Particle in a Box Part 1: Solving the Schrödinger Equation

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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[@Felixkeeg]

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]

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

[@makssachs8914]

Let the particle out of the box!😡😡😡

[@matrixate]

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.

[@zaynabhakim2023]

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

[@keithhammond5501]

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]

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

[@duck6100]

Brings back memories from my undergrad

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bro

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[@ishaangupta8589]

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

[@TheRogueRockhound]

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.

[@지구과학천문학]

oh god please tell me this is a series

[@ProfessorDaveExplains]

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

[@지구과학천문학]

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

[@meleardil]

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]

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

[@ashrafhossain4324]

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

[@pranavlimaye]

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!

[@Ai-ChingChen]

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

[@vihaangoel3157]

Simply lovely! Thanks Professor Dave!

[@zhanmaster6733]

Thank you very much! It became so much plain and understandable after this video.

[@light-and-thunder]

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

[@codecmac6232]

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

[@charlesnathansmith]

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]

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

[@jhrp1462]

@@ashrafhossain4324 kzbin.info/www/bejne/q17cnqOnfZeGeNU&ab_channel=TheMathSorcerer

[@williamcashion5262]

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

[@tonistar2000]

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]

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

[@cguy96]

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]

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]

@@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 kzbin.info/www/bejne/eqXCh36piMyad9E Important thing to know is the exponential identity e^(*/-)ix = cos x (+/-) i sin x

[@mehmethanbuyukdag3245]

@@cguy96 thanks a million Mark :) much appreciated

[@jackroche4531]

@@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.

[@cliffordwilliams9597]

These are my favorite Prof Dave quantum videos...

[@emlmm88]

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

[@stephanietarczynski8608]

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

[@Fatemehsoltani16]

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

[@0cgw]

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]

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

[@PrettyMuchPhysics]

Great explanation!

[@FeetMyWrathUwU-r8f]

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

[@biswakumardas8054]

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

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Amazing series. People like you are the real heros 👑

[@tommyottobisdee]

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 👍🏼

[@takyc7883]

Fantastic video! Far better than any lecture

[@yiwang2537]

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.

[@sillypoint2292]

I love professor Dave's videos!

[@athikhopfoze9761]

Great explanation.. I'm watching again and again

[@AvGeek_ixa2005]

U explained it very simply, Thanks

[@utkarshtripathi6777]

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

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

[@daisysmart8874]

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

[@kennethhicks2113]

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.

[@지구과학천문학]

this is the one i was looking for

[@OliverHoppe-iy5ly]

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

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

[@XBJASHITHABATULA]

That was such good explaination

[@johnmccrae2932]

Excellent as always

[@nadiames8061]

Thank you for all this useful explanation professor💙

[@fettuccine794]

Beautifully explained

[@ae5704]

Awesome video. Super clear and well explained. Love it

[@peterrobinherbert]

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

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legit, this man is something else

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

[@ZichengYing]

AMZING, U are so much better than my professors

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Thank you for creating this series

[@havocgaming438]

Finally I was able to understand

[@DebabrataDutta-by8ik]

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

[@nikkicambel1220]

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

[@Pateckharu3]

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]

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. 🙏

[@Jehannum2000]

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]

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.

[@خربشةإنسان]

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

[@mohamedhusam8189]

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

[@sashaannachemistryconcepts]

Nice presentation.

[@steveying1305]

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

[@alicialouv2218]

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

[@michellehu8264]

Good instruction and explanation

[@kiranbhaigaming6462]

Awesome. I like the way you explain. ❤

[@pritamnath9135]

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

[@Meow_yj]

Thank you, it's helpful !

[@matiassantacruz5487]

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

[@JM-nw5cn]

taking p chem right now, thank you professor Dave

[@chakravarthulapriyanka1652]

Thank you so much sir

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Professor thanks a lot

[@coolsammyg47-r]

Excellent

[@kega4062]

OOOOOOOOHHHHHHHH- so that'ssssssss what my textbook was trying to sayyy

[@Jin-ud8ij]

best QM video ever

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Thank you a lot

[@juniorcyans2988]

Just took a text on this. I wish I could watch it yesterday!

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thank you professor dave from ethiopia

[@muhepei]

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

[@alekderijan8348]

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

[@drvinylscratch1936]

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

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Thank You Sir

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it is my favourite video

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Thanks Professor!!!

[@5am484]

I never knew this was so simple

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Thanks alot... It helped me 😍😍😍

[@harshitamahajan1473]

thankyou

[@ButterHaus420]

Thanks mate

[@pretty.946]

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?

[@ujala6542]

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]

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

[@juicetime910]

finally differential equations make sense

[@5am484]

This is so helpful i wanna cry

[@aleezanoor8013]

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

[@shack1577]

well done , Thank you :)

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حقا أنا جد ضائعة في هاته الوحدة 😢

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[@خربشةإنسان]

@@qur4334 شكرا، و الحمد لله تمكنت من فهمها و سأسعى لتطوير نفسي من أجل خدمة وطني و العالم إن أتيحت لي الفرصة،

[@CB256]

❤

[@bingusiswatching6335]

Isnt e^ik a solution too? You get -k^2 e^ik + k^2 e^ik which is 0

[@pauljackson3491]

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]

yep i'll probably cover that a bit later

[@RendraNadeViro]

so it is the professor dave will be remembered for roasting flat earthers XD

[@ВладСетонов]

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

[@rossfriedman6570]

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