SOLVING the SCHRODINGER EQUATION | Quantum Physics by Parth G
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- เผยแพร่เมื่อ 9 มิ.ย. 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
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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 TH-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 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
You Sir are extremely talented. This is probably the clearest, most elegant description of solving the wave equation I have ever seen uploaded to TH-cam. Congratulations!
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’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!!
This is probably the most succinct and intuitive way I've heard the particle in a box explained :)
Extremely lucid discussion on an otherwise unwieldy topic, thank you Parth G! Also your footnotes are hilarious 👍👍👍
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.
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.
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!
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👍👍
Love your videos dude! 🤘
Sir, you are getting better and better each lesson.. Thank you so much!
Loved your patient description of what happens "at the walls". It leads naturally to the periodic solutions.
You made it look easier than what I thought, thx for the effort!
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!!
Parth , you are the best Physics "explainerer’ on the web! I need to watch all your videos now.
You do such a great job with concepts. And I don't know anyone who communicates "the joy of physics" better. Kudos Parth!
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’re absolutely unbelievably knowledgeable!
*As always, an incredible explanation. You simplified a complex looking expression. Great job.*
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.
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?
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.
This is the best explanation on the basics of the SE I've come across.
This video is a must for someone who just started learning the Schrodinger equations. Amazing explanation
This was such a good explainer I could actually guess the next step and do all the maths myself!!!
Nice and clean explaination!
Thanks Parth - I always wanted to understand how to solve the SE, now you made it easy to understand.
Taking quantum mechanics this quarter, so excited!
Brilliant video. Brilliant for motivation!
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 ☺️
AMAZING. Wow really intuitive. Thanks!
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.
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!
Wonderful explanation!
Very well done! Thanks for your time!
Really simple and usefuel af. Love the videos, subbed
Thanks for this video. Well done! Will be back for additional videos going deeper!
❗️Excellent explanation❗️
Best Physics explanaior and the best TH-cam channel for Physics in whole TH-cam .
I love your content! You make this stuff so clear
I wish I could give more than one like to your video, man. Awesome result, you are an amazing teacher.
Excellent video. Very interesting, informative and worthwhile video.
Excellent explanation! Thanks for sharing
Discovered ur channel and oh my! Loveee it!!!! ❤❤❤
this was super helpful, thank you so much!
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.
omg I kind of understand Schrodinger Equation now after 3 year of reading .... thank you!
Wonderful explanation.
I just discovered this channel. I feel like i have to watch all his old videos from scratch now!
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!
Is this why electrons fall into specific energy levels, due to their wave functions having discrete energies?
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.
@@wayneyadams Very intersting, thank you
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.
I think that this is your best video that i have watched
Thank you so much. Good work, as always.
Good job with the timestamps
your video gave me an MIT ad!!!
Amazing!
Best explanation so far 🙏
Thank you so much
thanks for this great explanation!
Best explanation ever!
I can't believe I actually understood that. Amazing teacher!
This channel is fantastic - great job
Very brief explanation THANKS ...!! please dirac equation next ?
brilliant explanation . liked every video. worth the time spent
Sir, this was nothing short of genius
Finally I understood that! Thank you soooo much 😭
Best video yet
I apologize for my comments (immediately below) from yesterday. I was 100% wrong. It is clear that "a" is the right hand boundary of the box. Your videos are fabulous.
Very cool! You're an awesome teacher!
Well done! Educational and Informative! 🎉 😊
Love your videos, please do more complex ones for schrodinger's equation.
Great, high quality video. Big thanks from an intro to quantum student
nice video! well explained!
Super.......hope to see more videos on this subject. Thank you
Subscribed. Thanks for the nice lessons 🥰
Great video sir.
I was thinking about this topic and you made a video on it thank you brother
Thank you very much! you're my savior
Superb - well done.
its a beautiful explanation...
Great great explanation sir... 😍
I am addicted to physics and this channel ❤️❤️
Best explanation
That’s great video you make everything easier to understand thank you brother. Would you please do Fourier transform ?
God! With that video you got my subscription! Excelent explanation!
Super. Trank you for the very clear explanation. Hope to See more,
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
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.
Thank you very much. This is very helpful!
I have no clue why was this in my news feed, watched it anyways, gj on this one, subbed
Thank you so much for your excellent explanation.......effective 😊😊
really nice one.
yess!! finally ! thank you. will be waiting for normalization video next!
At school my teachers never told me how to solve the equation..... Thanks for this video..... ❤️
I mean, that's not what they're supposed to teach you in school anyways.
"x + y = z."
The equation "x + y = z" is a simple algebraic equation. To solve for one of the variables, you'll need to isolate it on one side of the equation by subtracting or adding the same value to both sides.
For example, if you want to solve for x, you would subtract y from both sides:
x + y = z
y - y
x = z - y
So now you know that x = z - y. You can plug in any values you have for z and y to find the corresponding value of x.
I hope this helps. Yes, it can be challenging for teachers to manage students who do not study or listen in class.
many thanks!!
Woah....this was amazing
There's a request: can you make a detailed video on DIRAC HOLE THEORY??
Thank you!