Lorentz transform of a wavepacket

[bjnartowt]

Homework Statement

see attached .pdf. all parts of problem statement are italicized.

Homework Equations

see attached .pdf

The Attempt at a Solution

see attached .pdf


Actually: my question is pretty qualitative. You can look at everything I've done with this problem so far. However, the problem is asking for psi_PRIME, that is, the function in the new frame of reference. As you can see, I've proved the wave equation is invariant under Lorentz transform. Also, I don't think it's a mean feat to transform the x and t arguments of psi to make psi(x_PRIME, t_PRIME). However, what exactly *is* psi_PRIME, beyond the wave in the new frame of reference? I'm not sure how to "get" psi_PRIME. Do I "get" psi_PRIME when I transform its arguments? Then I think psi = psi_PRIME, because I'd be plugging transformed (primed) variables into the same ol' function...

 

[grey_earl]

$$\psi'(x', t') = A(\Lambda) \psi(x(x',t'), t(x',t')) A^{-1}(\Lambda)$$ where A(Λ) gives you the group theoretical factor which depends on the type of field you have (more exactly, its spin). If ψ is a scalar, then A = 1.

 

[bjnartowt]

Ok, thank you. So you're saying that if psi was a vector, the explicit functional form of psi would transform, and not just the variables x --> xprime and t --> tprime? Because my psi is a scalar (it's a one-dimensional wavepacket).

 

[grey_earl]

Exactly. If your ψ is a scalar, you only transform its arguments (x, t). If it would be a spinor, or a vector, or ..., you would have to multiply by some matrix A which depends on your Lorentz transformation Λ. This is very well explained in Peskin and Schroeder's book "An Introduction to QFT", if you have a library near you.

(BTW: Peskin & Schroeder is one of the most readable books on QFT I have found so far, if you plan to dig into QFT, get this one.)

 

[paranormal]

I can provide a response to your question about the Lorentz transform of a wavepacket. The Lorentz transform is a mathematical tool that allows us to relate physical quantities in one frame of reference to those in another frame of reference that is moving at a constant velocity relative to the first frame. In this case, we are interested in the Lorentz transform of a wavepacket, which is a localized disturbance in a medium that propagates through space and time.

The Lorentz transform of a wavepacket can be obtained by applying the Lorentz transform to the coordinates of each point in the wavepacket. This means that the position and time coordinates of each point in the wavepacket are transformed according to the Lorentz transform equations. This will result in a new wavepacket in the new frame of reference, with transformed coordinates and possibly different shape and size.

It is important to note that the Lorentz transform preserves the wave equation, which describes the behavior of the wavepacket in both frames of reference. This means that the wavepacket in the new frame of reference will still satisfy the wave equation, just with transformed coordinates. Therefore, the wavepacket in the new frame of reference, psi_PRIME, is essentially the same function as the original wavepacket, psi, but with transformed coordinates.

In summary, to "get" psi_PRIME, you would need to apply the Lorentz transform to the coordinates of the wavepacket, using the Lorentz transform equations. This will result in a new wavepacket in the new frame of reference, with transformed coordinates and still satisfying the wave equation. I hope this helps clarify the concept of the Lorentz transform of a wavepacket.