Right at the end I say we have derived Maxwell's Equations 🤦♂️ I've got Maxwell's Equations in the brain, it seems. I meant Schrödinger's Equation. Silly mistake, but I hope forgivable 😂 Also, I accidentally missed the square off the ℏ² on the final line! Sorry.
As a mathematician seeing you write what the "operators are equal to" makes my skin crawl. You physicists are on some black magic abuse of notation and I hugely admire the strength of your stomachs. Jokes aside this was a great video for someone who went down the pure maths path seeing what physicists see
I remember during my Physics studies that there were advanced Math majors attending beginners QM shaking in pain when they saw the "derivation" of the Schrödinger equation
Thank you very much! I think I've found a niche here; using A-Level accessible mathematics to push a tiny bit beyond the A-Level Physics specification to explore some of the more interesting aspects of Physics.
I know the mathematicans will go up the wall, but I read engineering and this is exactly how I use maths - for better understanding of something really tricky and non obvious. Great stuff, and a new subscriber. ☺️
Keith, I like your work. I do Zoom tutorials with high school students in Australia using a camera over a horizontal whiteboard on my desk. An improvement I made last year was to turn off the auto-focus so the camera doesn't switch focus to one's hand while writing and remains focused on the text on the board/paper. It depends on your camera set-up, but it might help.
I believe this is close to the reasoning that Schrodinger originally used (as opposed to Dirac's more fancy framework which is what is taught in more advanced QM courses). In the US, some top physics universities (Stanford, Caltech, MIT) teach a derivation like yours to incoming freshman students, who would be comparable to your A-level students. At most US universities this derivation is taught in a course called "modern physics / physics III" at the start of the 2nd university year.
In Scotland (SQA land) students at Advanced Higher (~A2 Physics level in England) do not cover Schrodingers' equation, potential wells etc. However tunneling is mentioned descriptively. I personally think that's a travesty up here, but recently the curriculum has morphed into Astrophysics territory. The Scots are so proud of Glasgow's connection with Gravity Waves! We also do not cover Maxwell's equations other than Ampere Law and Biot Savart etc in EM. No mention of thermodynamics either. J.C. Maxwell was Scottish, thought I would mention that, sort of our equivalent to I. Newton.😬
Similarly, Maxwell's Equations and the Schrödinger equation are only mentioned briefly in England too, but we teach them in a bit more depth in my school because they are A-Level accessible and education should be more than "teaching to the test". That being said, I work in a selective school, so I have the luxury of extremely bright students and our curriculum is about breadth and depth
I mean…..I watch all this stuff coz it amazes me….I was lost at E=hf…….and then all that follows…I mean who makes this stuff up…….Dell…..is that not a computer…..I mean seriously…..E=hf…..
@@marygaleac Max Plank (early in twentieth century) hypothesized (guessed) that so called BlackBody radiated energy in _"quanta"_ , each _quanta_ contributed one unit of energy. He was talking about EM radiation. In modern SI units the constant of proportionality is qiven by h = 6.63x10^-34. So if we plot radiance (energy emitted) v. frequency (if light = colour) then we get a straight line relation. Max Plank got 1915 Nobel Prize in Physics for his work. Einstein later used this to explain the photoelectric effect where light would release electrons from cold metal surfaces under certain circumstances (he got the Nobel prize for that).
Thank you very much. I first saw this 21 years ago, I think, but I'd forgotten it long ago and had to look it up again 😂. I think a video showing the time independent Schrödinger equation is a good idea next, and using it to calculate expectation values of energy or something
At the end you just have to rewrite exp(kx-wt) to exp(k.r-wt) to get the 4 dimension wave equation( r vector), and you just set the Hamiltonian as your […] to get HΨ=EΨ. Also the wave equation is only the real part of Euler formula I thought you could write that too.
I think it's far more sensible to shift to polar coordinates, but that'd be yet another new thing to add on top of A-Level Mathematics. Maybe another time 🙂. I was going to explore the time independent Schrödinger equation you mentioned in a later video 👍 and the physical significance for the wave function being complex
A bit back to front. The cos formula was one wave function (with no imaginary term). The exponential one (using Euler's formula) was a different wave function; a complex one. The reason we're opting for a complex wave function is because the product of wave function with its complex conjugate is the probability density of the particle's position
Well, this is not a very fundamental/rigorous derivation, its more of an heuristic argument. A derivation of the Schrodinger from the more fundamental postulates of quantum mechanics would require results from group theory and functional analysis
@@HilbertXVIJust because you have never seen it before, doesn't mean it hasn't been done. One thing physicists like to do is when a model or equation is hard to derive but gives good results in practice, is to just say its a postulate. Of course, when Schrodinger introduced it, it was indeed a postulate. However, its been many decades since then and it's actually possible to derive it now from more fundamental postulates of QM. The problem is that it takes a lot of work to derive it so physicists just continue to say its a postulate. If you would like to see the derivation check, Quantum Mechanics: A Modern Development by Leslie Ballentine, he derives it in chapter 3 and 4.
