Particle in a Box Part 1: Solving the Schrödinger Equation
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- เผยแพร่เมื่อ 14 ต.ค. 2024
- Now that we understand the Schrödinger equation, it's time to put it to good use, and solve a quantum problem. Let's find the eigenfunctions and eigenenergies associated with a quantum particle restricted to an infinite square well. This is also known as the famous "Particle in a Box" problem. It's not as hard as it sounds, check it out!
Script by Hèctor Mas
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Man, this brings back memories. This was one of the more satisfying things to do. Things being equal to zero and canceling out and ultimately arriving at a very clean solution is nice.
but still i didnt get the ans for my required ques.
I have to say...this is definitely well-scripted. Clearly, you've included the best parts of other videos and put it all together in one succinct, clear, and concise presentation. Honestly, it's about time someone did this. Thanks for making this available for everyone. I wish I would have had this when I took QM. It's all good...I had a lot of smart friends and grad students that were around to help clear things up.
I'll be starting my second quantum chemistry course in a few days, that was very helpful and couldn't have come at a better time. Thank you Professor Dave you rock as usual! Kudos from France
Hey from which university (if it's not lack of discretion)?
Je cherche un master également...
Cheers
Man I couldn't understand anything from the slides I had , and all that maths was frightening. Thanks to you I am confident enough to try this by myself again. You are doing a great work please keep it up
Prof Dave is truly amazing!!!
It's extremely clear and the steps are well-ordered to let us understand more easily.
You are literally a lifesaver. Perhaps I should send this to my professor.
Not afraid to admit I need to go back to less advanced videos.
Thanks for taking the time to make these, I hope to purchase a mug or something in the near future.
oh god please tell me this is a series
yes check the modern physics playlist, the last 4 i've posted have been part of this miniseries on a more rigorous approach to quantum mechanics
@@ProfessorDaveExplains I love you so much...♥
We followed this path in the physics studies... the real satisfaction was when we calculated our first Hydrogen atom and Hydrogen molecule, and the energy levels of electrons. and than we compared it to lab measurements to see, that all that mathematical abstraction were actually representation of something very real and very fundamental. A true "feeling like a god" moment in life.
@@meleardil That's the beauty of physics :)
@@ProfessorDaveExplains sir can you plz explain : psi=Asin(kx)+Bcos(kx)? How did it come? How to prove?
I missed a whole bunch of physics lectures because of medical reasons. *I've never been more glad to be your subscriber....*
Thanks Professor Dave!
Great video. You can prove that E≥0 and hence k is real by using that
E= = /2m + ≥ ≥ min V(x) =0.
"If E=0 we don't even have a particle to begin with" Huh??? That rather depends on the potential. Just as in classical theory, if the potential energy is negative (admittedly not the case here), then E=0 is possible (e.g., parabolic orbits). You should rule out E=0 by the boundary conditions at x=0 and x=a (or show E=0 implies that ==0 and =0 implies that ψ=0.)
The quick way of getting the normalization is to remember that the average value of sin²(x) over an non-zero integer multiple of the half period is ½, Thus = ½(length of interval)= ½a. Obviously, the reason why this works comes from the trig identity you quoted. This result is also useful when playing around with Fourier series.
Damn, I'm trying my best to decipher this coded message!
I would like to point out that the entire equation could be solved strictly mathematically, and it really isn’t that difficult. I assume Dave is quite capable of doing this.
However, the way he used, by looking at how the terms look and behave is much more useful to someone learning it, because it helps you to visualize it. If you just went ahead and solved it, there are multiple places that errors could have crept in, and you would have no idea, ecause you would have no appreciation for what you were looking for.
Nice job!
