How the Schrodinger Equation Predicts Real Life (and Why It's So Difficult) - Quantum Mech Parth G
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- čas přidán 13. 06. 2024
- Understanding the Schrodinger Equation theoretically is very useful... but the main aim of the equation is to predict what happens in real life!
In this video, we'll be looking at how the Schrodinger Equation can be used to predict the behaviour of a hydrogen atom. To do this, we'll first look at what the Hamiltonian is for a hydrogen atom, as well as how this is calculated. Then, we can substitute this into the Schrodinger Equation, and finally solve this to tell us the "allowed" wave functions for the system.
The Hamiltonian in the Schrodinger Equation is closely related to all the energy contributions in the system we happen to be studying. The easiest way to begin finding the Hamiltonian for the hydrogen atom is to think about the kinetic and potential energies in the system.
The kinetic energy of the hydrogen atom can be calculated using the reduced mass of the entire system (which is equivalent to finding the kinetic energy of the center of mass of the atom). This just makes life easier, rather than having to find the kinetic energy of the proton and electron separately.
en.wikipedia.org/wiki/Kinetic...
The potential energy of the hydrogen atom can be found by considering the electrostatic attraction between the positively charged proton, and negatively charged electron. We can directly import the classical expression for the potential energy for this system, which is equal to the product of the two charges divided by (4pi x epsilon 0 x the distance between the two particles). Epsilon 0 is a universal constant known as the "permittivity of free space". en.wikipedia.org/wiki/Electri...
Adding together the kinetic energy and potential energy for the system gives us a good first approximation for the Hamiltonian. We can then plug it into the Schrodinger equation and solve it to give us the allowed wave functions for the system. We find that the electron can be found in one of many discrete energy levels around the proton. The Schrodinger equation also accurately predicts the energies of these energy levels compared to what we measure experimentally.
But when we make more precise measurements, we find that these energy levels are further split up into very close energy levels. This is known as the fine structure of the atom. To predict this fine structure theoretically, we have to modify the Hamiltonian to account for other (smaller) energy contributions. There are three terms we need to add to fairly accurately predict the fine structure. These are the relativistic correction (since the electron can move pretty fast), a term relating to the spin of the electron (spin-orbit coupling), and an entirely quantum mechanical term known as the Darwin term. This deserves its own video.
Now finding the fine structure Hamiltonian is one thing, it is already an approximation as the relativistic correction is technically an infinite series of smaller and smaller terms. But solving the Schrodinger Equation with this new Hamiltonian is even harder. And this is just for a hydrogen atom. It only gets trickier when we consider larger atoms such as Helium, Lithium, Beryllium, and so on.
(Note: The methods used in this video are not always consistent with each other, they are mainly for illustrative purposes. For example, the reduced mass of the atom is sometimes used, but in other cases the proton is assumed to be stationary. Depends on the circumstance and the level of detail needed).
Timestamps:
0:00 - The Schrodinger Equation and the Hydrogen Atom
0:57 - The Hamiltonian as the Total Energy (Kinetic + Potential)
4:10 - Substituting the Hamiltonian into the Schrodinger Equation, and Solving
5:21 - The Fine Structure of the Hydrogen Atom
6:05 - The Three Extra Hamiltonian Terms
7:56 - Why Solving the Equation is Hard for Larger Atoms
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I would love for some of these videos to be 5-10 mins longer to go into slightly more depth
agreed
Seconded
Agreed. 10 minutes is nice but I would happily sit through a 20-minute video for a little more information.
Make part 2 videos!
Yes!
You broke my brain with "and we need more terms", as I barely had a handle on the first two terms! I am looking forward to the Darwin Term video.
I've been waiting for your video, the entire day. You are such a good teacher 🙆🙆
Thank you so much for the kind words
This video just connected dots in my brain with my understanding of the hydrogen atom unit in undergrad and perturbation theory unit in grad school (for example, I now have an idea of what the fine structure constant is, why it appears in certain terms/calculations, and why we cover perturbation theory lol). I probably would’ve gotten a PhD instead of a MS if I had your videos to help connect the math to the concepts 😂 (A little background conceptual knowledge kind of gives you conviction in your work).
I’ve never seen such a clear, concise explanation of how the maths and concepts relate. New grad students are very lucky to have this. Keep up the good work!
"Notation is inconsistent to keep things simple" is my new favourite sentence! I'll use that as rebuttal whenever someone criticizes a talk or presentation! :-D
Great stuff, Parth. Thanks a lot!
