SOLVING the SCHRODINGER EQUATION | Quantum Physics by Parth G

How to solve the Schrodinger Equation... but what does it even mean to "solve" this equation?
In this video, I wanted to take you through the steps for solving the simplest version of the Schrodinger Equation. As we may know from my old video on this topic ( • Schrodinger Equation E... ) the equation deals with a quantity known as the "wave function" of the quantum system we happen to be studying. More about the wave function in this video of mine: • Wave Functions in Quan...
Solving the Schrodinger Equation just means finding the wave function of a system, given the energy of the system as well as other factors in its environment. Now the equation is the governing equation of quantum mechanics, and it basically deals with the kinetic and potential energies of the system. And the right hand side can be thought of as the total energy of the system, meaning we're basically working with "KE + PE = E_total".
The important thing here is that we are working with the "Time-Independent" Schrodinger equation. This means that the total energy of the system does NOT change with time. We're doing this because the time-independent equation is much easier to solve than the time-dependent one.
The system in question is known as a one-dimensional particle in a box. Essentially, there is a particle that can move freely in one dimension (e.g. left and right, but not up and down or in and out of the screen). However, this free movement can only happen in a particular region (say between x=0 and x=a). This is because we have placed two barriers at these x positions. The particle cannot be found at x=0 or to the left of this position, and it cannot be found at x=a or to the right of this position. It can only be found between x=0 and x=a. In other words, the potential to the left of x=0 is INFINITE! The same is true for the potential to the right of x=a. What this means is that the particle would need to have infinite potential energy in order to be found either within, or beyond, the walls.
What we are looking to do, is to find out what our wave function looks like in the nice, friendly region of space where V=0, between x=0 and x=a. This tells us something about how likely we are to find the particle at various points within this range. To do this, we substitute V=0 into the Schrodinger equation, and then rearrange it so that we've got a second order differential equation that we can solve.
Solving a differential equation is tricky, but luckily the equation we're working with has a simple solution. The wave function, psi, is a sinusoid. It must be either a sine curve or a cosine curve. Because these functions, when differentiated twice, are equal to just a constant multiplied by the original functions themselves.
Additionally, we look at the boundary conditions that the wave function must be equal to zero at both walls. This is because we should not be able to find our particle at the walls. This ends up meaning that the wave function can only look like a half sine wave, or a full sine wave, or 3/2 sine waves, and so on. Integer multiples of half sine waves, basically. Because if this were not true, then the value of the wave function at the walls would be something other than zero - which is not allowed.
When we plug in the condition that the wave function must be zero at the wall where x=a, we get an interesting constraint on the ENERGY of the particle. We find that because the wave function can only be an integer multiple of half a sine wave, the energy can also only take particular values. This is a phenomenon known as quantisation (quantization in the US). In our setup, the particle cannot have any arbitrary value of energy - it can only have specific values! And even more intriguingly, the particle must have a minimum amount of energy in order to exist in the box! This minimum energy is larger than zero, and is often known as zero point energy.
Check this out if you're unfamiliar with radians btw: en.wikipedia.org/wiki/Radian
There is one further thing we need to consider, which is known as normalisation (normalization in the US)... but that's for a separate video!
Timestamps:
0:00 - Introduction!
0:18 - The Schrodinger Equation - Wave Functions and Energy Terms
2:15 - Time-Independent Schrodinger Equation - The Simplest Version!
2:45 - The One-Dimensional Particle in a Box + Energy Diagrams
4:44 - Substituting Our Values into the Schrodinger Equation
5:36 - The Second Derivative of the Wave Function
6:23 - 2nd Order Differential Equation
7:28 - Boundary Conditions (At The Walls)
8:52 - Quantization of Energy
11:29 - A Physical Understanding of our Mathematical Solutions
Thanks for watching, please check out my socials:
Patreon - patreon.com/parthg
Instagram - @parthvlogs

[@ArvinAsh]

Excellent episode! Masterful explanation of a very difficult topic. Couldn't have done it better myself. I won't need to make a video on this topic, as I will just refer people to yours. Ha.

[@ParthGChannel]

Arvin you're a legend, thank you for such a wonderful compliment!

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😀

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right, or he could just rise on his own since hes good enough to attempt that himself, and not have his efforts accredited to them

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[@ParthGChannel]

Haha thank you so much for your support :D

[@genericallyentertaining]

You're brilliant. Clearest explanation of this I've seen. I love how you didn't really assume any prior knowledge and explained every step; this was just what I needed.