Is there a link that solves it step by step, im a chemistru student and i do not have a clue how psi = a sin (kx) and B sin (kx) , every single video just assumes thats the baseline
@@mehmethanbuyukdag3245 there are, but this setup gives a fairly simple PDE that can be separated into x and t equations, and then transforms to an ODE. You may not get much from a step by step if you have little to no PDE (and ODE) experience. Here is one that will carry you much of the way th-cam.com/video/Dt_VKsSggAo/w-d-xo.html
Important thing to know is the exponential identity e^(*/-)ix = cos x (+/-) i sin x
@@cguy96 thanks a million Mark :) much appreciated
@@mehmethanbuyukdag3245 The wave equation is just a 2nd order homogenous linear partial differential equation ( I know that sounds insanely complex but its not as bad as it sounds) . This differential equation can be solved by using the "characteristic" equation method. It just requires subbing e^(rt) in for Psi ( you will see in doing differential equations how incredibly useful Eulers constant and Eulers formula is). Subbing this in, and doing the derivatives leaves you with a quadratic, where you may simply find the roots, and plug them into the characteristic formula. You will get complex roots and using Eulers identity (e^(ix)=cosx+isinx). After some algebraic manipulation, and dropping the "i" term (which you may do because the solution is a real value solution), you will eventually arive at Asin(x)+Bsin(x), then just plug in the proper variables.
Brings back memories from my undergrad
bro
wow, Lois by now post-covid u-must-b a Math wizz!
@Lucas Shepherd yea but u gadda 1st no their user name!
@Lucas Shepherd ive seen this comment spammed in every comment there is on existence
I've been looking at this and similar videos for maybe 6 years. Including the MIT series. Something about the way you presented this, I suddenly realized the integers ñ=2 ,n=3, etc, yielded the outer orbits. Thank you. Bill
This supplements the MIT OpenCourseware series really well. Both are great, but seeing the math broken down from a couple of different angles is really helpful
hi...can you plz explain : psi=Asin(kx)+Bcos(kx)? How did it come? How to prove?
@@ashrafhossain4324 th-cam.com/video/u-ympqH2SBs/w-d-xo.html&ab_channel=TheMathSorcerer
Let the particle out of the box!😡😡😡
Thank you so much ! This is exactly what I am studying right now, and you make everything so much clearer and easier! Keep it up! I am waiting for your future videos of the series. :)
can you plz explain : psi=Asin(kx)+Bcos(kx)? How did it come? How to prove?
how you made this simple is underrated ,you resparked my interest in physics. thumbs up
Perfect as Always, i hoped that you explain how we can find another eigen value (B) for the equation
That was the most comprehensive and well explained solution of the infinite square well problem I ever seen! Thank you Professor Dave!
I’m staggered by how much clearer this is to understand than my university textbooks?! If only I could find a book that explains all of QM so clearly I’d have my module done in a few weeks, honestly!
I notice others have commented on how disarmingly simple the maths methods ultimately are here, and I’m inclined to agree…
However, there are so many sudden leaps from one equation to another, seemingly, on a whim in this topic - dividing by i, BUT being expected to use (-i), instead of 1/i, is a good example - that, if the situation just isn’t explained well, it gets baffling fast!
Thanks so much, easy sub from me 👍🏼
Maaan, I can't tell you how much I wish I'd had your videos back in Pchem.
Simply lovely! Thanks Professor Dave!
Thank you very much! It became so much plain and understandable after this video.
These are my favorite Prof Dave quantum videos...
Great explanation!
Thank you Professor. You made these things so simple and easier to understand.
Great explanation! Sort of understood this in class but now I feel much more comfortable with it. Thank you so much.
Been watching the whole series, I have to say great job, sub'ed.
And on the topic, this is all that's really required to explain virtual particles.
But it all breaks when "a" < Planck length.
Fantastic video! Far better than any lecture
This has to be one of the smoothest qm video I have seen on youtube.
Amazing series. People like you are the real heros
👑
Thanks dear,
You saved my time.
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This is same explanation I got in physics class but here I understood far better
The part about the infinite potential always confused me. Nicely explained.
Really helpful video.....you explain really good at citical point where doubts and confusion can come
Great explanation.. I'm watching again and again
U explained it very simply, Thanks
I love professor Dave's videos!
Thank you so much, really save my life , my prof explain quantum in an sleepy way. Watch your video, I even don't have to pause and think.