I have just found your channel and just finished my degree in maths, had a module of quantum mechanics and a few other modules that are touched upon by other videos of yours, not bragging or anything just thought it was important to say with what i will say next
I really like your videos, while obviously there is more detail that could be worth going into I think you have done a fantastic job of making the material understandable to people who dont study modules with these concepts but are very interesting in learning. Please keep up what youre doing because I think your channel deserves so much more than 120k subscribers :)
Great content and friendly, outgoing presenter! Thanks, Parth.
Always refreshing to see such professionalism within physics. Respect! You're an inspiration to a new generation.
Thank you for the kind words! Really appreciate it
@@ParthGChannel your welcome.
Brilliant, as usual! It makes one want to know more, and more, and more… Thanks for doing these! Great stuff.
Please don't stop making these videos :')
I always learn not only something but many things from ur vedios.
Worth watching.
amazing video, really enjoyed applying all the equations you have taught us so well in the past to a real life scenario :)
Very interesting, informative and worthwhile video. I look forward to your sequel videos.
Really like the way you present this. You take the next step beyond a very high level explanation of the Schroedinger equation.
So fascinating. Even for someone who hasn't studied much physics your presentation succeeds in giving a glimmer of understanding into the subject and an appetite to know more. You are a very talented teacher. Thank you.
I love to see you reaching 1 million subscribers good luck
Thanks for your informative video.
awesome episode !
Yes please please make a video of the other 3 terms. I’m so intrigued.
Please make more like this love your work sir ....
Love from Bangladesh 🇧🇩 …really helpful…
This was an excellent explanation, thank you very much.!
Thank you. Powerful , clear , plain , ...
thank you! Love these videos.
Parth you’re insanely smart man! I love your videos so much!!!!
I absolutely love these kinds of videos, I'm looking forward to the videos on the Darwin term, relativistic correction and spin-orbit coupling terms.
Possibly the best yet....
Great stuff as usual. Totally loving your relatable content. I wonder what is the heaviest atom that has been successfully modeled using the Shrodinger equation? Even if solved numerically on super computers.
Easily among the top 5 Physics Channels. Videos are relatively brief yet comprehensive. Brevity is the soul of wit.
100th Comment :-)
Hello Parth,
Awesome Videos…
I have a question regarding superconductivity. Can you help me with it? I would be very thankful.
Dada Puro Agun💥💥💥
Nice and thanks!
Me: *finally grasps quantum mechanics principles after dozens of youtube videos throughout a year+*
Parth: "Okay here's the math"
Love the video. technically speaking there's still a long way to go for quantum devices/technologies? As those other elements are too complicated to begin computing with, we can hardly synthesize the results.
Great video. Bit sad you didn't include the shape of the hydrogen-like orbitals and went for a planetary diagram. I think they're well worth a detour since they're the whole reason chemical bonds have the shapes they do! This was a huge and very satisfying revelation for me.
Imagine we have supercomputers to do the mathematical calculations for the solution of the Schrödinger equation of a uranium atom
Is that even possible? I mean, even with a supercomputer?
maybe with a quantum computer
@@grapesalt You mean, its NOT possible today with supercomputers like the SUMMIT or FUGAKU?
@@jimmypk1353 no it's not I'm doing msc in physics and doing project on density functional theory and finding out properties of certain molecules. When I asked PhD student can we do the any radioactive element.he just started laughing and said even super computer will have trouble doing that
Hey parth pls make a video on the standard model and the boson-fermion interaction pls
Thanks, Parth for the amazing explanation. How far can supercomputers/quantum computers help here?
Good
love it
Parth please explain DIRAC EQUATION
Please make a video on Raman Effect.
What about making next video on Feynman path integrals?
Love from Bangladesh
Note it is possible to says things about energy levels, etc. of larger atoms without completely solving the Schrödinger equation. This is done by taking symmetries into account -> see group theory.
Can you make a video on DFT.
I learned a lot from your youtube channel, thank you for that!
I think I discovered something:
I combined the general relativity equation (Tuv=0) with quantum mechanics with a dimensional analysis. I translated de cosmological constant to a Planck area and combined it with the Schrödinger solution (energy).
See 4π or 8π
Im curious for youre reaction.
Wow!
Can you stop making me have a crush on you 😂😍😭, ugh. Such a great teacher, if only I had all this videos when I was studying for my radiology entry exam... I had to read 300+ pages on quantum mechanics only to understand how an MRI scanner works (before I could read another 300+ pages about the MRI scanner itself)
Bruh what u a dude
@@curiodyssey3867 Dudes can have crushes on other dudes, idk 🤷♂️
@@emiliocespedes3685 impossible!
