Physics Friday 107

The usage of four-vectors (vectors in Minkowski space) can simplify a number of problems where special relativity must be considered. For an example, we will consider the phenomenon of Compton scattering. Note, for the rest of this I will use the (+,-,-,-) signature; namely that the scalar product of four vectors is .

The energy-momentum 4-vector, also known as the 4-momentum, or sometimes by the (IMO atrocious) portmanteau “momenergy”), is the four-vector extension of momentum, the time component corresponding to energy. Specifically, I will use the convention
(as seen here). In this form, we see
thus the (invariant) length of the four-vector is simply the rest energy of the object.

Now, in Compton scattering, a photon of wavelenth λ collides with an electron at rest in our laboratory frame, and is scattered by an angle θ from its original direction, with new wavelength λ’. We desire to derive a formula for λ’ in terms of λ and θ.
The key is to apply the conservation of energy and momentum, which the use of 4-vectors makes simple: the combined energy-momentum 4-vector must be the same both before and after the collision. We choose our spatial coordinates so that the photon is initially moving in the x direction, and scattering is in the xy plane, with θ in the usual direction in that plane.
As discussed here, the energy and momentum of a photon are given by , and , giving us an energy-momentum 4-vector for our photon pre-collision of .
And letting m be the rest mass of an electron, then our electron has pre-collision energy-momentum 4-vector .
After collision, we have new energy-momentum 4-vectors and . From our energy and momentum relations for photons, along with the scattering angle θ, we have post-collision that .

Now, our conservation of energy and momentum is
Solving for the post-collision 4-momentum of the electron, we have
Taking the norm square of both sides, we see
But since the norm squared of a four-vector is its scalar product with itself, and the scalar product is a bilinear form, the right hand side of the above is
, and so
. But we recall that the length of an energy-momentum 4-vector is the rest energy; thus , as photons have no rest mass, and
. Thus we need only find the scalar products , , and . As for the first two, since all the spatial components of are zero, and . For the last scalar product,
Plugging in these, we see
Multiplying both sides by λλ’, we get
and then dividing both sides by 2hmc3 gives
where the quantity is the Compton wavelength of the electron.

Note that our use of the 4-momentum meant not only could we combine energy and momentum conservation into a single equation, by taking the norm square as we did, we eliminated the need to consider the individual components of the post-collision 4-momentum of the electron.


Tags: , , , , ,

One Response to “Physics Friday 107”

  1. Physics Friday 108 « Twisted One 151's Weblog Says:

    […] Friday 108 By twistedone151 Last week, I illustrated an example of momentum and energy conservation under special relativity, with the […]

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: