much from the bottom of my heart.
fantastic series! looking forward to whatever you release next
You are a legend. Please continue the series.
Beautiful video once again
Thank you so much for such an amazing series. Anxiously waiting for next episodes. Kindly upload them soon! Thanks again
Great lecture series, I am teaching a Molecular modelling grad level course this semester and I wish I could give students an indepth QM and CM intro before jumping into DFT and MD. Keep up the good work.
For someone who is just now starting a course on modern physics (basic concepts of quantum mechanics) I am loving this course although I wont need it until the next semester when I actually take Quantum Mechanics I. Beautifull job!
this is the craziest magic system for a fantasy series i've ever seen; keep up the good work!
Thank you so much for making these videos. Please keep making them. These videos helped me intuitively "realize" the connection between quantum world and the macroscopic world in a very gullible manner.
This is amasing, please, keep it up!
Very well presented. Thank you.
Thanks for the amazing content! :))))
It gave me so much understanding! I am currently taking a Theoretical Quantum Physics Class at TUM and this series really gave me more of an intuitive approach to quantum physics. Really great work, big Thanks and greetings from Austria! :)
PS: Looking forward to the next parts of the series :)
Loves these videos! When are we getting the rest? ❤
If you make a PDF I will buy it, and probably I will not be the only one. This really associates the formal math and the intuition in a striking way. Well designed, brilliant accomplishment.
Is this your first channel or do you have more content elsewhere? Your voice is a bit familiar :) This series is fantastic, I want to see everything you've ever made!
I love your videos. Please show us more!
This was amazing thank you
Gracias por la serie.
the best derivation video!
Loving it!
This is a great video!
Brilliant video.
Please allow save button some of us are normally busy so wed like to save and watch when we free from work please .and when we not on the wifi . Your series is the best out there .
Well done. 15’ of great physics
Deep Respect !
Great job
Waiting for your Next video
thanks for these great videos. maybe some general relativity video after this. you are amazing
Just Great !
7:12
what exactly is that symbol used to reference the future terms in the taylor expansion
at 8:48, when we multiply the two berackets, we got one identity operator from the multiplication of the two identitiy operators in both the brackets. don't we get another one from the multiplication of the dagger \dot U with \dot U? so shouldn't there be two identity operators on the right side of the equation at 9:01?
6 days without a video, I'm dying on the inside
Absolutely mind-blowing. If I may ask, did you figure out this elegantly understandable path of quantum mechanics all by yourself or was there a book (books) that mentioned these brilliant ideas?
Hello! Thank you for watching.
In truth, everything in this series came from a combination of textbooks, lectures, and self derivation. For example: the proof that observables eigenstates must be orthogonal was inspired by a similar argument in Nielsen and Chuang’s quantum computing book. The derivation of Born’s rule was something I figured out myself after messing with the math for a while. And the derivation in this episode is largely inspired by a lecture given by Prof. Patrick Hayden, in which he did most of this derivation in the first lecture of my graduate QM course (you can imagine how amazed I was! The very first lecture and he shows us this!).
So over the course of ~5 years, I’ve learned bits and pieces about QM from all over the place. And this series is just me putting it all together to share to the world.
So I can give you some textbook recommendations, but my biggest recommendation is to learn from everyone and everything. Each source teaches you something different. After some time, all these viewpoints will give you an incredibly wide and thorough understanding of QM. Hopefully that somewhat answered your question!
-QuantumSense
@@quantumsensechannel Wow, thanks a lot, not only for the reply but also the effort into these videos. Extensive learning on one topic greatly widens one's comprehension. I totally agree with you. I am a chemistry freshman now, and am really interested in quantum mechanics, so I would very much appreciate it if you could recommend some literature or sources!
more quantum when? relativistic quantum when?
Love this videos as a bachelor physicist
What's the intuition behind time evolution being the same as multiplyng by a linear operator and not some arbitrarily chosen function?
To rephrase, why do we expect the time evolution function to be linear?
Great series! - many thanks for this. One question:
If I understand correctly, in the classical case, it is the rate of change of energy which is proportional to the change in the Lagrangian over time. Here in the quantum case, it seems like it is energy itself (not its rate of change) that is proportional to the change in the quantum state over time. Why the difference?
Hello, thank you for watching.
In truth, I don’t really have a good answer to this besides “that’s what the math led us to.” I’ve tried to find a better reason, but the best I have is that I think it’s an artifact from when we transitioned to thinking about physical observables as operators. Operators are transformations in our space, they “do” something to our quantum state by acting on it. But in classical mechanics, physical quantities are functions, and functions don’t really act on anything. So the way they “act” is by changing through time, hence we get a derivative with time in classical mechanics.