[@toadalmoji5853]

this was more than a clear explanation, he took the arrogant-physics-genius-higher-intelligence-explanation ingredient that's baked into learning physics, out

[@v.gedace1519]

Studied Chemistry in the second half of the 80´s. Phyiscs and this topic was also part of it. We solved it for Hydrogen. It was really fun (no sarcasm). But still today (around + 35 years later) I could´nt get my head around the fact that time has to be eleminated / ignored in the first step. Seems to me that time does does not exists. Waiting for someone who figures out that the dimensions are curled one quantum level and that this is the reason why we experience time, the (non-force) effect of gravity, black matter and energy and strange things, like time bending, when "we go closer" to (massive) objects / black holes ... Pretty sure some one will find out that black holes are not singularities. Black holes are the oposite. Starting with the event of horiziont the dimension are so far "un-curled" (= straight) that (from our point of view) we could not "see" beyond our > 3 dimensions because at this point there are way more dimensions. So black holes aren´t holes they are the opposite, they are columns. lol . greek mythology came in my mind ... Didnt they say that the world is carried by Atlas and columns?

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PARTH ROCKS LIKE ALWAYS!😎

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@@localtitans4166 LoL I think it's related to quantum physics not astronomy

[@Lixmage]

You Sir are extremely talented. This is probably the clearest, most elegant description of solving the wave equation I have ever seen uploaded to KZbin. Congratulations!

[@farhannoor3935]

Phys Major here, really glad you posted this. just shows whats up ahead for me in junior year. The walk-through was good, really. Makes one excited for their 1st QM course.

[@graememorrison333]

Wish we had a resource like this back in my day. We were chucked in straight at the deep end with heavy-duty bra-ket notation, operators and the rest of it. Needless to say, I never got a good qualitative understanding and grounding in of QM, and essentially had to blag my way through the remainder of my course.

[@mcrrocks897]

This is probably the most succinct and intuitive way I've heard the particle in a box explained :)

[@ryanlyle9201]

I love how you explained the connection between quantization and the usage of radians in the mathematics. Professors and books tend to skip over those aspects assuming it's trivial. Thanks for the help!

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Your Videos are by far the best Physics Videos out there mate!! Ur vids are the main ones which pop up in my recommendations Thx a lot!!

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Hi Parth, I feel compelled to say what an absolutely BRILLIANT job you do in explaining each lesson. I have been a secondary teacher for some 40 or so years; I marvel are your style. You are a 'natural'. Whatever you do, don't change a thing.

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You do such a great job with concepts. And I don't know anyone who communicates "the joy of physics" better. Kudos Parth!

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Sir, you are getting better and better each lesson.. Thank you so much!

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Parth , you are the best Physics "explainerer’ on the web! I need to watch all your videos now.

[@bonelessbooks9263]

I’m in modern physics 2 and we deal with the Schrödinger equation all the time. This is the best and most concise explanation of the Schrödinger equation, boundary conditions, particle in a box, and quantization. Thank you!!

[@stevewhitt9109]

I have seen a lot of videos on the Schrodinger Equation, but this video brings them all full circle, 2 pi radians :) Lots of details, great job! A compliment from Mr Ash says it all.

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I like how you don't just explain how great is Schrodingers equation and But you really give them a feel of how Schrodingers equation actually work.great job👍👍

[@user-ie9tb4pr1c]

thanks a lot. I'm a computer scientist and haven't studied physics for 7 years. but everything was explained so well that I was able to understand.

[@AnthonyDavid59]

Loved your patient description of what happens "at the walls". It leads naturally to the periodic solutions.

[@Agent_Foxtrot]

How the Schrodinger Equation is actually used is a concept that always escaped me until now. Thank you for a fantastic explanation as well as for not shying away from more advanced concepts like derivatives and radians. Subscribed.

[@hedgehogchaser2494]

You just explained to me in 13 minutes what my prof couldn’t in two weeks. Please do a video just like this for the time-dependent Schrödinger equation.

[@fortytwo5357]

Beautiful explanation!! The step-by-step development of the theory is really awesome!

[@muddassirghoorun4322]

Great video! Amazing job. The world needs more of these explanations.

[@abenezersida8618]

This video is a must for someone who just started learning the Schrodinger equations. Amazing explanation

[@motonationarmy9083]

This is the absolute best explanation of solving this equation! Thank you so much and I will be referring to this in my upper level physics classes for my major ☺️

[@rock3tcatU233]

This is the best explanation on the basics of the SE I've come across.

[@AngelTorres-cc6cp]

Extremely lucid discussion on an otherwise unwieldy topic, thank you Parth G! Also your footnotes are hilarious 👍👍👍

[@user-sf5kd8ld2f]

You made it look easier than what I thought, thx for the effort!

