EXTREMELY clear explanation. I always had vague ideas of phase velocity and group velocity, but this is the first time I've ever actually followed the derivation for e.g. the phase velocity. THANK YOU!! ❤ Super grateful!
16:09: Dr. Remillard says "the group velocity is LESS than the phase velocity", which I believe was a mistake, since it's written that the group velocity is GREATER than the phase velocity, which is true for d v_p / d omega > 0.
At 11:02, why did we take the phase of the cosine to be directly equal to zero instead of the differential of it (with respect to t) to be equal to zero like how we did for phase velocity.
It's a little bit different from what I taught myself using quadratics. That one in the middle (9:46) is a Second Kind type, or Fibonacci-like discrete homogenous sequence, even though it has that plus sign. That would make the magnitude equal to one, but that can be manipulated to r^((t-1)/2). I don't know how that changes, given the outside term. Is that a cubic? Are they triangle waves in 3d? The 'e' on the outside is vector angle addition. The magnitude is 1. That one is easier, r^t. There should be another function that pairs with this. Ψ(n+1)+Ψ(n-1)+f(n)=0 (I'm too lazy to do notation correctly). I might be wrong though, given that it is cubic. Anyway, that's wrong. You can't do the 2cos(dwt-dkx) thing. ChatGPT always simplifies that function, but it's wrong, I checked. It will work if time and displacement is an integer. It becomes a complex number when they are rational (fractional). 2cos(dwt-dkx) doesn't appear to become a complex number. Standup Maths: "Complex Fibonacci Numbers" kind of addresses it. Ψ(n+1)/Ψ(n) where n = -2 -i*2.. 2+i*2 should be a magnetic field. Three poles, I'm guessing, project it on a sphere. You might need to customize the tool you use to graph it because it's cubic.
On the Group velocity slide i read "This can exceed the speed of light". I think it's the fase velocity that can exceed the speed of light, and never the Group velocity, because it's the envelope that brings energy/information. (In the case of a particle it's also physically the probability of finding it somewhere, so in some sense the position, that moves at the Group Velocity... Sure a particle can't move faster than light, right?)
Good question. I'll try my best here. The group velocity of an electromagnetic (EM) wave can exceed the speed of light in vacuum. But energy does not travel with it. There are a few ways to visualize this. It might help to think about a shadow being cast by an object moving near the speed of light. The shadow on the ground can exceed the speed of light in vacuum, c. Things such as shadows and wave group profiles can move faster than c as long as matter and energy don't move with them. Now imagine the front of an EM wave. All waves have a beginning, and that wave front propagates at the speed of light (phase velocity) in the medium. That's the speed of energy/information. The shape, or modulation, of the wave is the result of interference between frequency components, which can have different phase velocities in dispersive media. The destructive interference nodes that define the group, just like a shadow, might be moving faster than the energy - maybe even faster than c. But they don't arrive at the destination earlier than the wave front. Each component carries spectral energy which travels at the speed of light in the medium. It isn't the energy that can travel faster than c. Rather the interference between components of the wave is what can travel faster than c.
Well explained ... But if the interference between waves travels > c , then doesn't it imply that information has travelled > c ... And that's again an impossibility @@stephenremillard1
Although the interference is the information, it is carried by the energy, which cannot exceed c. You will notice that as a wave pulse travels, the phase fronts might be moving faster than the pulse itself, but they die out at the edge of the pulse. The interference may be moving around faster than c within the pulse, but it will not get there faster than the pulse can get there.
k=2*pi/Lambda is the "wave number". Inside of a sinusoid in x, sin(kx), it's a spatial frequency. When used in certain topics, such as Fourier optics and band theory, 2*pi/Lambda might be better referred to as the "spatial angular frequency of the wave".
True. But the point being made at 11:20 is that a free particle moves at its group velocity. I prefer not to complicate that matter with a protracted aside about the relativistic dispersion equation, which is a topic in itself. So, sure, you're right, this discussion can only conclude that a nonrelativistic particle's wave function travels at the group velocity.
EXTREMELY clear explanation. I always had vague ideas of phase velocity and group velocity, but this is the first time I've ever actually followed the derivation for e.g. the phase velocity. THANK YOU!! ❤ Super grateful!
SO DAMN CLEAR EXPLANATION I CAN'T BELIEVE, THANKS HELL A LOT
So clearly explained. I get it now. 20 minutes well spent.
20 minutes and it solves my 2-day problem. Big thanks!
Very nice explained! All your vedios!
I've been trying to design, essentially, a microwave cavity resonator and this has helped me conceptualize the wave pattern so much! Thank you!
Thank you so much for your hardwork. I hv finally understood this.
