I was reading a SciFi book which had some particle suddenly becoming a photon and taking off at the speed of light.
So I wake up this morning and, lying in bed, think about the consequences of that event.
That suggested the question: What's the wavelength of the photon? Would the "light" be visible?
So I figure I might be able to calculate the frequency of that photon:
>You're kidding, right?
Gimme a chance, eh?
The momentum of a particle is (usually) given by:
[1] p = mv
where m is the particle mass and v its velocity.
Further, according to Einstein's Theory of Relativity:
[2] E = m_{o} c^{2} / [ 1  v^{2} / c^{2} ] ^{1/2}
where E is the Energy of the particle, c is the velocity of light and m_{o} is the particle mass when it's at rest.
If you juggle [1] and [2], you can generate:
[3] E^{2} = m_{o}c^{4} + p^{2}c^{2}
where p = m v and m = m_{o}v / [ 1  v^{2} / c^{2} ] ^{1/2}.
Okay, so we stick m_{o} = 0 in [3] and get an Energy which we can associate with a massless particle like the photon, namely:
[4] E = pc
Okay, so if there's Conservation of Momentum, when a particle with mass m and velocity v turns into a photon, the massless photon momentum would be determined as:
[5] p = mv = E/c
using [3] with m_{o} = 0 and mv is to the momentum of the particle (before it turned into a photon).
That is, the photon would have Energy determined by the particle's momentum via:
[6] E = (mv)c
Aah, but there's a relationship between the Energy of a lightphoton and its frequency, namely:
[7] E = h f
where f is the frequency of the light and h is Planck's constant: h = 6.626 x 10^{34} Jouleseconds.
Note: In [7], you multiply Jouleseconds by f (measured in 1/seconds) and you get Energy ... measured in Joules.
So where are we?
The particle that turned into a photon has a frequency associated with it, namely:
[8] f = (mv)c / h Hertz
where (as before) mv is to the momentum of the particle before it turned into a photon.
Suppose we consider an electron turning into a photon. Suppose, further, that it's travelling at 1% of the speed of light. We'd have:
m_{o} = 9.1 * 10^{31} kg and v = c/100 which would give (via [8]):
[9] f = (9.1 * 10^{31}) (c/100) / 6.626 x 10^{34} = 1.2 x 10^{16} Hz or 12,000,000 GigaHertz or 12,000 TeraHertz or 12 PitaHertz (ultraviolet radiation)
... using c = 3 x 10^{8} m/s.
The wavelength of that radiation would be in the range of molecular sizes:
The human eye can see up to 340 TeraHertz.
I guess we couldn't see it, eh?
>zzzZZZ
 
Note:
Are photons really without mass?
Since Einstein's magic equation, E = m c^{2}, generates an equivalence between Energy and mass, presumably we can associate a mass with a photon.
>zzz  huh?
A photon has Energy (according to [7]) so it has an "associated" mass, namely h f / c^{2}.
But, tho' it's sometimes called the "relativistic mass", can we weigh it on a bathroom scale?
Is it anything like ordinary, garden variety masses (like that of an electron)?
Apparently the debate
continues as to whether a photon has "real" mass.
Of course, since a photon moves with the speed of light, it don't make much sense to talk of its "rest mass".