[@mrfinesse]

Thanks Parth - I always wanted to understand how to solve the SE, now you made it easy to understand.

[@JustMoseyinAround]

*As always, an incredible explanation. You simplified a complex looking expression. Great job.*

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Best video on youtube on Schrodinger eqn❤

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Best Physics explanaior and the best KZbin channel for Physics in whole KZbin .

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Thank you, Parth. I remember struggling mightily with the purpose of studying trigonometry. What my instructors failed to mention was how essential it would be to understanding the nature of the universe.

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Really simple and usefuel af. Love the videos, subbed

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This was such a good explainer I could actually guess the next step and do all the maths myself!!!

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Good job with the timestamps your video gave me an MIT ad!!! Amazing!

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Taking quantum mechanics this quarter, so excited!

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This was extremely helpful! I’m a junior physics major and I’m taking my qualification exam tomorrow to be allowed to write a thesis in physics, and this is a really helpful explanation and refresher for the quantum mechanics part of the exam

[@davidwright8432]

Decades back, as part of an admissions procedure to a graduate physics course, i had to write a term-type paper on the Schrodinger wave equation. About which, as a humble pure mathematician, I knew nothing. It took me a week's hard work. If I could have seen your video, it'd have taken me a lazy afternoon! (Yes, my efforts persuaded the admissions committee!) I now always refer people to your videos if hey want a very clear explanation of basic (not the same as simple!) points in physics.Thanks, Parth - from all your happy customers!

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I think that this is your best video that i have watched

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omg I kind of understand Schrodinger Equation now after 3 year of reading .... thank you!

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Thanks for this video. Well done! Will be back for additional videos going deeper!

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Great, high quality video. Big thanks from an intro to quantum student

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Me after studying atomic physics in high school:

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Same here,After Studying Atomic Structure of 11th Grade

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I can't believe I actually understood that. Amazing teacher!

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i love this man, i waited from 9:00am (IST) till 9:30 pm (4:00pm UK time ) to watch his vids

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Thanks very much for your support!

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❗️Excellent explanation❗️

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Nice and clean explaination!

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Love your videos, please do more complex ones for schrodinger's equation.

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Best video yet

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I'm eagerly waiting for your video on Bose Einstein condensate 😀 By the way I've not seen the video yet but I can surely say that it will be wonderful 😊❤️

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Excellent explanation! Thanks for sharing

[@rbettsx]

You're so good at this. I would love to see something like this that really explains how the wave-finction is related to the vectors in state-space

[@schmetterling4477]

The wave function describes the behavior of an infinite number of repetitions of the same experiment, while a state describes the actual measured behavior of one experiment. Think of the wave function as an entity similar to a probability distribution. The probability distribution can't tell us what number an individual dice throw will generate, but it can tell us that, on average, all numbers will be equally likely.

[@ismailfadul6759]

That’s great video you make everything easier to understand thank you brother. Would you please do Fourier transform ?

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thanks for this great explanation!

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Such an awesome explanation. Keep em coming. You got a sub from a physics major who already knew the solution.

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Mordern day Richard Feynman! Your explanation was so eloquent and simple dude! Thanks for that

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Parth you are really a physicist........ Just take our lecture the next Tuesday!😎

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I hv been trying to understand this part of the griffiths book for so long but cudnt get it, But today my man, you hv taught me this concept so thank you.

[@mrdr4934]

Is this why electrons fall into specific energy levels, due to their wave functions having discrete energies?

[@wayneyadams]

Exactly. In simplistic terms, the most probable locations of the electrons are like three dimensional standing waves. The first energy level has only one type of region, called an s-orbital, which is essentially spherical. The second energy level allows for both the s-orbital and a more complex set of regions called p-orbitals. Remember that these are regions of space where the probability of finding the electron is high, not absolute boundaries.

[@mrdr4934]

@@wayneyadams Very intersting, thank you

[@helmutalexanderrubiowilson6835]

This is why some scientists have the theory that we leave in a matrix. Because the only way to simule our universe is quantizing energy it means the most basic form of energy in the universe have finite states.

[@thejuniorastrophysicist]

Hey Parth! The equation was explained in simple terms and the wavefunction was explained beautifully. Can you make a video on the shape of the universe? Λ>1, Λ=0, Λ

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Well done! Educational and Informative! 🎉 😊

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Very brief explanation THANKS ...!! please dirac equation next ?

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Thank you sir, more clear and precisely mention .. excellent visualization. Kindly make the videos for tunneling. Thanks again

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Superb - well done.

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nice video! well explained!

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