Gay.
@@KRATOS-dq6sd Verily
So I have a question.. schrodinger eq. is not used frequently cause of its complexity? It’s usability is less practiced cause of the difficulty?
Cool
Damn, you went deeper than my MS Quantum Mechanics prof ever did at UMA. He never mentioned the fine structure. In todays age, we should be pushing these into super computer with AI to solve for complex reactions with isotopes, I'd imagine it wouldn't be too hard to implament the weakforce equation. I theorize there are some very technological advances in radioactive decay and quantum mechanics while utilizing EM wave guides to direct this energy. Double edge sward I know, but limitless advances in energy, computation and directed force emissions all in one unit/network.
Would you be willing to bring back the worksheets with the video
Are there mathematical equations for more complex things like orbital hybridisation?
Fine structure will most likely be only approximated. As structures approach the Plank limit the effects of the spacetime they are contained in and its expansion has a quantity and gravity element as well as any electro-magnetic. A stretching pushing, shrinking, yanking. Action to such a small quanta.
Come a long way from all those Cambridge physics student videos
Hi, I really like your videos! However I find it extremely distracting if there is a cut after almost every sentence. It would be nice if there were a little bit less cuts in the video :)
Haha I was working on reduced mass a few weeks ago..made no sense to me. I’m sure this vid will help
Are these calculations for heavier atoms done?
nice vid, thanks! ooh, in real life as in within an atom, I see! it's complicated of course. what I thought this was going to be about is quantum mechanical connections to huuuman life, like consciousness, and choice. think there are any? if not, then everyone is on a roller-coaster, with the ride and end predetermined. because of uncertainties on the smallest scales, I like to think that there are elements of choice, at least of chance B)
How the shapes of orbitals are deduced from Schrodinger equation if it is not solved for bigger atoms ?
A quantity which takes on the value zero is said to vanish. For example, the function f(z)=z^2 vanishes at the point z=0. For emphasis, the term "vanish identically" is sometimes used instead, meaning the quantity in question does not merely vanish by all appearances, but is mathematically identically equal to zero. A quantity that is nonzero everywhere is said to be non vanishing. Now finding the fine structure Hamiltonian is one thing, it is already an approximation as the relativistic correction is technically an infinite series of smaller and smaller terms. Zero and infinity is meaningless unless finding and measure to finest able before it disappearance. Infinite series of smaller and smaller (point like) particles are much more bizarre and are sometimes said to have zero size. This statement has raised more than one eyebrow. How can something have no size at all? And if it has mass, does the zero size mean it has infinite density? Let's start with the easiest point-like particle we know, the electron. Assume it has zero size. Although we know that the quantum realm differs from the familiar world, in which things are measured in inches and feet, we can still get a reasonable mental image of what happens as we imagine looking at an electron with a perfect microscope. To begin with, since it has zero size, you can never actually see the electron itself.
However, you notice the electron does have an electric charge, and that sets up an electric field around it. That's the first crucial point. The second crucial point is an idea called the quantum foam, which refers to the fact that empty space isn't actually empty. Matter and antimatter particles appear and disappear with utter abandon, willfully flouting what seems like a principle of common sense. Empty space is actually pretty complicated.
Now if you combine those two ideas-that there is an electric field and that space consists of a writhing, bubbling mix of particles-then you can imagine what a point particle is like. At a large distance from the particle, its electric field is weak and doesn't much affect the quantum foam. However, as you get closer to the point particle, the field becomes stronger. The stronger field affects the ephemeral virtual particles to a greater and greater degree, eventually lining up other particles with its point particle. (For example, the field of a positively charged point-like particle will push away other positive particles and hold negative particles close.)
Thus if you collide two point-like particles, while the two particles might never actually collide, the cloud of particles surrounding them will likely interact. The point-like particle is the mathematical abstraction at the center of the particle, but the extended field in essence makes even a point particle not so point-like. "Reality" domain and the subatomic domain that we could measure as not zero (fine tuned to 0.0000000000000000000000000000000000000000000000000000000000001 etc.) and disappear into infinite series of smaller and smaller (finest-tuned). Other levels of multi independent domains etc. may appear and disappear. Oblivion and meaningless.
