Particle in a Box Part 1: Solving the Schrödinger Equation

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3 жыл бұрын

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!
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Пікірлер: 196

@Felixkeeg 3 жыл бұрын

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.

@shenbagavallia9858 5 ай бұрын

but still i didnt get the ans for my required ques.

@matrixate 3 жыл бұрын

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.

@user-vu9st6fd7v 3 жыл бұрын

Prof Dave is truly amazing!!! It's extremely clear and the steps are well-ordered to let us understand more easily.

@keithhammond5501 3 жыл бұрын

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

@cafe-tomate 2 жыл бұрын

Hey from which university (if it's not lack of discretion)? Je cherche un master également... Cheers

@ishaangupta8589 3 жыл бұрын

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

@pranavlimaye 3 жыл бұрын

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!

@user-qx8zg4de6p 3 жыл бұрын

oh god please tell me this is a series

@ProfessorDaveExplains 3 жыл бұрын

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

@user-qx8zg4de6p 3 жыл бұрын

@@ProfessorDaveExplains I love you so much...♥

@meleardil 3 жыл бұрын

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.

@frede1905 3 жыл бұрын

@@meleardil That's the beauty of physics :)

@ashrafhossain4324 2 жыл бұрын

@@ProfessorDaveExplains sir can you plz explain : psi=Asin(kx)+Bcos(kx)? How did it come? How to prove?

@zaynabhakim2023 3 жыл бұрын

You are literally a lifesaver. Perhaps I should send this to my professor.

@TheRogueRockhound 3 жыл бұрын

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.

@tonistar2000 3 жыл бұрын

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. :)

@ashrafhossain4324 2 жыл бұрын

can you plz explain : psi=Asin(kx)+Bcos(kx)? How did it come? How to prove?

@emlmm88 3 жыл бұрын

Maaan, I can't tell you how much I wish I'd had your videos back in Pchem.

@duck6100 3 жыл бұрын

Brings back memories from my undergrad

@benni-1157 3 жыл бұрын

bro

@ivornworrell 3 жыл бұрын

wow, Lois by now post-covid u-must-b a Math wizz!

@ivornworrell 3 жыл бұрын

@Lucas Shepherd yea but u gadda 1st no their user name!

@dihydrogenmonoxide6748 3 жыл бұрын

@Lucas Shepherd ive seen this comment spammed in every comment there is on existence

@williamcashion5262 8 ай бұрын

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

@stephanietarczynski8608 2 жыл бұрын

Great explanation! Sort of understood this in class but now I feel much more comfortable with it. Thank you so much.

@tommyottobisdee 9 ай бұрын

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 👍🏼

@biswakumardas8054 3 жыл бұрын

Thank you Professor. You made these things so simple and easier to understand.

@codecmac6232 8 ай бұрын

how you made this simple is underrated ,you resparked my interest in physics. thumbs up

@light-and-thunder 6 ай бұрын

That was the most comprehensive and well explained solution of the infinite square well problem I ever seen! Thank you Professor Dave!

@saadhassan9469 2 жыл бұрын

Amazing series. People like you are the real heros 👑

@daisysmart8874 3 жыл бұрын

Thanks for making these videos, they're really helping with my revision!

@cliffordwilliams9597 2 жыл бұрын

These are my favorite Prof Dave quantum videos...

@sillypoint2292 3 жыл бұрын

I love professor Dave's videos!

@takyc7883 8 ай бұрын

Fantastic video! Far better than any lecture

@0cgw 3 жыл бұрын

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.

@paulwalker1617 3 жыл бұрын

Damn, I'm trying my best to decipher this coded message!

@cguy96 3 жыл бұрын

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!

@mehmethanbuyukdag3245 2 жыл бұрын

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

@cguy96 2 жыл бұрын

@@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 kzfaq.info/get/bejne/etqPia6kuM3LcaM.html Important thing to know is the exponential identity e^(*/-)ix = cos x (+/-) i sin x

@mehmethanbuyukdag3245 2 жыл бұрын

@@cguy96 thanks a million Mark :) much appreciated

@jackroche4531 Жыл бұрын

@@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.

@ae5704 Жыл бұрын

Awesome video. Super clear and well explained. Love it

@GimbertLane 3 жыл бұрын

Thank you for creating this series

@utkarshtripathi6777 3 жыл бұрын

Really helpful video.....you explain really good at citical point where doubts and confusion can come

@kennethhicks2113 Жыл бұрын

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.

@yiwang2537 3 жыл бұрын

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.

@charlesnathansmith 2 жыл бұрын

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

@ashrafhossain4324 2 жыл бұрын

hi...can you plz explain : psi=Asin(kx)+Bcos(kx)? How did it come? How to prove?

