Question

A hypothetical particle has a mass 512eV/c^2.If such a particle at rest decays into two gamma-rays photons, what is the wavelength of each photon?

Answer 1

Mass of hypothetical particle `=512\ eV*c^-2`
Using Einstein's Mass-Energy relation, its energy `mc^2=512\ eV`

Particle decays in two `gamma`-photons, while it is at Rest.

Momentum of a photon is equal to Planck's constant divided by the wavelength of the photon

`p=h/lambda`

Using law of Conservation of momentum, we see that momentum of one gamma ray photon must be equal and opposite to the momentum of other photon.

`=>` Wavelength of both photons must be equal.

As such each photon must have energy `E_gamma=512/2=256\ eV`
Now from Planck's-Einstein equation `E_gamma=(hc)/lambda`.

`=>lambda=(hc)/E_gamma`
where Planck's constant `h=4.136 xx 10^-15 eV s` and velocity of light `c=3xx10^8\ ms^-1`

Inserting given values we get

`lambda=(4.136 xx 10^-15 xx3xx10^8)/256`
`lambda=4.846875\ nm`