@@PhysicswithKeith I spotted it instantly because it is a common forgetting when writing developped Shrödinger's equation, even for us researchers using Quantum Mechanics on a daily basis
Why IS the function complex? All I know is that the modulus equals to the probability of the of wave being at a specific region of space given the specific period of time, but why exactly is that so?
The Schrodinger equation without the "i" in front of the time derivative is called the heat equation (circa ~1850 so the quantum founders knew it well). If you have d psi / dt = real constant*psi and seek a real solution for psi, then you will have either a growing or decaying exponential. This models the spread of heat accurately, but it does not describe an oscillating wave. So the complex numbers come from wanting an equation that is (1) first order in time, to fit E = hbar omega, and (2) that had wave like solutions.
The most contrived way I’ve ever seen of showing 1/i = -i. Just multiply top and bottom by i. Further I get this is a “heuristic argument”, but this requires so much, oh ignore the fact we’re using equations but not relating them to any physical system. Ignore the fact that using the De Broglie wavelength is begging the question. Ignore the fact that your operators look like that on my because you chose a specific function and then you just claim the same relationship will hold for any other. I get it. It’s a toy way to get to the Schrödinger Equation, but it requires such unhealthy habits to “follow” that I think it’s just pedagogically harmful. To students you’re making them think there are ever circumstances where this sort of thing may be justified, which it isn’t. And otherwise it makes it look like physicists are so bad at maths it’s a miracle we ever get a single correct result.
And as a mathematician, I'd add that his "proof" of 1/i = -i isn't just contrived but also invalid. Expressions like (-1)^(1/2) are in general multivalued, since a^b is defined as exp(b*log(a)) but log is itself multivalued. The classic example of this, which goes back to Euler himself, is "-1 = ((-1)^(1/2))^2 = ((-1)^2)^(1/2) = 1^(1/2) = 1", showing just how easily contradictions can occur when you don't properly handle the different branches of the complex logarithm.
The DeBroglie wavelength function was conjectured in the first place, from E=pc =hf = hc/λ for photons, DeBroglie just made the intuitive jump to particles (his day electrons) p = h/λ thus he predicted electron interference/ diffraction. Because the wave function can be applied to simple Euler form e^i(kx - ωt) it can also be applied therefore to any function using its Fourier Series ∑ₙ {Cₙe^i(kₙx - ωₙt)} or transform. There are no unhealthy habits here, on the contrary its a good way to model real phenomena. I mean you use calculus, I don't think Newton bothered too much about what is fellow mathematicians thought of his 'elements', it worked so was good enough for him.
@@amritlohia8240 but 1^(1/2) = 1, -1 (depending on which branch you select the root to be) so your argument is mute. Because x^2 =1, x = 1, -1 does not mean 1 = -1.
@@tomctutor The point was precisely to show that you have to be careful about which branch of the complex logarithm (or equivalently, the power function) you're dealing with, and so you're actually just making my point for me.
Man these A level students put me to shame. Even though I have a PhD in math I never could understand the physicist mindset, seems they either have unusual intuition about the world or make stuff up as they go.
Right at the end I say we have derived Maxwell's Equations 🤦♂️ I've got Maxwell's Equations in the brain, it seems. I meant Schrödinger's Equation. Silly mistake, but I hope forgivable 😂
Also, I accidentally missed the square off the ℏ² on the final line! Sorry.
As a mathematician seeing you write what the "operators are equal to" makes my skin crawl.
You physicists are on some black magic abuse of notation and I hugely admire the strength of your stomachs.
Jokes aside this was a great video for someone who went down the pure maths path seeing what physicists see
😂 You should hear all the bad things we physicists say about mathematicians 😂 it's all good fun. Thanks for the kind comments
I hope you never see how he treat derivatives... as quocients...