Please make a video on "Noether's Theorem"
@Daniel Salami 😊😊
Thanks for the recommendation, it's already on my list - currently working on finding a clean and satisfying way to explain it
@@ParthGChannel thanks i really wanted to understand the equations, and I'm happy that i will finally understand it ❤️
If the terms contain an infinite seies, doesn't that mean that we could have infinite levels (I mean, if we allow ourselves to go smaller than the plank length)?
Hi parth
But are there some sort of numerical solutions to the equation?
Sir.think deeply
Meaning of negative sign in electron
I think what you are presenting here as Schödinger's equation is nothing more than the definition of the energy eigenvalue of a system. Since the Hamiltonian is defined as a linear, hermitian operator, it is guaranteed to have a real eigenvalue, and this is the equation for finding it. However, to my understanding, Schrödinger's equation is different. It replaces the right-hand side of this equation with the application of an operator that would extract the energy out of a wave representation (thus saying that the energy of a wave representation of a system and its particle representation are the same). This, incidentally, results in a time derivative of the wave function, a result that allows us to know how the wave function evolves with time.
pointing vector pls
I have a doubt about kinetic energy-
From the formula (1/2)mv2, what would be the physical meaning of 1/2 in it ?
It's nothing but the integral of - mv dv
It has none, it's just part of the math.
The 1/2 comes from calculating the acceleration, which is an integral of m*v*dv, which is the momentum of the particle times its infinitesimal difference in speed over a path summed up physically. The result you get when you perform the integral is 1/2*m*v^2.
KE is just the area of a right triangle with one side length of mv and the other of v. Since that triangle has just half the area of a square of same sides, you get half the area.
@@ThatCrazyKid0007 yes,but when we look at other formulae like Newton's sencond law f=ma or of potential energy e=mgh, we can imagine these formulae and can get their physical meaning,
But what about kientic energy formula, how to imagine that
how comrs you said towrds the end ,helium is two protons and two electrons...what happened ro the two neutrons ....????
I think this topic is called quantum chemistry and a lot of research is going on in it
Anyway, great vedio😃😃
I have an exam on it tomorrow :v.
Don't we need to add vibrational energy ??
Life is like the table of elements, you never know how they will combine for something wonderful. Or horrible for that matter.
That's a lot of terms.
Interesting
Can you please explain my question. That I have written above ...sis please..😊
@@mahamkamal6190 first of all I am a boy... Neha is not my name... Secondly I cannot explain it to u through Text... So, sorry
@@nehaseth2793 lol... , do you understand the whole concept that leads to theses mathematical formulas..???
If so I would like to learn from you through WhatsApp or something..
@@mahamkamal6190 lol... Yes I do understand.... And Obviously I am not going to give my number... If u want to understand this check out some more videos on Web OK?
@@nehaseth2793 thank you for that rude reply...💀
Somehow I went from watching theories on the loki show to this video. Not disappointed.
Sir can we neglect the gravitational potential energy between the two particles?
Well we all know that they have very little mass and that gravitational potential energy will be almost negligible.
But just imagine if those particles are travelling very fast and get very high relativistic mass now surely we can't ignore gravitational interactions.
@@yashdadhwal3034 thanks bro
@@arjyadebsengupta8159 but still just imagine if we have theory of quantum gravity we can consider those interactions too
@@yashdadhwal3034 exactly
When you say “solve the schrodinger equation”, what are you solving for? E? Psi? Is the answer a function?
I think that solving the Schrodinger equation means finding all of the quantum states, |ψ⟩, that the system could possibly have, given the Hamiltonian that we use. That's why adding more and more complexity to the Hamiltonian reveals a finer structure--ie, a greater number of possible states, |ψ⟩. Someone correct me if I'm wrong
Since the Schrödinger Equation is a differential equation, solving it yields a function, the wave function.
It's important to understand what objects H, psi and E are. Psi is a function - it takes in numbers (here, the numbers are your three spacial co-ordinates, and the time), and spits out a number. H is an operator - it takes in a function (ie psi) and spits out another function. E is just a simple number - it's the energy of the state associated with the wavefunction psi.
So solving the equation means finding functions in three dimensions of space and one of time, so that when a certain operator acts on those functions, the effect is the same as multiplying by a constant. The constants E that we get for the various different solutions psi correspond to the energy levels in the system.
At some point, my brain smacked into this contradiction: the hydrogen atom electron has no angular momentum, yet it's traveling in a spherical orbital. How can both be simultaneously true? After a LOOOONG time, the answer dawned on me: that spherical orbital describes only where you will find the electron, not the electron's travels. Not that we can really think of an electron as a tiny hard nugget of matter orbiting a larger hard nugget of matter, but if you'll allow the inaccuracy, the electron is really zipping in towards the nucleus and zipping out away from the nucleus at all times. The math is such that there is a zero chance of the electron actually being AT the nucleus (within Heisenberg limits anyway), while the chance of finding the electron at one particular distance is pretty high. Which direction ... ? All are equally likely, and that's why we describe the orbital as spherical.