16:09: Dr. Remillard says "the group velocity is LESS than the phase velocity", which I believe was a mistake, since it's written that the group velocity is GREATER than the phase velocity, which is true for d v_p / d omega > 0.
You are right. What is written is correct. Thanks for pointing that out.
Great! Lots of visual examples. Thx a lot!
At 11:02, why did we take the phase of the cosine to be directly equal to zero instead of the differential of it (with respect to t) to be equal to zero like how we did for phase velocity.
What an amazing lecture! Thanks a lot!
You're very welcome!
Fantastic explanation. Many thanks!
Great video and truly helpful, thank your!
I wish you had all your QM lecture organized in your QM playlist. But thanks for amazing lecture.
Thank you for this video! Very helpful!
Excellent
It's a little bit different from what I taught myself using quadratics. That one in the middle (9:46) is a Second Kind type, or Fibonacci-like discrete homogenous sequence, even though it has that plus sign. That would make the magnitude equal to one, but that can be manipulated to r^((t-1)/2). I don't know how that changes, given the outside term. Is that a cubic? Are they triangle waves in 3d? The 'e' on the outside is vector angle addition. The magnitude is 1. That one is easier, r^t. There should be another function that pairs with this. Ψ(n+1)+Ψ(n-1)+f(n)=0 (I'm too lazy to do notation correctly). I might be wrong though, given that it is cubic.
Anyway, that's wrong. You can't do the 2cos(dwt-dkx) thing. ChatGPT always simplifies that function, but it's wrong, I checked. It will work if time and displacement is an integer. It becomes a complex number when they are rational (fractional). 2cos(dwt-dkx) doesn't appear to become a complex number. Standup Maths: "Complex Fibonacci Numbers" kind of addresses it.
Ψ(n+1)/Ψ(n) where n = -2 -i*2.. 2+i*2 should be a magnetic field. Three poles, I'm guessing, project it on a sphere. You might need to customize the tool you use to graph it because it's cubic.
This is fantastic, thank you so much.
Amazing lecture! Very efficient, thanks a lot!
Thank you, very interesting.
Where could i find info on the "extreme normal dispersion" ?
Thank you for this!
My pleasure!
14:22 not the product rule, it's an inverse application of the quotient rule
On the Group velocity slide i read "This can exceed the speed of light". I think it's the fase velocity that can exceed the speed of light, and never the Group velocity, because it's the envelope that brings energy/information. (In the case of a particle it's also physically the probability of finding it somewhere, so in some sense the position, that moves at the Group Velocity... Sure a particle can't move faster than light, right?)
Good question. I'll try my best here. The group velocity of an electromagnetic (EM) wave can exceed the speed of light in vacuum. But energy does not travel with it. There are a few ways to visualize this. It might help to think about a shadow being cast by an object moving near the speed of light. The shadow on the ground can exceed the speed of light in vacuum, c. Things such as shadows and wave group profiles can move faster than c as long as matter and energy don't move with them. Now imagine the front of an EM wave. All waves have a beginning, and that wave front propagates at the speed of light (phase velocity) in the medium. That's the speed of energy/information. The shape, or modulation, of the wave is the result of interference between frequency components, which can have different phase velocities in dispersive media. The destructive interference nodes that define the group, just like a shadow, might be moving faster than the energy - maybe even faster than c. But they don't arrive at the destination earlier than the wave front. Each component carries spectral energy which travels at the speed of light in the medium. It isn't the energy that can travel faster than c. Rather the interference between components of the wave is what can travel faster than c.
Well explained ... But if the interference between waves travels > c , then doesn't it imply that information has travelled > c ... And that's again an impossibility @@stephenremillard1
Although the interference is the information, it is carried by the energy, which cannot exceed c. You will notice that as a wave pulse travels, the phase fronts might be moving faster than the pulse itself, but they die out at the edge of the pulse. The interference may be moving around faster than c within the pulse, but it will not get there faster than the pulse can get there.
But what is the concept of K(function of lamda)?
k=2*pi/Lambda is the "wave number". Inside of a sinusoid in x, sin(kx), it's a spatial frequency. When used in certain topics, such as Fourier optics and band theory, 2*pi/Lambda might be better referred to as the "spatial angular frequency of the wave".
Great video. But I have a question.
I wonder why you mentioned E=p**2/2mv. This is classic, not relativistic formula.
True. But the point being made at 11:20 is that a free particle moves at its group velocity. I prefer not to complicate that matter with a protracted aside about the relativistic dispersion equation, which is a topic in itself. So, sure, you're right, this discussion can only conclude that a nonrelativistic particle's wave function travels at the group velocity.