😂
I remember during my Physics studies that there were advanced Math majors attending beginners QM shaking in pain when they saw the "derivation" of the Schrödinger equation
This is my idea of entertainment! Can't wait to see the future video where you use Schrodinger's Equation.
Thank you for easy explanation. Like really, i havent seen something similar on a youtube. This just give more motivation to learn physics in general.
Thank you very much! I think I've found a niche here; using A-Level accessible mathematics to push a tiny bit beyond the A-Level Physics specification to explore some of the more interesting aspects of Physics.
I know the mathematicans will go up the wall, but I read engineering and this is exactly how I use maths - for better understanding of something really tricky and non obvious. Great stuff, and a new subscriber. ☺️
Keith, I like your work. I do Zoom tutorials with high school students in Australia using a camera over a horizontal whiteboard on my desk. An improvement I made last year was to turn off the auto-focus so the camera doesn't switch focus to one's hand while writing and remains focused on the text on the board/paper. It depends on your camera set-up, but it might help.
Same, getting lighting right on shiny whiteboard is not easy.
I set up with lots of diffuse, reflected light from multiple directions. Direct light is a no-no for shadows and shiny spots.
@@Hiltok [ 2x = 🌟 >💡< ] 😎
I believe this is close to the reasoning that Schrodinger originally used (as opposed to Dirac's more fancy framework which is what is taught in more advanced QM courses). In the US, some top physics universities (Stanford, Caltech, MIT) teach a derivation like yours to incoming freshman students, who would be comparable to your A-level students. At most US universities this derivation is taught in a course called "modern physics / physics III" at the start of the 2nd university year.
What a channel ! Gold !!
qualility video looking forward to the 3 future videos you hinted at
In Scotland (SQA land) students at Advanced Higher (~A2 Physics level in England) do not cover Schrodingers' equation, potential wells etc. However tunneling is mentioned descriptively. I personally think that's a travesty up here, but recently the curriculum has morphed into Astrophysics territory.
The Scots are so proud of Glasgow's connection with Gravity Waves!
We also do not cover Maxwell's equations other than Ampere Law and Biot Savart etc in EM.
No mention of thermodynamics either.
J.C. Maxwell was Scottish, thought I would mention that, sort of our equivalent to I. Newton.😬
Similarly, Maxwell's Equations and the Schrödinger equation are only mentioned briefly in England too, but we teach them in a bit more depth in my school because they are A-Level accessible and education should be more than "teaching to the test". That being said, I work in a selective school, so I have the luxury of extremely bright students and our curriculum is about breadth and depth
I mean…..I watch all this stuff coz it amazes me….I was lost at E=hf…….and then all that follows…I mean who makes this stuff up…….Dell…..is that not a computer…..I mean seriously…..E=hf…..
@@marygaleac Max Plank (early in twentieth century) hypothesized (guessed) that so called BlackBody radiated energy in _"quanta"_ , each _quanta_ contributed one unit of energy. He was talking about EM radiation. In modern SI units the constant of proportionality is qiven by h = 6.63x10^-34. So if we plot radiance (energy emitted) v. frequency (if light = colour) then we get a straight line relation. Max Plank got 1915 Nobel Prize in Physics for his work.
Einstein later used this to explain the photoelectric effect where light would release electrons from cold metal surfaces under certain circumstances (he got the Nobel prize for that).
Thank you!!
This is already fabulous :D
Thank you very much. I first saw this 21 years ago, I think, but I'd forgotten it long ago and had to look it up again 😂. I think a video showing the time independent Schrödinger equation is a good idea next, and using it to calculate expectation values of energy or something
At the end you just have to rewrite exp(kx-wt) to exp(k.r-wt) to get the 4 dimension wave equation( r vector), and you just set the Hamiltonian as your […] to get HΨ=EΨ. Also the wave equation is only the real part of Euler formula I thought you could write that too.
I think it's far more sensible to shift to polar coordinates, but that'd be yet another new thing to add on top of A-Level Mathematics. Maybe another time 🙂. I was going to explore the time independent Schrödinger equation you mentioned in a later video 👍 and the physical significance for the wave function being complex
Really enjoy your derivations. Why is the image quality lower then last time, because it's sometimes hard to read. But well explained :D
Thank you for your kind comment. It's still processing by UA-cam. It will be 4K in a few hours 😁
Ok thank's 😀@@PhysicswithKeith
8:27 what happened to the isin(x) part of eulers formula?
