A moving pi-meson decaying into 2 photons - finding their energy

[physconomics]

Homework Statement

How is it possible to find the energies of two photons from a moving pi-meson that has a given rest mass and kinetic energy. It decays in flight and the photons are emitted in paths along the direction of motion of the meson.

Relevant Equations

E' = E(sqrt{1+beta/1-beta})
TotalE{meson} = E{1} + E{2}

I've tried using gammamc^{2} = E1 + E2 but how do i find gamma?? If i try to use the kinetic energy then I just get gammamv^2 = 1gev but i don't know v? very confused

 

[PeroK]

Homework Statement:: How is it possible to find the energies of two photons from a moving pi-meson that has a given rest mass and kinetic energy. It decays in flight and the photons are emitted in paths along the direction of motion of the meson.
Homework Equations:: E' = E(sqrt{1+beta/1-beta})
TotalE{meson} = E{1} + E{2}

I've tried using gammamc^{2} = E1 + E2 but how do i find gamma?? If i try to use the kinetic energy then I just get gammamv^2 = 1gev but i don't know v? very confused

What else is conserved apart from energy?

 

[George Jones]

but how do i find gamma??

Maybe I have misunderstood something, but can't you find $\gamma$ from

a moving pi-meson that has a given rest mass and kinetic energy.

 

[physconomics]

What else is conserved apart from energy?

Well, okay, so in the meson rest frame the equations would be

Momentum: 0 = hf1 + hf2
Conservation of Energy: 135MeV = hf1 + hf2

Right? But then this clearly contradicts itself. I don't know what I'm missing

 

[PeroK]

Well, okay, so in the meson rest frame the equations would be

Momentum: 0 = hf1 + hf2
Conservation of Energy: 135MeV = hf1 + hf2

Right? But then this clearly contradicts itself. I don't know what I'm missing

Momentum is a vector.

 

[physconomics]

Momentum is a vector.

Yeah but the questions states that the two photons are emitted along the direction of motion of the meson? Wouldn't this mean they are both transmitted in the positive direction?

 

[PeroK]

Yeah but the questions states that the two photons are emitted along the direction of motion of the meson? Wouldn't this mean they are both transmitted in the positive direction?

No. It means they are not emitted at an angle to the original direction.

 

[physconomics]

No. It means they are not emitted at an angle to the original direction.

Ah okay.
So it would become:
0 = hf1cosx1 - hf2cosx2
sinx1 = sinx2
so f1cosx1 = f2cosx2?
Then E = 2hf = 135Mev so f = =1.63x10^22 and lambda = 6.14x10^-23

But then how would I get this in the lab frame when I don't know lambda?

 

[PeroK]

Ah okay.
So it would become:
0 = hf1cosx1 - hf2cosx2
sinx1 = sinx2
so f1cosx1 = f2cosx2?
Then E = 2hf = 135Mev so f = =1.63x10^22 and lambda = 6.14x10^-23

But then how would I get this in the lab frame when I don't know lambda?

There's no reason to complicate things by using the rest frame of the meson. Can you write down the conservation of energy and momentum equations in the lab frame?

 

[physconomics]

There's no reason to complicate things by using the rest frame of the meson. Can you write down the conservation of energy and momentum equations in the lab frame?

This is kind of what I'm confused about, how would I get the momentum of the meson in the lab frame?

 

[PeroK]

This is kind of what I'm confused about, how would I get the momentum of the meson in the lab frame?

Momentum is related to mass and energy. Do you know that equation?

 

[physconomics]

Momentum is related to mass and energy. Do you know that equation?

Well, yeah, but wouldn't it have to be relativistic?
so something like gamma mu = hf1/c + hf2/c
And then energy would be
E = 1135Mev = hf1+hf2
I've found gamma as 8.407 from the KE but how about v?
Sorry I'm finding special relativity quite difficult

 

[PeroK]

Well, yeah, but wouldn't it have to be relativistic?
so something like gamma mu = hf1/c + hf2/c
And then energy would be
E = 1135Mev = hf1+hf2
I've found gamma as 8.407 from the KE but how about v?
Sorry I'm finding special relativity quite difficult

The most important equation in SR, probably, is:

$E^2 = p^2 c^2 + m^2c^4$

That holds for both massive and massless particles.

One tip for SR problems is to focus on energy and momentum ($E$ and $p$). In general, don't try to use velocity and gamma unless you have to. The conservation equations in this case are:

$E_1 + E_2 = E$ (where $E$ is the energy of the meson)

$p_1 - p_2 = p$ (where $p$ is the magnitude of the momentum of the meson etc.)

Can you relate the magnitude of the momentum of a photon to its energy?