@jhrp1462 Жыл бұрын

@@ashrafhossain4324 kzfaq.info/get/bejne/q5OpoNOirZi3cqc.html&ab_channel=TheMathSorcerer

@user-qx8zg4de6p 3 жыл бұрын

this is the one i was looking for

@OliverHoppe-iy5ly Жыл бұрын

Fantastic explanation, very easy to follow along with. Thank you!

@user-lk9hi3ey7t 2 ай бұрын

This has to be one of the smoothest qm video I have seen on youtube.

@nadiames8061 3 жыл бұрын

Thank you for all this useful explanation professor💙

@nikkicambel1220 3 жыл бұрын

i hope a series for differential equations soon! vv nice vids

@matiassantacruz5487 2 жыл бұрын

Awesome video, helped me a ton. Thank you Professor Dave!!!

@PrettyMuchPhysics 3 жыл бұрын

Great explanation!

@johnmccrae2932 3 жыл бұрын

Excellent as always

@Fatemehsoltani16 Ай бұрын

Perfect as Always, i hoped that you explain how we can find another eigen value (B) for the equation

@sofiaabidi7607 3 ай бұрын

literally bless u. cant thank you enough for all ur help !

@athikhopfoze9761 5 ай бұрын

Great explanation.. I'm watching again and again

@alicialouv2218 3 жыл бұрын

Literally have my first exam today, thank you sooooo much!

@AvGeek_ixa2005 7 ай бұрын

U explained it very simply, Thanks

@miarizakafabien3452 3 жыл бұрын

Hello Prof, why not using the Laplace transform to resolve the differential equation ??

@mohamedhusam8189 2 жыл бұрын

Amazing video very straight forward derivation and mathematically accurate without being too advanced

@serenen4630 5 ай бұрын

This man is the savior. Bout to pass pchem now 🎉

@savitamalhotra9180 2 жыл бұрын

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]?

@pretty.946 6 ай бұрын

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?

@DebabrataDutta-by8ik Жыл бұрын

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

@pauljackson3491 3 жыл бұрын

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.

@ProfessorDaveExplains 3 жыл бұрын

yep i'll probably cover that a bit later

@Meow_yj 3 жыл бұрын

Thank you, it's helpful !

@sashaannachemistryconcepts 3 жыл бұрын

Nice presentation.

@youtubeshortsviral1361 3 жыл бұрын

Hello Sir Iam a beginner and i have a doubt, so you found out the energies, correct me if Ian wrong but according to Heisenberg uncertainty ∆E∆t is greater or equal to h cross/2π so that means we cannot figure the exact energy of particle right?

@kiranbhaigaming6462 9 ай бұрын

Awesome. I like the way you explain. ❤

@kall228 Жыл бұрын

legit, this man is something else

@pritamnath9135 Жыл бұрын

Wow it was so well explained! Quantum mechanics is easy if you explain all!

@fettuccine794 Жыл бұрын

Beautifully explained

@XBJASHITHABATULA 9 ай бұрын

That was such good explaination

@Pateckharu3 3 жыл бұрын

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.

@imaginaryuniverse632 2 жыл бұрын

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. 🙏

@aleezanoor8013 3 ай бұрын

How can we solve for boundaries x=-a/2 and x=a/2?

@alekderijan8348 3 жыл бұрын

This was a really nice video. I am 7th grade and I found it very helpful. :)

@ujala6542 3 жыл бұрын

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

@jackhamerordewey9552 Жыл бұрын

Please look into “Auxiliary differntial equations”. Luckily, this is one of the easiest types PDEs to solve. Good lick

@user-hn1ol9sv1x 5 ай бұрын

Tanks for evry thin in this vid really i can understund some thing

@michellehu8264 Жыл бұрын

Good instruction and explanation

@muhepei 10 ай бұрын

can you explain why the partical has zero probability been outside of the well ?

@thejaswinihc1246 2 жыл бұрын

What happened to the schrodinger wave equation when the walls if the box is opened?

@JM-nw5cn 3 жыл бұрын

taking p chem right now, thank you professor Dave

@a_8764 2 жыл бұрын

At 3:20, what happened to the potential energy operator V? Shouldn't there be +V*psi on the left side of the time-independent Schrödinger equation?

@bobsemple7382 2 жыл бұрын

Hello, sorry if I'm a bit late. Our equation is for when the potential energy is 0. As shown by the particle in the box. Hence the potential energy operator V is 0 and has been removed from the equation, leaving only the kinetic energy operator.

@peterrobinherbert 3 жыл бұрын

The part about the infinite potential always confused me. Nicely explained.

