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 wave mechanicslenth λ collides with an electron at rest in our laboratory frame, and is scattered by an angle θ from its original direction, with new wave mechanicslength 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 wave mechanicslength 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.

I just finished ~80% of my application essays. In around two hours. (Thank you, very similar essay prompts). I’ll add something about the AMC after I take it, and I still need to write the HCSSiM essay. But apart from that, I’m done. I was on quite a roll, huh?

Anyway, NACLO. It may have been a little too easy, if somewhat frustrating, but that’s all I’m saying about the contest itself. We also got free food! I ate 1.5 bagels and drank a water bottle during the exam. I didn’t have time to get dressed for PE when I got back, but whatever.

I took a two hour nap, but, as a result, didn’t practice piano. Nonetheless, I did OK at the piano lesson (even though I only practiced for ~45 minutes this week). I’ll practice some more tomorrow for Saturday’s recital. I’ll also do some more physics theory and maybe work on application problems.

Seperti biasanya, gigapedia.com menjadi sumber utama bagiku dalam mengunduh beragam e-book gratis yang berkaitan dengan mata kuliah di jurusan Fisika. Kali ini, saya mengunduh Schaum Series Fourier Analysis. Kegiatan mencuri-waktu selama 20 menit yang lalu ternyata ada gunanya. Dengan buku ini, saya berharap dapat mengerjakan seluruh tugas dari dosen mata kuliah Fungsi Khusus dan mampu memahami aplikasi deret Fourier dalam menyelesaikan masalah fisika dan teknik.

dionisius montja

schaum_fourier_analysis.pdf

Today, I received a letter in the mail for the first time being in college. Inside was a 3-page comic storyboard of a dream my friend had. I laughed out loud the whole time I was reading, so I thought I would showcase it here, where others can appreciate it too.

I hope my narration and pictures do this justice. It tells the tale about a young lad named Alan who, while at the movie theater, discovers he has Hydrogen and Oxygen bending capabilities. He is unsure whether he is “evil or just power hungry” (Image 2 above), but still thrusts his powers on a nearby city by “air/water/snow/ice” (Image 2 above) powers. However, he is in a city and decides to fly around in his collapsible laundry hamper which the, get this, “downward thrust creates levitation perpendicular to the ground. Water on ground combined with airbending allows for ideal acceleration with zero friction” (Image 3 above). Yeah, my dreams are never this sophisticated either. So he’s flying around when he sees a mask stand and decides to be a vigilante. But he has no money. He also attempts to gain a pet sidekick, but fails twice. The story ends here, but I think this is just the beginning of an awesome tale I hope to continue to hear and weave…That, or I’m just hungry for an awesome story.

There are obvious influences from Avatar: The Last Airbender as well as physics theory class. What I fail to ascertain is the fact that the character faces a moral dilemma within himself. Does he want to be evil, or does he want to be good? The end of the tale foreshadows the fact that Alan tries to be a good guy, but seems to fail every time. On the other hand, when Alan blanketed the city in water elements, he succeeds without interference. Hm. We can only see what happens next (unless we don’t).

Well, that was fun and a perfect distraction!

By the way, if you ever want to send me drawings of dreams you had, my address is:

4061 Mesa Road #301D
Irvine, CA 92617

(which no one will, but just in case! (I wish I was borderline famous, so that when I put out things like this, I would actually receive stuff and read everything and love everyone!))

Anyway, thank you so much, Alan, for the great read! It will now be placed right above my bed, so as I sleep, I will be able to have awesome dreams like you.

Di sela waktu istirahat, saya menyempatkan diri mengirimkan file hasil unduhan dari internet yang berkaitan dengan topik presentasi kuliah spektroskopi I. Ada dua puluh menit waktu yang tersedia sebelum pukul 10.00 pagi ini. Saya pikir ada cukup banyak waktu yang dapat digunakan untuk mengirim file-file hasil unduhan tersebut. Sementara itu, ada satu file lagi, yaitu e-book “Fourier Analysis”, yang saya unduh dari gigapedia.com. Saya berharap “download” buku tersebut dapat segera selesai sebelum masuk ke kelas Seminar. Saya sedang “mencuri” waktu.

dionisius montja

The Science Spot is a site that was created by an actual science teacher.  I feel like this site has everything a teacher would ever need to provide their students with information on science topics.  Mrs. Trimpe not only provides complete lesson plans for teachers, she also provides activities that can be paired nicely with them.  When you click on the lesson planning section, the lessons are broken down into different science subjects such as general science, ecology, biology, physics theory, chemistry, astronomy, earth science, and forensic science.  Under each topic there are at least ten to twenty lessons, in which she provides you all the information you need to teach the material.  I was using this site to find information under chemistry about atoms and molecules.  She has this great activity entitled “The Atoms Family” a spin of the movie “The Adams Family.”  The students find information on their element and then place their finding in the Adams Family song.  I thought this was really creative and a great way to help kids remember information through music. 

She has examples of science experiments students may want to try, along with a number of experiments teachers may want to use during lessons.  There are tons of games, puzzles, trivia questions and numerous other activities that students can use at their leisure.  There is an “Idea Factory” which is almost like a discussion board for teachers to tell about their experiences teaching science and even information on lesson plans and activities that they have used from this very site.  There is also a link to show ways to incorporate technology into your science lessons.  I found this to be very helpful and informative, especially considering how tech-savvy even some of our youngest students may be.

The Science Spot