Again, this is just me speaking off the top of my brain, and it’s just a loose heuristic. But who knows, maybe there is a deeper reason! Definitely worth thinking about.
-QuantumSense
This makes quantum sense!
Thank you for watching, and I’m glad it made sense!
As a side note, I wonder how many people caught on to the idea that “quantum sense” is supposed to be a play off of “common sense”. Either way, I hope this series is making quantum mechanics feel like common sense!
-QuantumSense
@@quantumsensechannel It’s a lovely play on the phrase “common sense”! I hope your presentations reach many!
@@quantumsensechannel I didn't get that until this comment and your answer 😂 I think it is language dependent (mexican here)
While checking hermiticity of time evolution operator you take adj.(U*.U)* as U*.U but it should be U.U* until they commute. So do time evolution and unitary operators commute ?
Hi will be chapter 14 of your amazing series?
This series is great, which book should I read to further understand those concepts ?
GRIFFITHS
What book do you use my friend?
can someone help mi out with calculation at 9:00 ? Why we only got 4 arguments left. Does smth cancel out?
There are infinite arguments actually. The big O just shows that the rest of them are a coefficient times a dt² or dt³ e.t.c. So when you calculate that expression and you multiply for instance U(dagger)(0)dt with U(0)dt this becomes a term of order dt²
Another great video. I think you have to be very careful with your proof of unitarity in the way you chose to prove it. While it is easy to see that the operator you are working with is Hermitian, one actually has to show it is self adjoint in order to use the spectral theorem as you do. It may be better to start from probability conservation as a requirement for time evolution from the start and the fact that unitary operators preserve inner products, and hence preserve probability. I personally prefer the semi group property to show the exponential behavior of the evolution operator (for constant Hamiltonians) because it does not require using the differential equation. Sure that does not then derive the Schroedinger equation, but one does not really need the Schroedinger equation if you want to work in its “integrated” form. Then using the Trotter product formula, you obtain a clean “derivation” of the time-ordered product.
how do you make these videos?
ALL I wanted to say is, "I am Amazed as to how you were able to show HOW the imaginary number 'i' got into the Schrödinger equation. The closest explanation was via Partial Differential Equations theory, where if you put complex/imaginary 'i' into the Diffusion Equation, the solutions would yield Sinusoidal type functions, just like the Wave Equation of classical Partial Differential Equations. But I see from your approach, you came at it from Linear Algebra from idea of Hermitian Operators. I can't believe I did not see this way of thinking about it, it's due to the fact that for some reason I forgot that the Hamilitonian is a Hermitian or Anti-Hermitian operator. SO it makes sense that we need to include this basic property, that we learn from Linear Algebra. This complex/imaginary number, was something that seemed to kind of come out of no place, it was a strange mystery to me. Thanks for clearing this up, and clearing it up so "Naturally", it is really really great what you have shown here! Thanks so much!
Hey there, I'm making a 5 minute on Shor's algorithm for college and I'd like to use manim to do so. I'm brand new to the library and I'm trying to find some resources on using bra-ket notation and I haven't found anything. Could you help me?
Hello! Thank you for watching.
In using Manim for this series, I didn’t do much fancy for bra-ket notation. I wrote kets as | \psi
angle and bras as \langle psi | . There do exist some packages, but I didn’t find them necessary. In terms of moving things around in bra ket notation, I just used the tools within Manim by shifting and fading things in/out. Hopefully this answered your question!
-QuantumSense
Could i ask; Why are there so many complex numbers in quantum mechanics?
Bro when you multiplied U dot by i my brain almost exploded. BOOM
When is the next video coming out?
Can we derive the equation of understanding? I started at chapter 1 with the definite state 'understand'. But after some time watching more chapters, I evolved into a superposition of 'understand' and 'not understand'.
Most of the arguments can be applied to a real vector instead of a quantum state, until the final part where the operator matrix becomes anti-symmetric instead of anti-hermitian. Anti-hermitian operator can be written as i*H which H is hermitian and hermitian matrix has real eigenvalues. But anti-symmetric matrix cannot be written as something*symmetric matrix with real eigenvalues. Is this the origin of the i from reality?
At 2:05, "we want to use the intuition that we expect that time evolution is reversible." Why I do not have that intuition? And, our intuitions are shaky most of time.
Just wow
HI I think it would be great if you could make a video on bound and scattering states and the infinite potential well.