I look forward to the tutorial on the physical implications of this situation. I'm realising that once you've got the maths down, it's beneficial - and enjoyable - to think through what the maths is telling you. This is how you develop intuition that might help in more difficult problems.
You'd probably want to watch a video on finite potential wells and quantum tunnelling then. There's not much real life in a infinite square well.
Thanks for making these videos, they're really helping with my revision!
Fantastic explanation, very easy to follow along with. Thank you!
Beautifully explained
This man is the savior. Bout to pass pchem now 🎉
Very organised and Good explanation Of concepts Appreciate the effort I came when I didn't understand a Topic In My paid Course for IITJAM
Excellent as always
Your videos are very good and easy to understand. Ive seen those concepts so many time but without knowing maths and equations... Its so fascinating. in super mario rpg world, there is that interpretation of matter and antimatter, see past, present and future, consuming time and consuming you. you defeat it fair and square then you get reward quartz charm in a crystal form giving you immunity to the quantum of the world. kind of... it's AI is based on calculation and what is the best action possible for any turn based... he looks like the root square symbol too hahah. wow Im blown away.
That's very interesting! I feel certain it's based on the book the Revelation of Jesus Christ to St John from someone who understands that it's all about us individually. The quartz crystal is the new Jerusalem which is a transparent cube with 12 pearl gates. I happened to see on the weather channel a couple of weeks ago how salt cubes are being generated by the dead Sea because it's drying up.
I think there is a lot of meaning in the choice of making him a square root symbol. The odds are you think I'm off my rocker but if you are interested check out the last 5 or so pages of the book of the Revelation.. I think the pearly gates relate to our twelve cranial nerves and 12 perfected cell salts as salts seem to automatically seek their perfect match leaving irritant matches for matches that can only be broken in water to loosely paraphrase Walter Russell. Salts are made from the Halogen elements with a metal. I'm sure this is connected to halos, may sound crazy. The river Jordan represents our cerebral spinal fluid which is the river of Life. This river empties into the dead Sea. The 12 cranial nerves flow through and out the Spine in a way I have to say grosses me out a bit because I don't like the look of how this looks like roots coming from a tap root but that is what they are, the roots of the tree of Life. Anyway, don't worry if you aren't interested in what I say for any reason. I am interested if there is anything else you think I might find interesting about that game. 🙏
legit, this man is something else
can you explain why the partical has zero probability been outside of the well ?
Nice presentation.
Amazing video very straight forward derivation and mathematically accurate without being too advanced
AMZING, U are so much better than my professors
this is the one i was looking for
Thank you for all this useful explanation professor💙
Literally have my first exam today, thank you sooooo much!
That was such good explaination
Wow it was so well explained! Quantum mechanics is easy if you explain all!
literally bless u. cant thank you enough for all ur help !
Awesome video. Super clear and well explained. Love it
Good instruction and explanation
Thank you for creating this series
taking p chem right now, thank you professor Dave
i hope a series for differential equations soon! vv nice vids
Awesome. I like the way you explain. ❤
Thank you, it's helpful !
Low key this video became so useful after I’ve learned differential equations lol
Are you going to, or did you, make a video on quantum tunneling with the math involved?
Maybe how it allows transistors and fusion (apparently gravity and heat isn't enough) to work.
yep i'll probably cover that a bit later
thank you professor dave from ethiopia
Tanks for evry thin in this vid really i can understund some thing
best QM video ever
Can you please explain the mathematical procedure to solve the equation (d2¥/dx2) + k.k¥ =0
Or please give me the resources where the whole thing is explained instead of hit and trial method
Please look into “Auxiliary differntial equations”. Luckily, this is one of the easiest types PDEs to solve. Good lick
Finally I was able to understand
This was a really nice video. I am 7th grade and I found it very helpful. :)
Awesome video, helped me a ton. Thank you Professor Dave!!!
