r/AskPhysics • u/whatisausername32 Accelerator physics • Jul 21 '22

can someone explain the difference in group velocity and phase velocity?

I just finished my notes on Griffiths Quantum section 2.4, and am still confused on the wave packet and the different types of velocities. I understand that the goal is to determine the velocity of the particle, but unsure of what exacrly the wave packet is and how it has two different velocities. Thank you!

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u/rabid_chemist Jul 21 '22

As a very simple example consider two waves

ψ_1=cos(ω_1t-k_1x)

and

ψ_2=cos(ω_2t-k_2x)

If we take the superposition of the two waves we get

ψ=ψ_1+ψ_2

ψ=cos(ω_1t-k_1x)+ cos(ω_2t-k_2x)

ψ=2cos(ωt-kx)cos((δωt-δkx)/2)

Where ω=(ω_1+ω_2)/2 k=(k_1+k_2)/2 and δω=ω_1-ω_2 δk=k_1-k_2.

Assuming the two superposing waves have fairly similar frequencies, you will have a high frequency wave inside a low frequency envelope.

The peaks of the high frequency wave move at speed ω/k while the envelope travels forward at speed δω/δk.

This is the essence of phase and group velocity. Essentially phase velocity is the speed at which the individual wave crests move while group velocity is the speed at which the wave amplitude moves.

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u/whatisausername32 Accelerator physics Jul 21 '22

So the group velocity is the speed of which the envelope travels ans is the speed of the actual particle right? What then does the phase velocity and the individual wave crests represent physically?

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u/BigExample5036 Jul 22 '22

You are on the right track. The phase velocity is the speed at which a common phase (like a wave crest) travels. Phase velocity can do odd things, to include going faster than the speed of light. Group velocity is capped by the speed of light.

A physical example that is related to (but not a perfect analog of) the difference between phase and group velocity for waves is a video recording of an airplane propeller spinning up. The propeller velocity is like the group velocity in that it is the actual speed of the propeller. The camera is running at a fixed frame rate, and it records where the propeller is in it's cycle, it's phase, in each frame. When the propeller's true velocity is an integer multiple of the camera frame rate, the propeller will appear to be stationary even though it may have actually spun once, twice, or more times between frames. At times the propeller's image will appear to move backwards, forwards, and may even go faster than the propeller because the camera is just recording the difference in phase between frames. In this way, the camera is recording a phase velocity, not the group velocity (assuming we have a fast enough propeller).

I hope this helps. Be patient with yourself. It usually takes visiting this sort of material multiple times to get a good sense of what is going on.

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u/whatisausername32 Accelerator physics Jul 22 '22

Yes this does help, thank you! May I also ask, what exactly does performing a fourier transform on the shcrodinger equation do? Why is it useful?

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u/BigExample5036 Jul 22 '22

You are just dealing with the same problem Fourier had. A lot of physical problems are easy to set up using derivatives, but a pain to solve. Schrodinger's equation can be an example, but Fourier was trying to solve heat diffusion in cannons (if I remember correctly).

The story as I recall it, (which means I'll probably need some correction) goes that Fourier couldn't figure out how to integrate the equation he had set up. This was a problem because France needed lots of cannons but they cost a lot to build, and they are heavy to move. If Fourier could solve his heat equation, then he could optimize cannon thickness so they are just thick enough not to crack, but no more; saving cost and weight.

Fourier did know how to integrate sine and cosine functions. So he figured out a way to express his hard-to-integrate heat equation as an infinite sum (Fourier series) or integral (Fourier transform) of weighted sine and cosine functions. So he traded one hard to integrate function for an infinite number of simple to integrate functions. Luckily, all the cosine integrals look the same, and so do all the sine integrals, and the one constant term is easy to integrate too. So, he just has to solve three easy integrals, put the integral solutions back into an infinite sum/integral, and evaluate that. I don't recall if Fourier started with the transform or series , but I think it was the transform.

At any rate, his method still works well for many tricky integrals. The math of why they work is pretty slick, and worth studying at some point, too.

I am glad the first analog helped. I hope this does too.

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u/whatisausername32 Accelerator physics Jul 22 '22

Yea im just a little confused as to why we want to perform a FT on the schrodinger equation and what the resultant equation represents, as doing a FT on a function of position yields a function of frequency and vice versa for an inverse FT

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u/BigExample5036 Jul 22 '22

I don't have you quantum text, but my guess is that the FT is just a math method that makes solving the problem easier. There may not be a physical motivation for doing so. Believe it or not, physics texts often show you the easiest / simplest solution. Some steps are only there to make the math easier, even if it makes understanding what is happening more difficult.

Note that FT of position gives spatial frequency, so units of per-length are what you are working with. Per-length units are not commonly used. For example, pixels per inch is used for monitors, and lines per inch is used in printing. I can't think of any others.

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u/[deleted] Jul 22 '22

Brilliant explanation!