Proving Massless Particle's Speed: 0.5c

[im_slow]

Homework Statement

Massless Particles
A neutral pion traveling along the x-axis decays into two photons, one being ejected exactly forward and the other exactly backward. The first photon has three times the energy of the second. Prove that the original pion had speed 0.5c.

Homework Equations

for m=0, E=p*c
conservation of Energy E^2=(c*p)^2+(m*c^2)^2
gamma=1/sqrt(1-Beta^2)
Beta = v/c
p=gamma*m*v
E=gamma*m*c^2

The Attempt at a Solution

sum(momentum photons) = sum (momentum pion)

 

[kamerling]

The photon energy is easy to compute in a frame where the pion is at rest.
Use the formula for the relativistic doppler effect to find the frequency of both fotons in a frame that moves with a speed v. The frequency of the forward photon must be 3 times the
frequency of the backwards photon.

 

[kdv]

Homework Statement

Massless Particles
A neutral pion traveling along the x-axis decays into two photons, one being ejected exactly forward and the other exactly backward. The first photon has three times the energy of the second. Prove that the original pion had speed 0.5c.

Homework Equations

for m=0, E=p*c
conservation of Energy E^2=(c*p)^2+(m*c^2)^2
gamma=1/sqrt(1-Beta^2)
Beta = v/c
p=gamma*m*v
E=gamma*m*c^2

The Attempt at a Solution

sum(momentum photons) = sum (momentum pion)

Are you familiar with four-momentum conservation?


You may use [tex] P_\pi = P_1 + P_2 [/tex] where 1 and 2 refer to the photons. After squaring and using P^2 = m^2 c^4 , so that P_1^2 = P_2^2 = 0, you get

[tex] m_\pi^2 c^4 = 2 E_1 E_2 - c^2 {\vec p}_1 \cdot \vec{p_2} [/tex]

now, using the fact that the two photons move in oppposite directions, you find that [tex] 4 E_1 E_2 = m_\pi^2 c^4 [/tex]. Using the fact that one photon has three times the energy of the other one, you then have the energy of each photon in terms of the pion mass.

Then use [tex] E_1 + E_2 = \gamma m_\pi c^2 [/tex] to find the speed.

 

[bjewels]

I don't understand this. can you try explaining it in more detail?

 

[paranormal]

= 0
p1*cos(theta1) - p2*cos(theta2) = 0
p1*sin(theta1) + p2*sin(theta2) = 0

Using p1=3*p2 and theta1=0 and theta2=pi, we get p1=3*p2=3*p2*cos(theta1) and p2*sin(theta2) = 0. This means that p1=p2=0, which means that the photons have no momentum. This also means that the pion had no momentum before decaying, since momentum is conserved. Therefore, the pion had no velocity and was at rest, which means that its speed was 0c. This contradicts the statement that the pion was traveling along the x-axis, so we can conclude that the pion must have had some velocity before decaying.

Since the pion is massless, we can use the equation E=p*c to relate its energy to its momentum. Plugging in the values for the two photons, we get E1=3*E2 and p1=3*p2. Using the conservation of energy equation, we can solve for the pion's energy before decaying:

E^2=(c*p)^2+(m*c^2)^2
E^2=(c*p1)^2+(0)^2
E^2=(c*3*p2)^2
E=3*c*p2

Substituting this into the equation E=p*c, we get:

3*c*p2=p*c
3*p2=p
p2=p/3

Using the equation p=gamma*m*v and solving for v, we get:

v=p/(gamma*m)
v=p/(1/sqrt(1-Beta^2)*m)
v=p*m/sqrt(1-Beta^2)

Since the pion is massless, its mass is equal to 0, so we can simplify the equation to:

v=p/sqrt(1-Beta^2)

Substituting in the value for p2, we get:

v=p/3/sqrt(1-Beta^2)

Since the photons have no momentum, we can set p=0 and solve for the velocity of the pion:

v=0/3/sqrt(1-Beta^2)
v=0

Therefore, the pion had a velocity of 0 before decaying, which means its