Phasers on stun
Sorry, this is a bit advanced - I won't do too many like this.
In the inertial rest frame of a dust cloud of identical particles, at some time t'=0, all the dust particles spontaneously and simultaneously begin to emit light.
What physically happens in an inertial frame where the dust is moving with speed v?
Compare this with the phase velocity of the associated de Broglie wave.
In the inertial rest frame of a dust cloud of identical particles, at some time t'=0, all the dust particles spontaneously and simultaneously begin to emit light.
What physically happens in an inertial frame where the dust is moving with speed v?
Compare this with the phase velocity of the associated de Broglie wave.
Labels: funphysics
posted by Chris at
11:12 AM
18 Comments:
I don't know but thanks for asking.
As I said in my "three gods" post, I haven't a clue about advanced mathematics and physics, but I have enjoyed trawling the internet finding out about the de Broglie wave equation, Planck's constant and if light can behave as a particle, particles should be able to behave as waves.
Good luck everyone.
Agrajag(my blogger account wont work)
Hi Agrajag. The blogging seems very hit and miss. Some blogs are OK, some are near useless (for posting). That's partly why you'll notice I've deleted loads - most of them were something like, "kicking the post through".
What caused the "spontaneously and simultaneously begin to emit light"?
I think they exploded ... particles spread everywhere. Faster than I could clock them.
It was something to do with killing their own grandfathers ;~)
For those not familiar with special relativity and/or quantum mechanics, Google "Lorentz transform" and "De Broglie equation". You might then feel brave enough to tackle the problem.
Even if you don't feel up to answering, you may enjoy reading the info you find.
why should lonely dust particles start to emit light? are they in an excited state??
some funny laser-resonance?
I think they get doppler-shifted, so the wavelngth appers either blue-shifted or red-shifted, depending where v is pointing.
i don't know enough about matter-waves, but maybe that shift expresses in some gain of energy or stuff -> they gain weight. that would be cool with a mass gain by a higher momentum. on the other side the second they emit light they are accelerated in the opposite direction of the light (conservation of momentum) and they actually loose energy -> they get lighter.
my two cents ;)
Hi t:b:H. I hadn't thought of what happened to the mass of the dust particles. They will be losing mass if they are emitting energy or they will gain a small mass if they are being cunningly illuminated.
While writing this, I worked out how to actually make the dust particles really turn on as indicated. You just need to send a signal to each one from a carefully timed bank of lasers - timed so that the laser wavefronts get to each particle at the same time in the rest frame. Do it with two oppositely directed laser banks, so no net momentum is imparted. The lasers could either set off a photochemical reaction inside the dust particles or even constantly illuminate the (stationary) particles.
All. The problem doesn't require an explanation of how or why the particles light up. The fact the the "event" involves light at all is irrelevant. The maths involved is very simple - you are not being asked to derive the de Broglie equation or the de Broglie phase velocity equation using quantum mechanics.
For convenience, I assume the inertial frames are in standard configuration, and that the dust cloud's left end is at the origin in its rest frame. We need the Lorentz time transform, t' = γ*(t - xv/c^2) where γ is the Lorentz factor and ' denotes rest frame coordinates. In the rest frame, at the moment the lights turn on, t' = 0, (and x' = 0 to length of the dust cloud) so t=xv/c^2 => u = x/t = c^2/v => the front of the flash moves with speed u (which is greater than c), in the same direction as v. Further away parts (to the right) of the dust cloud illuminate later than the nearby parts.
This is precisely the de Broglie phase velocity of a particle whose (group) velocity is v. In fact De Broglie was led to the ground-breaking p=h/λ equation (which expresses the wave behaviour of particles) through considerations of compatibility of quantum mechanics with special relativity.
Some authors, dramatically, say that c^2/v is the speed of propagation of time!
Uuhhh, 42?
Hi Chris, unfortunately I was very busy last time and couldn't go here. I've just read new (for me) problems and found this one. Actually I don't think your solution and problem itself correspond each other.
Do you mean cloud is spatially local? From your solution I see it's in x'=0 in the rest frame K'. If so, and it's a light source, which emits light in the t'=0, and you want to determine time and coordinate in our frame K of the event of emission, you should use not transf. K->K', but K'->K. It means you should write
t=gamma(t'+x'v). If you do so, you'll get t=0. The same way you'll get x=0.
In the future I hope you will formulate your problems more clear.
Hi qantense, most of the books I see seem to use K' as the rest frame and K as the lab frame. So I put the cloud of dust in the K' frame. I then wanted to see how the flash developed in the lab frame, so I used the most direct equation there was. If I'd used t = γ(t'+x'v) and x = γ(x'+vt') it would have been a lot more work as x' has a range, but I'd have got the same result. I deliberately wasn't using the LT in the usual way. For you though, at x' = t' = 0 get x = 0, t = 0. At t' = 0, x' = L' get X = γ(L') and T = γ(L'v). Now I want u = (X-0)/(T-0) = γ(L')/(γ(L'v)) = 1/v. As you see, my original way is much slicker.
I had imagined the cloud as being finite (0 <= x' <= L') or semi-infinite (0 <= x'), but even that was only to make it a bit easier to discuss.
I think the question is clear (but with the understanding that ' denotes the rest frame).
I've just re-read your post. I'm not sure that you realise that the whole of the dust cloud flashes simultaneously in its rest frame. The speed of the "wavefront" is infinite there. So, the speed addition equation u=(v+u')/(1+u'v)gives the same 1/v, by letting u'->infinity..
qantense. A trivial matter: I'm used to using S and S' (rather than K and K') to label the frames. I guess S means system; what does K stand for?
Sorry everyone, but I hate loose ends - I wasn't clear in the problem that all the dust particles are stationary in the rest frame.
The idea of the light turning on was just a device to focus the mind on something concrete to examine. If I'd talked about the speed of simultaneity of the rest frame as observed in the lab frame, it would have been much harder to understand what was being asked for (to put it mildly ☺ )
Hi Chris, after all these posts I realise what you wanted to ask.
Actually I hadn't got any clear picture of problem formulation and was mislead.
I don't know what K stands for. Maybe Someone once wrote K and everybody began to write the same.
Another petty detail. Chris, note you don't write my nickname correctly.
Hi quantense. I'm sorry about the spelling. I must be going mad, I was sure I had learnt to spell it correctly ages ago, and haven't rechecked. The only name I deliberately spell incorrectly is t:b:H as he changes it at every post. Is quantense a mix of quantum and tensor? I used photino here for my first posted problem (card staircase 02/08/09).
I'm glad you got to understand the problem: I was gutted when you said it wasn't clear - it seemed very clear to me. I actually remembered the idea from a book by Wolfgang Rindler called "Essential Relativity: Special, General and Cosmological". I think it's buried in my attic somewhere. Greetz.
I forgot to say that the de Broglie wave of a particle, can be seen to be, no less than, the wave of simultaneity of the particle. That seems pretty cool to me.
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