@chakravarthulapriyanka1652 8 ай бұрын

Thank you so much sir

@user-wj8rz6hh7h Жыл бұрын

AMZING, U are so much better than my professors

@Jin-ud8ij 3 жыл бұрын

best QM video ever

@RogerTerrill 3 жыл бұрын

Thanks Professor!!!

@rossfriedman6570 5 ай бұрын

why is there a time independant form of the Scrodinger equation?

@Jehannum2000 3 жыл бұрын

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.

@duck6100 3 жыл бұрын

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.

@azimmridha8772 10 ай бұрын

Thanks dear, You saved my time. Love from Bangladesh 🇧🇩🇧🇩🇧🇩🇧🇩🇧🇩🇧🇩

@majdiabdalla3505 Жыл бұрын

Thank you a lot

@ArgumentumAdHominem 3 жыл бұрын

Can you be a bit more precise when moving between operators and eigenvalues? In this particular case, the form of the potential function is clear, but what is the explicit form of the potential operator?

@ProfessorDaveExplains 3 жыл бұрын

it depends on the type of potential, so that will be expanded upon later

@ArgumentumAdHominem 3 жыл бұрын

@@ProfessorDaveExplains Type of the potential? An infinite potential well can have multiple types of potentials? I'm confused. Anyway, looking forwards to follow-up videos :)

@oliviamativi7954 3 жыл бұрын

@@ArgumentumAdHominem It's not important for the 1-D particle in a box problem what the form of the infinite potential is, since it's assumed to be zero within the well, which nullifies the V(x) term in the Schrödinger equation. The full form of the potential operator will depend on what kind of potential field you are imposing (electric/gravitational/etc) and its properties, so there is no consistent form for the potential operator.

@ArgumentumAdHominem 3 жыл бұрын

@@oliviamativi7954 We are currently in the realm of mathematics. Please give me any example of a mathematical operator that would correspond to an infinite potential well in Schrödinger equation. If it helps you to call it electric or gravitational go ahead. I'm just trying to understand how one can impose boundary constraints on a PDE by means of an operator, regardless of its meaning

@pallavidwivedi1308 11 ай бұрын

Why we can not solve Schrodinger wave equestion in n dimensional space?

@drvinylscratch1936 2 жыл бұрын

ty for the carry for the return to uni. my ability to memorize is bad so this helps a lot

@saviolaaldino 3 жыл бұрын

thank you it means a lot to me

@user-vj4rs7fh1o Жыл бұрын

Professor thanks a lot

@user-uv1uc1sb7m 2 жыл бұрын

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

@steveying1305 8 ай бұрын

Low key this video became so useful after I’ve learned differential equations lol

@jjw3046 3 жыл бұрын

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.

@shankarnarayan1234 7 ай бұрын

Why can't we take psi as epsilon^rx as the ansatz

@anynamecanbeuse 2 жыл бұрын

3:20 why did you drop the V_caret term in your time-independent equation?

@abhyuraj7985 Жыл бұрын

As our problem states that the potential is 0.

@HarshilKapadiya-ii9gp 4 ай бұрын

Can anybody write the subtitles of first 5 seconds..... I didn't get that initial part....

@jinsa.a6760 3 жыл бұрын

Thanks alot... It helped me 😍😍😍

@priyaks6562 2 жыл бұрын

Thank You Sir

@ivornworrell 3 жыл бұрын

*Hello prof., @ about **02:48**, wudnt the Probability density func.also=0 if psi*(X)=0?*

@ProfessorDaveExplains 3 жыл бұрын

yes, it's just the complex conjugate though, for a real function it's the same

@user-pk9qo1gd6r 3 жыл бұрын

Psi and Psi* have the same magnitude even if they're complex numbers, so one is equal to zero if and only if the other is also equal to zero.

@ivornworrell 3 жыл бұрын

@@user-pk9qo1gd6r Ah, now I understand, thank you.

@ivornworrell 3 жыл бұрын

@@ProfessorDaveExplains Ah, now I understand, thank you Prof.

@sandylissy1323 3 жыл бұрын

Sir can u please send code for particle in a box through matlab

@salaheddinesb7312 3 жыл бұрын

😍😍😍😍😍😍🤗 thank you so much!

@hitheshreddy6601 9 ай бұрын

but in this expression, A=(2/a)^1/2 , A is amplitude of the wave and a is boundary length. If u verify it using dimensional analysis, then it is not satisfying. then how can u say that the relation is true??

@golovolog 7 ай бұрын

Could you explain how is it not satisfying exactly? Isn't psy-function scuared a linear density in this case?