- #1
Haorong Wu
- 417
- 90
- Homework Statement
- The problem is from Problem book in relativity and gravitation by Alan P. Lightman et al.
2.2 When a photon scatters off a charged particle which is moving with a speed very nearly that of light, the photon is said to have undergone an inverse Compoton scattering. Consider an inverse Compton scattering in which a charged particle of rest mass ##m## and total mass-energy (as seen in the lab frame) ##E\gg m##, collides head-on with a photon of freqency ##\nu (h\nu \ll m)##. What is the maximum energy the particle can transfer to the photon?
2.12 Consider the elastic collision of a particle of mass ##m_1## with a stationary particle of mass ##m_2<m_1##. Let ##\theta_{max}## be the maximum scattering angle of ##m_1##. In nonrelativistic calculations, ##\sin \theta_{max} =m_2/m_1##. Prove that this result also holds relativistically.
- Relevant Equations
- The conservation of 4-momentum
Solutions are given in the book, but I could not understand some part of them.
For problem 2.2, denote the 4-momentum of the photon by ##\mathbf P_\gamma##, that of the particle by ##\mathbf P## and the values after scattering by primes.
Then by the conservation of momentum, we have ## \mathbf P_\gamma+\mathbf P=\mathbf P_\gamma^{'}+\mathbf P^{'}##. Also, ##\mathbf P^2=-m^2##. But then the solution gives a equation that $$\mathbf P_\gamma \cdot \mathbf P_\gamma^{'}=\mathbf P \cdot (\mathbf P_\gamma-\mathbf P_\gamma^{'}) .$$
I could not figure out what laws this equation stands for. Other than this equation, I could understand the solution.
For problem 2.12, the solution first consider the C.M. frame. It reads, since the collision is elastic, ##E_1^{CM}=E_{1'}^{CM}##, i.e., $$ \mathbf P_1 \cdot \mathbf u_{CM}= \mathbf P_{1'} \cdot \mathbf u_{CM}$$ which can be evaluated in the lab frame yielding $$-E_1 +\mathbf p_1 \cdot \mathbf v_{CM}=-E_{1'}+\mathbf p_{1'} \cdot \mathbf v_{CM} $$ where ##\mathbf v_{CM}=\mathbf p_1/(E_1+m_2)##. The last equation is my problem.
In classical mechanics, I know ##\mathbf v_{CM}=(\mathbf p_1+\mathbf p_2)/(m_1+m_2)##. But I have difficulties to generalize this result to SR.
PS. I am not sure what the convention to type 4-vectors, the spatial components of them, and ordinary vectors. So the variables may look scrambled. I apologize for that.
Thank you for your time.
For problem 2.2, denote the 4-momentum of the photon by ##\mathbf P_\gamma##, that of the particle by ##\mathbf P## and the values after scattering by primes.
Then by the conservation of momentum, we have ## \mathbf P_\gamma+\mathbf P=\mathbf P_\gamma^{'}+\mathbf P^{'}##. Also, ##\mathbf P^2=-m^2##. But then the solution gives a equation that $$\mathbf P_\gamma \cdot \mathbf P_\gamma^{'}=\mathbf P \cdot (\mathbf P_\gamma-\mathbf P_\gamma^{'}) .$$
I could not figure out what laws this equation stands for. Other than this equation, I could understand the solution.
For problem 2.12, the solution first consider the C.M. frame. It reads, since the collision is elastic, ##E_1^{CM}=E_{1'}^{CM}##, i.e., $$ \mathbf P_1 \cdot \mathbf u_{CM}= \mathbf P_{1'} \cdot \mathbf u_{CM}$$ which can be evaluated in the lab frame yielding $$-E_1 +\mathbf p_1 \cdot \mathbf v_{CM}=-E_{1'}+\mathbf p_{1'} \cdot \mathbf v_{CM} $$ where ##\mathbf v_{CM}=\mathbf p_1/(E_1+m_2)##. The last equation is my problem.
In classical mechanics, I know ##\mathbf v_{CM}=(\mathbf p_1+\mathbf p_2)/(m_1+m_2)##. But I have difficulties to generalize this result to SR.
PS. I am not sure what the convention to type 4-vectors, the spatial components of them, and ordinary vectors. So the variables may look scrambled. I apologize for that.
Thank you for your time.