What ! Electron has no angular momentum in H atom ! But we say mvr= nh/2pi
@@ManojKumar-cj7oj I know, it's counterintuitive! But "mvr" is classical mechanics. It just so happens that the Bohr model -- derived via classical mechanics -- produces results consistent with quantum mechanics. But that doesn't mean that the electron is really remaining a fixed distance from the nucleus, only that you're most likely to find it there as it follows something like a vanishingly-tight elliptical orbit around the nucleus.
If I recall correctly, you stated that in the coulomb law there is a single r in the denominator. However I’m fairly certain that should be a r^2
r² is for force, r is for potential.
Your videos are just as beautiful as your name
Bro , but that's all maths ... can you please it in logical terms.. .. Cuz I and maths don't get along very well..
I understand the classical version of atom( its wrong though).. That electrostatic and centrifugal forces balance to give us stable orbits.. How are electron stable in orbitals..can you please explain that..
It's easier if you think of orbital electrons as a wave like on a vibrating guitar string. A certain number of wavelengths is needed to fit around the orbit in order for the wave to join up with itself properly.
Well unfortunately it _is_ just maths, we derive the logic and interpretation from the maths, not the other way around. That is why it's so difficult.
But to try and explain in physical terms to help you understand the result of the maths more. Systems tend to stay in or approach stable configurations if they aren't in one already. These are always states of minimal energy the system needs to use to maintain itself, otherwise it takes more energy to maintain less stable configurations and thus they decay into more stable configurations. This is basically what entropy is, systems in the universe just seem to have this tendency to reach states of minimal energy.
So applying that principle, we use this equation to see in which states the wave will be stable in. In other words, in which configurations does the system maintain itself. We use the energies of the components of the system (the electron and the proton) to calculate this.
When you calculate the equation taking the correct terms into account (so energies that affect the system, like relativistic motion, the spin of the particles, its kinetic and potential energies and so on), the result you get is that the electron has to be in discrete spheres around the proton for the system to be stable. These are the levels you learn about in school, the so called orbitals. If an electron were to find itself outside these orbitals it would just use the energy that knocked it out of it to settle into another orbital around the proton.
Now a common misconception is that electrons orbit protons. This is not true. The electron takes positions from a diffuse gas distribution around the proton which has a much more concentrated distribution of positions in space. So the electron moves around in the proton in these specific distributions, but it's not a literal orbit like the moon around the earth, hence the name orbital. Just a collection of points you could find the electron in as it propagates itself around the proton.
To conclude, the electron has to be in very specific distributions around the proton in order for the system to be stable. If you place it somewhere else it will just stabilise into another orbital or just break the system completely separating the proton and electron entirely. The maths tell us this because the total energy of different effects dictates these are its lowest energy states as a system and thus the most stable. Hope that clears it up a bit.
@@ThatCrazyKid0007 thankypu soooo much for answering... But that is still kinda hard for me to digest. 😩
Artinya yg naik hukum entropy
Itu sebabnya Fisika menggunakan pendekatan macro cosmos
The beginning looks ok, but I thinks he makes a big jump from 4:38 to 4:40
cant we use computers to solve the complex equations for other atoms????
We do. I think numerical solutions are sought to these differential equations.
I always hit the bellybutton, but notifications never come. Can someone help me?
Kalau dua mah hukum Fisika turun jadi tidak bisa dipakai
Why people don't like?
Itu disebut kiamat
Are you from Indian origin?
Bro you didn't go live..
A human made the equation and humans are struggling to solve it🤣🤣
Ah.. so much for "real life". Hmm
Remove the music from the intro please or lower it ☺
This video showed that...
chemistry is hard :(
but not that hard since you can't solve these equations anyway so you don' t worry about them. You let mathematicians and physicists worry about them :D
That's just running away from things
@@maxwellsequation4887 Welcome to chemistry :D
This is brilliant.
Tremendously clear.
Thank you so much.
Is the Lamb Shift part of the fine structure or the hyperfine structure?
I think I am turning to be a gay after watching your vid lessons. I love you mate.
FALSE ADVERTIZING. I DID NOT TAP ON A COMMERCIAL!!!!!!!!!!!