Brant Carlson
IIIIIIII nnnneeeeeeeeddddddd mmoooooorrreeeeee of your videos pleaseeeeeeee!!! I hate how this subject is taught in my class
Instead of Joule seconds, which is kind of hard to picture, I more intuitively understand h-bar as Joules per Hertz (energy per frequency or energy / (1/time)), that is, the amount of energy the system gets for each cycle around the complex plane that that "i" is contributing to the state in the time evolution. I think it's something like that. But I think we need to wait for a future video to see more explicitly how a frequency gets into the picture.
Hello, thank you for watching.
And this is an interpretation of hbar that I’ve never considered, but I think is very intuitive! I’d have to think a bit more to make it more concrete, but it makes for good intuition (especially once the time evolution phase factor is introduced, as you say). I’ll remember this interpretation moving forward!
-QuantumSense
study noncommutativity. Penrose points out the origin of the de Broglie-Einstein principle is noncommutative nonlocality. Same with Basil J. Hiley and Alain Connes. negative frequency should not be discounted. Quantum algebra is more foundational as a process than geometry. Math Professor Lou Kauffman explains this also. thanks
What do you mean by “conserving probability”? Does it mean that the norm of the quantum state is preserved by 1, or does it mean that the literal probability to measure the eigenvalue of a certain eigenstate is preserved?
Hello, thank you for watching.
And both! To see this, I recommend watching my chapter on unitary operators, where we derive this fact.
-QuantumSense
@@quantumsensechannel I am a little suspicious about the step at 08:14. Wouldn't taking the limit of the left-hand side simply lead to U (dot)(0)|psi>, not the derivative of the quantum state at an arbitrary time? If you think about it, what you are doing is simply reversing the Taylor expansion at point 0, so how could you get the general derivative from this derivation?
Hello,
Remember what the time evolution operator does: it time evolves our quantum state by amount t. So U(dt) will time evolve our state forward by time dt, no matter what our quantum state is. So let’s take our quantum state to be |psi(t)> (and imagine this is what we have at that point in the video). Acting on this with U(dt) will give us |psi(t+dt)>, and therefore we would indeed get a derivative in the left hand side for arbitrary time t.
So the time evolution operator doesn’t take your state to time t, it just evolves it forward by time t. The distinction is subtle, but it allows this proof to work for arbitrary time t. Let me know if this didn’t clear it up!
-QuantumSense
Quick question, does the existence of Û^-1, or the whole thing where time is reversible imply that time is symmetric? How does that not break the concept of entropy?
Born rule breaks time symmetry, hope this helps 👍
Of course a quantum mechanical system in contact with a thermal bath is not reversible, but where do you see a temperature bath in here? For that you need, at least, a density matrix.
What amazes me is how pure math and pure/applied physics sort of coalesce into something that closely models reality.
But I guess that is in the nature of mathematics plus! a couple of questions if I may.
1theimportanceofspaceandpunctuation or rather 1 - the importance of space and punctuation (if you see what I mean)?
EDIT: or more strictly speaking fsctioneampndunctppceottaa1heiornu the importance of ordering space and punctuation if you see what I mean 🙂
and
2 - does the value and units attributed to Plank's constant imply the event space is not a continuity but a very dense quotient space? (sorta helps with a leap from micro-physics to macro-physics and allows for things clumping together ~ a continuity is a continuity - quotient in time and also quotient in energy)
okay 3 (sheesh)
3 - space seems too well behaved to be a no-thing so maybe relatively matter free space is a relatively non-thing space therefore not a no thing space? (difference between an almost empty set and a null set)
Although it should be obvious but I doubt that if I really correct.
In the whole series of the video, you talk about physical observable and could be call an operator.
one of the example would be the Ê.
So, what exactly is the operator Ê?
What is the physical meaning of Ê. This is one of the things that I still cannot get it through your video.
Does it means that if you measure the Energy of a quantum state, you apply some Ê on |ψ> and you should get the result ( probability ) of that particular Engery State Ê?
It is iℏ∂/∂t. The energy is associated with the evolution in time, like in classical mechanics. The Hamiltonian is the energy as a function of the coordinates and the conjugate momenta, so that naturally H |ψ> = iℏ∂/∂t |ψ>
I guess this was meant to be answered in the next video but I don't see it coming..
My reward is that I get to watch this 3 more times and then maybe understand a bit of it
Hello! Thank you for watching!
And don’t be afraid to ask questions! I want to make the derivation understandable, so let me know if I can help you with anything that didn’t quite make sense.