Hi Solve this for me please,
For the particle of mass m in the one dimensional box with width a, the wave function of the particle at time (t = 0) inside the box is Ψ(x) = Asin(3πx/2a)cos(πx/2a) 1- Find Ψ(x, t > 0). 2- A measurement is made of the energy. What energies can be found? What is the probability of obtaining each value of the energy?
ty for the carry for the return to uni. my ability to memorize is bad so this helps a lot
This video is a bit old, but I've been curious for a while as to where your videos on differential equations are. Were you planning on making them and haven't done that yet, or were they hidden/deleted for some reason?
haven't managed to do it yet sadly
@@ProfessorDaveExplains Oh, that's disappointing as your math videos have been very helpful when I'm studying before taking a class, and I'm going to be taking ODEs not too far in the future. I also just wanted to thank you, as before I found your mathematics playlist I was struggling to relearn HS algebra, but I'm now having a pretty easy time in calculus and am planning on majoring in mathematics. So do please know that you're having a very direct positive impact on the education of a huge amount of people 💜.
Thank you so much sir
Professor thanks a lot
Isnt e^ik a solution too? You get -k^2 e^ik + k^2 e^ik which is 0
it is my favourite video
If the conjectured wavicle - dual particle and wave - cannot occupy the implicitly universal, infinitely potential energied 'boundary,' then Schrodinger's equation is a universal portal model for it's application here, with the law of non-contradiction : nothing is both x and not-x, which is implicitly installed interuniversally by reason, establishing a logical necessary condition for energy conservation everywhere.
However, some Schrodinger solution argument premises violate the law of non-contradiction, for example, "integers" (10.00 s), ..., -1, 0, 1, ... , includes positive and negative natural numbers, 0, 1, 2, ... and "set" (14.30 s) is demonstably inconsistent :
'the set of all sets that are not elements of themselves, is a set that is not an element of itself and is not, a set that is not an element of itself' (Russell, B.) and
"2. Axiom of the Unordered Pair: For any a and b there exists a set {a,b} that contains exactly a and b." ("Zermelo Fraenkel Axioms," WolframMathWorld), but suppose the single two element set, c = { x , not-x }, inherits t, because x is of type t, then c also inherits not-t and from this, c violates the law of non-contradiction.
Excellent
Thank You Sir
Just took a text on this. I wish I could watch it yesterday!
Thank you a lot
Thanks Professor!!!
Thanks alot... It helped me 😍😍😍
How can we solve for boundaries x=-a/2 and x=a/2?
It seems to me that the quantization only appears because we've defined sin as a periodical function and hence it can be zero infinitely many times. But if we were to define sin only from zero to 2pi for example then we could do the same math and get the same psi function, but this time with n can only be equal 1 or 2.
Sooo... I don't really understand this point
Sorry for my english btw
If the particle is in a box with 0 potential energy, and it cannot escape the box, then how did it even accumulate kinetic energy to be able to move in the first place?
Also even with kinetic energy what force is acting on it to keep it moving indefinitely if it has already reached a location with no potential?
By analogy, if you drop a rock from a certain height it gains kinetic energy, but that's only because it had a nonzero gravitational potential energy to begin with.
When it hits the ground the potential energy is 0 and it stops moving.
thank you it means a lot to me
Hello Prof, why not using the Laplace transform to resolve the differential equation ??
😍😍😍😍😍😍🤗 thank you so much!
I never knew this was so simple
finally differential equations make sense
Can anybody write the subtitles of first 5 seconds.....
I didn't get that initial part....
حقا أنا جد ضائعة في هاته الوحدة 😢
Aww, 😊 Don't Feel Lost Nor Lonely. You Are Not Ever Alone, Especially If You VOLUNTEER YOUR TIME TO PEOPLE OR THE,🌏🌎🌍🌐⛑️🌊
@@qur4334
شكرا، و الحمد لله تمكنت من فهمها و سأسعى لتطوير نفسي من أجل خدمة وطني و العالم إن أتيحت لي الفرصة،
❤
why do we use the time independent scrodinger equation instead of the schrodinger equation[does the 1D box model involve the assumption that the system doesn't evolve with time]?
Thanks mate