SOLVING the SCHRODINGER EQUATION | Quantum Physics by Parth G
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- Опубліковано 6 чер 2024
- 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
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.
Arvin you're a legend, thank you for such a wonderful compliment!
You're both great, I love watching your videos
😀
Finally here comes my favorite youtubers.
Wait what arvin ash is here
I think a combination of you and up and atom and 3 blue 1 brown doing a deep dive into quantum mechanics to possibly the best series ever created
Add some apostrophes to names bro
bro. that would be incredible
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
All my UA-cam recommendations are videos made by Parth😁😁🤩
Haha thank you so much for your support :D
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.
this was more than a clear explanation, he took the arrogant-physics-genius-higher-intelligence-explanation ingredient that's baked into learning physics, out
This is probably the most succinct and intuitive way I've heard the particle in a box explained :)
You Sir are extremely talented. This is probably the clearest, most elegant description of solving the wave equation I have ever seen uploaded to UA-cam. Congratulations!
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!
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.
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.
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.
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.
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👍👍
Sir, you are getting better and better each lesson.. Thank you so much!
Parth , you are the best Physics "explainerer’ on the web! I need to watch all your videos now.
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.
You do such a great job with concepts. And I don't know anyone who communicates "the joy of physics" better. Kudos Parth!
This is one of my favourite channels! Tysm😭❤️
PARTH ROCKS LIKE ALWAYS!😎
U r from India I guess
It's ur dream to be at NASA
@@localtitans4166 LoL I think it's related to quantum physics not astronomy
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.
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!!
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!!
Loved your patient description of what happens "at the walls". It leads naturally to the periodic solutions.
This video is a must for someone who just started learning the Schrodinger equations. Amazing explanation
*As always, an incredible explanation. You simplified a complex looking expression. Great job.*
Beautiful explanation!! The step-by-step development of the theory is really awesome!
Great video! Amazing job. The world needs more of these explanations.
You are the best at explaining such complicated topics. I can almost pretend that I understand what you are talking about. In other words, the probability that I can understand this stuff when I watch your videos is very high.
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 ☺️
You made it look easier than what I thought, thx for the effort!
Best Physics explanaior and the best UA-cam channel for Physics in whole UA-cam .
This is the best explanation on the basics of the SE I've come across.
Thanks Parth - I always wanted to understand how to solve the SE, now you made it easy to understand.
You’re absolutely unbelievably knowledgeable!
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!
Extremely lucid discussion on an otherwise unwieldy topic, thank you Parth G! Also your footnotes are hilarious 👍👍👍
Good job with the timestamps
your video gave me an MIT ad!!!
Amazing!
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.
Love your videos dude! 🤘
Excellent video. Very interesting, informative and worthwhile video.
Taking quantum mechanics this quarter, so excited!
Thanks for this video. Well done! Will be back for additional videos going deeper!
I wish I could give more than one like to your video, man. Awesome result, you are an amazing teacher.
Great, high quality video. Big thanks from an intro to quantum student
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
I just discovered this channel. I feel like i have to watch all his old videos from scratch now!
i love this man, i waited from 9:00am (IST) till 9:30 pm (4:00pm UK time ) to watch his vids
Thanks very much for your support!
Very well done! Thanks for your time!
Really simple and usefuel af. Love the videos, subbed
Finally I understood that! Thank you soooo much 😭
Woah....this was amazing
There's a request: can you make a detailed video on DIRAC HOLE THEORY??
AMAZING. Wow really intuitive. Thanks!
I can't believe I actually understood that. Amazing teacher!
I love your content! You make this stuff so clear
Best explanation so far 🙏
Thank you so much
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 😊❤️
Nice and clean explaination!
I have no clue why was this in my news feed, watched it anyways, gj on this one, subbed
Excellent explanation! Thanks for sharing
Brilliant video. Brilliant for motivation!
omg I kind of understand Schrodinger Equation now after 3 year of reading .... thank you!
Wonderful explanation!
Thank you so much. Good work, as always.
This was such a good explainer I could actually guess the next step and do all the maths myself!!!
This channel is fantastic - great job
brilliant explanation . liked every video. worth the time spent
Love your videos, please do more complex ones for schrodinger's equation.
yess!! finally ! thank you. will be waiting for normalization video next!
I think that this is your best video that i have watched
I am addicted to physics and this channel ❤️❤️
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
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.
this was super helpful, thank you so much!
Discovered ur channel and oh my! Loveee it!!!! ❤❤❤
❗️Excellent explanation❗️
Super.......hope to see more videos on this subject. Thank you
Very cool! You're an awesome teacher!
Wonderful explanation.
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
God! With that video you got my subscription! Excelent explanation!
Such an awesome explanation. Keep em coming. You got a sub from a physics major who already knew the solution.
Oh can u give me advices on how to major In physics (& how to understand it easily) 😊
Parth you are really a physicist........
Just take our lecture the next Tuesday!😎
I was thinking about this topic and you made a video on it thank you brother
Well done! Educational and Informative! 🎉 😊
This channel is on a whole new level 🙏
Hats off to you sir.
thanks for this great explanation!
Bro, I took 3 quantum mechanics classes, and you explained everything far better in 13 minutes.
Very brief explanation THANKS ...!! please dirac equation next ?
Mordern day Richard Feynman! Your explanation was so eloquent and simple dude! Thanks for that
Best explanation ever!
Parth - love your videos! You are a gifted teacher.
TOPIC SUGGESTION - please show us how symmetry is really used by physic to manipulate an equation or limit the scope of alternative answers to an equation. There are tons of videos "explaining" symmetry as something "symmetrical" but not on how symmetry comes into play in the way physicists use symmetry in calculated expected values, or solve equations, etc.
That’s great video you make everything easier to understand thank you brother. Would you please do Fourier transform ?
Best video yet
Sir, this was nothing short of genius
Really nicely explained, thank you. Just a little technicality at the end when you said that the wave function "is the probability", it's not actually : the wave function is the probability amplitude, the squared modulus is the probability density, and the squared modules times "dx" (or d^N.r N being the dimension) is the actual probability.
WOOOOOOOOW Thank u so much, like your videos are actual treasures of space time to my eyes x)
these subjects sounded so big but even if they don't go very deep, they appear a lot more accessible it's incredible
i solved the schroddinger equation of my gratitude in the comments section, and in all the solutions it is effectively not 0 in my comment (lel but actually true 😁)
Thank you so much sir 😊 great video as always
Superb - well done.
Great video sir.
nice video! well explained!
Thank you sir, more clear and precisely mention .. excellent visualization. Kindly make the videos for tunneling. Thanks again
Great great explanation sir... 😍
Super. Trank you for the very clear explanation. Hope to See more,
Subscribed. Thanks for the nice lessons 🥰