-QuantumSense
@@quantumsensechannel My comment was more about my own information absorbing process. First pass is usually pretty superficial. I can follow along and get the gist. Second pass I see some of the things I missed or misinterpreted the first time, and fill in details. Third pass is when it really starts to click.
That said, there were a couple moments earlier in the series where an explanation was followed with "I hope you can see how that leads to X". I couldn't really at the time, but maybe just need another pass. If I notice them again, I'll point them out.
And as a counterexample, I was also surprised to find myself anticipating some of the conclusions in this video before you worked through them. (like when moving the operators to the middle of the ket, how it must equal zero)
What about time dependent Hamiltonian? It seems that this derivation is not valid for time dependent Hamiltonian.
Perfect series!
I am left with one question, if i understand this correctly, conserving probabilities makes only sense in an unchanged quantumstate? Ass soon as something observable happens like emission of a photon, the waveform collapses, probabilities change and the whole thing can not be applied?
So quantum theory is a good theory as long as nothing happens, we can predict energy levels and probabilities, but ass soon as me measure something, or anything interacts with the system it’s completely useless?
Thats very unsatisfying.😰
Hello! Thank you for watching!
And you bring up an interesting point, and you have just derived the fact that *measurement in quantum mechanics is not a unitary process* . Exactly as you say, collapsing the wavefunction modifies all probabilities to coincide with being in a particular eigenstate, so measurement can’t be unitary.
That being said, the whole framework of QM still applies. You just start applying the Schrodinger equation after you’ve made the measurement, using the eigenstate as your initial state. Is it weird that we have to do this? Absolutely, and much debate still exists in physics about the nature of collapse. But the framework and theory itself works just fine.
So don’t let that one fact deter you from seeing QM as a wholly satisfying theory!
-QuantumSense
Will you do a "Maths of Quantum Field Theory" ?
study Professor Jean Bricmont on Bell's Inequality first. He proves it debunks QFT.
How can be verified that probability remains invariant (conserved) over time evolution?
That's an axiom of the theory. In an actual experiment you will inevitably see losses and we account for those with concepts like "quantum efficiency".
I have a question, why did we bother ourselves with proving that Udager is U^-1 isn't it enough to say if UUdager=I then Udager= U^-1
Wonderful series. Will you expand it to N particle systems and relativistic QM?
By evolving backwards in time, i believe you meant we can back track to find where is was.not we go back in time.
So i see! Breaking time symmetry means we can not back track Which state it was in!
Which state what is in? A single quantum system is in no state whatsoever. The ensemble is in a state or, more generally, in a mix of states.
For me I need the steps between seen.
where's episode 14 :')
PLEASE COME BACK
This is considering the Schrödinger equation in hindsight, not where it came from. The probabilities have been introduced only after the equation, by Born because of interpretation difficulties. Actually, the Schrödinger equation came from the analogy between classical mechanics and optics, both relying on Hamilton's principle of least action, and above all from De Broglie's phase wave to explain the quantum condition. But it has been a long and winding road. Of course today with much more knowledge, which stems principally from wave mechanics besides, it is simpler.
To surmise what the objective of this series is: let me use the following comparison: Let's say you want to show the structure of the London Underground system. The historical sequence of tunnel construction is not necesssarily the best way to show the structure of the system.
I surmise: the intention of the author of this video series is to start with individual mathematical elements, and from there show how those elements combine to form the higher order concepts of the theory.
By its nature the historical sequence of progress is in the opposite direction of that. The Bohr atom was an instance of reverse engineering, using various sources of data, such as the Balmer series and the de Broglie relationship. I like to think of formulating the Schrödinger equation for the first time as reverse engineering the Bohr atom.
My personal preference would be a hybrid approach to exposition of physics theories. The historical sequence was arduous and with many blind alleys. With the benefit of hindsight a narrative can be constructed with the reverse engineering steps presented in a flowing manner.
The common factor of course is: not using the benefit of hindsight is not an option.
@@cleon_teunissen You have to chose an approach, and no approach is perfect. But it should be clear that it is not the only one. The historical approach brings an insight that no other can. The history of quantum mechanics is fraught with errors that carry on still today: classical mechanics is a limiting case, there is indeterminacy because a measure perturbes the system etc. It is not perfect either, but complementary.
Shrodinger equate Hamiltonium
❤👍
just one word - rotation
What would happen if the time evolution operator were NOT unitary? 0:04
Then we would lose (or pick up) quantum systems along the way. Experimental physicists account for that with terms like "quantum efficiency". There is no simple theory for these losses, so it's pointless to drag that into the theory itself, but it's being extensively discussed and dealt with by experimentalists.
