Solving a Physics Mystery: An Explanation of Landau's Collision Problem

In summary: Prob(V < v < V + dV) = Prob(V < v < V+dV) ...In summary, the conversation revolved around a physics student's struggle to understand a problem in the "Course of Theoretical Physics" textbook. The problem involved the disintegration of identical particles and the distribution of resulting particles in direction. The student had questions about the meaning of certain angles and velocities, as well as the derivation of a specific equation. The conversation ended with a request for feedback on the student's understanding.
  • #1
MManuel Abad
40
0
Hi there.

I'm a second-year physics student. I'm from Mexico (so please, forgive my english), and I'm on winter holidays. I've been studying Mechanics with the first volume of the "Course of Theoretical Physics", by Landau and Lifgarbagez, that awesome though hard book series. My study plan is to do all exercises and to explain every mathematical detail Landau skips in his exposition. Two days ago, everything was alright, but then I found, in chapter IV "Collisions between particles", something I cannot fully comprehend.

In the first section (from the last paragraph in page 43 in the 3rd english edition), Landau considers the disintegration of many identical particles, isotropically oriented in space and wants to know the distribution of the resulting particles in direction etc.

Please, I can't get the picture of the problem. ¿Can someone explain it to me?
I think it's this:

You have this bunch of particles, all identical, randomly oriented. Each of them disintegrates in two resulting particles of masses m1 and m2, with equal momentums in the C (center of mass) frame of reference, and the direction of m1 making the same angle phi (say) with that of m2 for every original particle. As a result of this, there's a uniform distribution of the directions of the resulting particles. But then there's the problem:

So far in the text, the angle [tex]\theta[/tex]0 was the angle the velocity (seen from the C frame) of one of the particles made with the direction of the velocity V of the original particle as seen from the LAB frame of reference (see fig. 14). Now, what does [tex]\theta[/tex]0 mean?? When you consider that system of particles disintegrating in the LAB system, are they moving with velocity V? In what direction? Is V the velocity of the center of mass of the system of particles as seen from the LAB reference frame?? Then [tex]\theta[/tex]0 is the angle, say, particle 1 makes with the velocity of the center of mass? But then [tex]\theta[/tex]0 varies with the resulting particle of mass m1 you choose, because they're resulting particles from different original particles differently oriented. If this is so, then how do they measure the solid angle d[tex]\omega[/tex]? And another question: how do they obtain eq. 16.8?? I mean, I know they differentiate v2 and all that (paragraph between eq 16.7 and 16.8), but d(cos([tex]\theta[/tex]0)) is -sin([tex]\theta[/tex]0)d[tex]\theta[/tex]0! WITH A MINUS SIGN! When you substitute in eq. 16.7 and obtain eq 16.8, what happened to the sign?

Please, I need your help!
 
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  • #2


Caveat: I may also have misunderstood key points, but I gave it a shot:

MManuel Abad said:
You have this bunch of particles, all identical, randomly oriented. Each of them disintegrates in two resulting particles of masses m1 and m2, with equal momentums in the C (center of mass) frame of reference, and the direction of m1 making the same angle phi (say) with that of m2 for every original particle. As a result of this, there's a uniform distribution of the directions of the resulting particles.

I agree with your understanding of the sitation described in the book.

So far in the text, the angle [tex]\theta[/tex]0 was the angle the velocity (seen from the C frame) of one of the particles made with the direction of the velocity V of the original particle as seen from the LAB frame of reference (see fig. 14). Now, what does [tex]\theta[/tex]0 mean??

[tex]\theta[/tex]0 is now the angle of a chosen disintegration particle with respect to [tex]\mathbf{V}[/tex], the centre-of-mass velocity of the system of primary particles.

When you consider that system of particles disintegrating in the LAB system, are they moving with velocity V? In what direction? Is V the velocity of the center of mass of the system of particles as seen from the LAB reference frame??

Yes, boldface [tex]\mathbf{V}[/tex] is the velocity of the c-o-m frame of the system of primary particles and V is its abolute value.

Then [tex]\theta[/tex]0 is the angle, say, particle 1 makes with the velocity of the center of mass?

Yes

But then [tex]\theta[/tex]0 varies with the resulting particle of mass m1 you choose, because they're resulting particles from different original particles differently oriented.

Yes

If this is so, then how do they measure the solid angle d[tex]\omega[/tex]?

I assume you mean what is called [tex]do[/tex]0 in the book, the solid angle differential element for distribution of disintegration particles in the centre-of-mass frame? By symmetry, you already know they will be isotropically oriented in the c-o-m frame, so in that frame it is simply as stated in the text. In the c-o-m frame, [tex]\theta[/tex]0 is simply an angular coordinate in [tex][0,\pi)[/tex], so you can find the distribution with respect to that coordinate as in the text.


And another question: how do they obtain eq. 16.8?? I mean, I know they differentiate v2 and all that (paragraph between eq 16.7 and 16.8), but d(cos([tex]\theta[/tex]0)) is -sin([tex]\theta[/tex]0)d[tex]\theta[/tex]0! WITH A MINUS SIGN! When you substitute in eq. 16.7 and obtain eq 16.8, what happened to the sign?

