Photons (Homework 8, Physics 323)

3. A photon has energy 0.2 GeV with velocity in the x-direction. A second photon with energy 0.1 GeV has velocity in the y-direction. a. What’s the total energy of the photons? b. What’s the total momentum of the photons? c. Suppose these two photons annihilated to create a single particle in the final state. Find the mass of this single particle. d. Find the direction this single particle’s velocity. e. Find the single particle’s velocity factor β.   

Total energy of photons is
$E=E_x+E_y=0.2+0.1 GeV=0.3 GeV$

Total momentum is

$p=\sqrt{(p_x^2+p_y^2 )}=\sqrt{((E_x/c)^2+(E_y/c)^2 )}=$

$=1.6*10^{-19}*\sqrt{(((0.2*10^9)/(3*10^8 ))^2+(0.1/0.3)^2 )}=1.19*10^{-19}  kg*m/s$

c)

$E_(γ_1 )+E_(γ_2 )=mc^2$     or $E^2=p^2 c^2+m_0^2 c^4$  
$m_0^2=(E^2-p^2 c^2)/c^4 =((0.3*10^9*1.6*10^{-19})^2-(1.19*10^{-19})^2*(3*10^8 )^2)/(3*10^8 )^4 =$

$=1.27*10^{-55}  kg^2$

$m_0=3.565*10^{-28}  kg$

$p=γm_0 v$   or $p^2=(m_0^2 v^2)/(1-v^2/c^2 )$    or $p^2=(m_0^2)/(1/v^2 -1/c^2 )$   or $1/v^2 =1/c^2 +(m_0^2)/p^2$
$1/v^2 =1/(3*10^8 )^2 +(9.1*10^{-31} )^2/(1.19*10^{-19} )^2$ $=1.1111169588619761630142252979623e-17$

$v=2.9999921055674933215651466562172e+8$

$β=0.99999736852249777385504888540574$

$tan⁡ α=E_x/E_y =0.5$    $α=26.57 deg$