A bit back to front. The cos formula was one wave function (with no imaginary term). The exponential one (using Euler's formula) was a different wave function; a complex one. The reason we're opting for a complex wave function is because the product of wave function with its complex conjugate is the probability density of the particle's position
Very cool
Loved this
my teacher just told me to remember it as the derivation is beyond our scope. But I understood and enjoyed it
Well, this is not a very fundamental/rigorous derivation, its more of an heuristic argument. A derivation of the Schrodinger from the more fundamental postulates of quantum mechanics would require results from group theory and functional analysis
@@whonyx6680 Well you can't really "derive" the schrodinger equation anyway, it's a postulate of QM
@@HilbertXVIJust because you have never seen it before, doesn't mean it hasn't been done. One thing physicists like to do is when a model or equation is hard to derive but gives good results in practice, is to just say its a postulate. Of course, when Schrodinger introduced it, it was indeed a postulate. However, its been many decades since then and it's actually possible to derive it now from more fundamental postulates of QM. The problem is that it takes a lot of work to derive it so physicists just continue to say its a postulate. If you would like to see the derivation check, Quantum Mechanics: A Modern Development by Leslie Ballentine, he derives it in chapter 3 and 4.
In the last equation you wrote hbar on the left instead of hbar²
Whoops, well spotted! I'll add that correction to my pinned comment. Thank you for pointing it out!
@@PhysicswithKeith I spotted it instantly because it is a common forgetting when writing developped Shrödinger's equation, even for us researchers using Quantum Mechanics on a daily basis
Why IS the function complex? All I know is that the modulus equals to the probability of the of wave being at a specific region of space given the specific period of time, but why exactly is that so?
The Schrodinger equation without the "i" in front of the time derivative is called the heat equation (circa ~1850 so the quantum founders knew it well). If you have d psi / dt = real constant*psi and seek a real solution for psi, then you will have either a growing or decaying exponential. This models the spread of heat accurately, but it does not describe an oscillating wave. So the complex numbers come from wanting an equation that is (1) first order in time, to fit E = hbar omega, and (2) that had wave like solutions.
@@iyziejane I see, thank you so much for clearing that up!
The most contrived way I’ve ever seen of showing 1/i = -i. Just multiply top and bottom by i.
Further I get this is a “heuristic argument”, but this requires so much, oh ignore the fact we’re using equations but not relating them to any physical system. Ignore the fact that using the De Broglie wavelength is begging the question. Ignore the fact that your operators look like that on my because you chose a specific function and then you just claim the same relationship will hold for any other. I get it. It’s a toy way to get to the Schrödinger Equation, but it requires such unhealthy habits to “follow” that I think it’s just pedagogically harmful.
To students you’re making them think there are ever circumstances where this sort of thing may be justified, which it isn’t. And otherwise it makes it look like physicists are so bad at maths it’s a miracle we ever get a single correct result.
And as a mathematician, I'd add that his "proof" of 1/i = -i isn't just contrived but also invalid. Expressions like (-1)^(1/2) are in general multivalued, since a^b is defined as exp(b*log(a)) but log is itself multivalued. The classic example of this, which goes back to Euler himself, is "-1 = ((-1)^(1/2))^2 = ((-1)^2)^(1/2) = 1^(1/2) = 1", showing just how easily contradictions can occur when you don't properly handle the different branches of the complex logarithm.
The DeBroglie wavelength function was conjectured in the first place,
from E=pc =hf = hc/λ for photons, DeBroglie just made the intuitive jump to particles (his day electrons)
p = h/λ thus he predicted electron interference/ diffraction.
Because the wave function can be applied to simple Euler form e^i(kx - ωt) it can also be applied therefore to
any function using its Fourier Series ∑ₙ {Cₙe^i(kₙx - ωₙt)} or transform.
There are no unhealthy habits here, on the contrary its a good way to model real phenomena.
I mean you use calculus, I don't think Newton bothered too much about what is fellow mathematicians thought of his 'elements', it worked so was good enough for him.
@@amritlohia8240 but 1^(1/2) = 1, -1 (depending on which branch you select the root to be) so your argument is mute. Because x^2 =1, x = 1, -1 does not mean 1 = -1.
@@tomctutor The point was precisely to show that you have to be careful about which branch of the complex logarithm (or equivalently, the power function) you're dealing with, and so you're actually just making my point for me.
Man these A level students put me to shame. Even though I have a PhD in math I never could understand the physicist mindset, seems they either have unusual intuition about the world or make stuff up as they go.
Oh, we definitely make stuff up as we go 😂
And then argue for decades about it 😂