I kind of suspected that Schrodinger equation is based on intuition. And I hardly would recommend taking a plank of wood and try to hit someone with it and say that I had done it intuitively by physics that tells me that it is all an illusion because I used the imaginary number ''i''. By this I guess intuition can be used everywhere.
He explains in the video where the unitarity comes from: it's the requirement to preserve the number of systems in the quantum mechanical ensemble. If you understand the identical concept in probability theory, then you can understand this. At the formal level the difference is minimal. In probability theory we write p +(1-p) =1 and in quantum mechanics we use sin^2(phi) + cos^2(phi)=1 to express the partition of unity that underlies the description of uncertain systems. To an experienced mathematician this is probably a matter of a few seconds of intuition. Having said that, mathematicians have been doing probability theory for a century or so and it never occurred to them to use a different partition of unity formula. ;-)
There is a video from Science Asylum showing that this equation is not a wave equation, actually it's a diffusion equation, what is diffusing? The probability! check that video is very helpful
Yes, but this video is wrong. Many proficient physicists have pointed out that the imaginary unit changes everything, but he keep on with his rubbish. There is no diffusion equation for the probability density, there is an equation of continuity together with the probability current, which is of first order both in space and time. He censored me for having corrected him, he is not honest. Perhaps Quantum Sense will confirm.
12:45 I find it not convincing and arbitrary to introduce constants out of nowhere inorder to make our equations work. If our deductive math steps were correct, why haven't we arrived at an equation that has units that match? isn't this a sign that there's something wrong with our steps?
Hello, thank you for watching!
And ah, I think you are thinking too much like a mathematician! In terms of the math, yes, it is somewhat arbitrary and “illegal” feeling.
But as a physics equation, this “constant” insertion should actually be comforting. First, math really doesn’t care about units, so we can have 100% correct math and still have incorrect dimensions. As far as pure math is concerned, a mathematical object is just some abstract object.
But it goes deeper than that. Inserting a constant actually does something incredibly important: it sets the scales of quantum mechanics. We have energy and time in the schrodinger equation, but how large/small are these time-energies? Joules seconds? Mega joule seconds? Pico joule seconds? On its own, math doesn’t know the overall scale of quantum mechanics, this is our job to figure out as physicists, and is the reason experimental physicists still have a job. So by inserting plancks constant, we’re saying that the scale of Schrodingers equation is ~10^-34 Joule seconds, which is really really really small. So inserting that constant actually sets the scale of quantum mechanics to be really really small, which is what we know to be true. Math doesn’t know that on the outset, math couldn’t care less about what the scale of quantum mechanics is. But we care, and hence that constant is actually incredibly important (hopefully now intuitive) as physicists. Let me know if this didn’t clear it up!
PS: This is also somewhat why in advanced theoretical physics, we work in “natural units”, where we set hbar = c = 1. We work in nature’s physics scales, and just let the math run free without these annoying constants. But they are still there! We’re just assuming that we are working in whatever scale is set by those constants.
-QuantumSense
H operator we got here, without reduced plank's constant, correspond not to energy, but to frequency. It's de Broglie relation that connects energy and frequensy by a constant, and it was verified experimentaly.
@@pavelrozhkov3239 hello, thanks for watching.
And in that case, we are still assuming that we have H/hbar, and hence we are still doing some constant introduction. And we want H to really be the energy operator, so in that case, we do indeed need that extra constant. As we have framed quantum mechanics, we do not have a “frequency” operator, frequency is only introduced through dividing the energy eigenvalues by hbar, as you say.
-QuantumSense
@@quantumsensechannel Thank you for replying!
I understand that it's has important consequences for setting the scale with which we work, but I'm stil wrapping my head around why it is a reasonable thing to do.
@@pizzarickk333 hello,
I completely understand why it feels weird. The trouble comes from the fact that we suddenly introduced dimensioned quantities in an undimensioned theory, and so we are sort of retroactively fixing it. If we wanted to introduce hbar “rigorously”, we would define a “dimensionless” time parameter t/tau, and dimensionless energy operator H/eta. So this way we’d keep with dimensionless QM the entire time. Putting this in the schrodinger equation would give us the product tau*eta in the equation, which we would then call a new constant that has to be experimentally measured.
This is a bit more rigorous, and we never introduced dimensioned quantities at all, so we stay consistent. But this is every bit as valid as our approach, and they’re both worth understanding.
-QuantumSense
Chapter 14?