They are looking for the probability distribution, which is supposed to give positive probabilities when integrated. The difference in sign just comes from the face that as [tex]\theta_0[/tex] increases, T decreases. Therefore an extra sign has been introduced, so that you don't have to do all integrations in T-space in the negative direction to get positive probabilities.

Please comment on this, e.g. I'd like to know if I am wrong on any point :smile:

I hope my latex expressions are not jumbled/incorrect. Seems to be quite problematic to construct a post containing several latex-expressions, since they are mixed togethereach time i change the post and click on preview. Seems to not even help using the browser reload :confused:
 
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  • #3


MManuel,

Please observe that this paragraph contains text together with formulas.
To translate that in pure mathematical notations, you need to translate properly the words "the fraction is" and "the required distribution" that appear in front of equations 16.7 and 16.8 .

To do that you will write something like

Prob(T < T' < T+dT) = Prob(T < mv²/2 < T+dT) = ...

When you finally arrive at the last expression involving the speeds, you need to extract what is called "a distribution" in the proper way. To do that remember that a distribution f(x) is defined by:

Prob(x < x' < x+dx) = f(x) dx for dx>0

The sign that you are looking for is hidden in the fact that the above definition is valid for dx>0.

If you have other questions, please be a little bit more specific. (I have a headache because of another thread here)
At first sight, this is just a question changing coordinates from L to C and assuming isotropic disintegration in the C system for each particle. It is indeed interesting that the energy distribution in L can be derived and that the energy of the disintegration can be determined by statistical observations in L.

I would be interrested to know a practical application of this.
I don't see an example of disintegration leading to two equal mass fragments.
Spontaneous fission is already much more complicated.
The "gamma rays disintegrate" by electron-positron creation, but do that by transfering momentum to an existing charged particle. Could the Landau example be extended to this pair creation process??
Anyway, as usual, Landau is an excellent reading.
Solving all the exercices is an excellent idea, but you may need to postopone some to use your time more efficiently, unless you are very gifted.
 
  • #4


Thank you so much! You both were very helpful, indeed!

Torquil:

Yes, I did the integration of the distribution in terms of T, and, as you say, I saw that when [tex]\theta[/tex]0=0 then T=Tmáx, and when [tex]\theta[/tex]0=[tex]\pi[/tex] then T=Tmín; then when you do the integration, the minus sign changes the limits of integration and voilà: you get +1, that is the whole distribution :D THANK YOU! And well, then! I wasn't that far from the right explanation of the situation! :D Yeah, what you explained to me makes so much sense!

And lalbatros:

Thank you so much too! You're explanation was something didn't occur to me, and a very clever one! And yeah, solving all the problems is indeed very good, but I think I'll take your advice and skip/postpone some of them: I've been "losing" so much time in them! D:

As you say, there might be practical applications of this, but I think it's too simplified, so it's more like "introductory". Not to mention that this is Classical Mechanics, and disintegration is a quantic process. As you say, particle disintegration is actually much more complicated. I don't know if Landau explains those phenomena deeper, but I believe he does, in his 3th volume: Quantum Mechanics. And about that electron-positron creation, I'm pretty sure Landau and Lifgarbagez have something to say in their 4th volume of these series: Quantum Electrodynamics.

Thank you so much again :)
 

1. What is Landau's Collision Problem?

Landau's Collision Problem is a physics mystery that was first proposed by Lev Landau in 1936. It involves two particles approaching each other with a certain initial velocity and colliding in a head-on collision. The mystery lies in the fact that, according to classical mechanics, the particles should completely stop after the collision. However, experiments have shown that the particles actually scatter at large angles after the collision.

2. Why is Landau's Collision Problem important?

Landau's Collision Problem is important because it challenges the traditional understanding of collisions in classical mechanics. It shows that there are phenomena that cannot be explained by classical mechanics and requires a deeper understanding of the underlying principles of physics.

3. How was Landau's Collision Problem solved?

Landau's Collision Problem was solved by physicist Igor Tamm in 1950. He proposed that the scattering of particles at large angles was due to the presence of a third particle, known as a "virtual particle", which temporarily exists during the collision and allows for the transfer of energy and momentum between the colliding particles. This solution was later confirmed by experiments and is now known as the "Landau-Tamm mechanism".

4. What are the implications of the Landau-Tamm mechanism?

The Landau-Tamm mechanism has important implications in quantum mechanics and the study of subatomic particles. It shows that the behavior of particles at the quantum level cannot be fully explained by classical mechanics and requires the use of quantum theory. This mechanism also provides a deeper understanding of the processes that occur during particle collisions and has applications in fields such as nuclear physics and astrophysics.

5. Are there any other unresolved physics mysteries similar to Landau's Collision Problem?

Yes, there are many other unresolved physics mysteries that continue to challenge our understanding of the universe. These include the nature of dark matter and dark energy, the origin of the universe, and the behavior of matter at extreme conditions such as in black holes. These mysteries serve as a driving force for scientific research and discovery, pushing us to constantly expand our understanding of the world around us.

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