First collisions at the LHC with unprecedented Energy! (Ask a particle physisicist)

[Iroscato]

That gave me a fuckin' nosebleed.

[pocaracas]

How cool! The integral of x^n * exp(x), from 0 to infinity, is n!
Thanks Alex!

[Alex K]

@poca
You mean x^n * exp(-x), surely, otherwise it's not finite.

Look up the Gamma function which is directly related to n!, the above is more or less its definition.

But I'll tell you a secret (the most useful trick in the book when working with moments of exponentials): such identities and similar ones can for example be proven by introducing an auxiliary variable in the exponent:

exp(-a x)

and realizing that n times the derivative with respect to a and then setting a=1 gives you (-1)^n x^n exp(-x).
You can therefore simply calculate the integral

int_0^infty exp(-a x)

and *then* perform the derivatives. All you have to take care of is to put the appropriate minus sign to compensate for the (-1)^n.

But it turns out that simply,

int_0^infty exp(-a x)= a^-1

Taking the derivatives of that with respect to "a" the required number of times automatically gives you the combinatorial factor n! from the rule for calculating derivatives of monomials, and the alternating minus sign which cancels the extra (-1)^n above.

[pocaracas]

[Alex K]

Table shmable. What if you're stranded on a desert island and really need to know the integral of x^n e^-x^2 to divert a sabre tooth tiger? What then, huh? Will the Russians come to the rescue? I think not!

[pocaracas]

Table shmable. What if you're stranded on a desert island and really need to know the integral of x^n e^-x^2 to divert a sabre tooth tiger? What then, huh? Will the Russians come to the rescue? I think not!

I build a power station to power my computer and, since it is known that all desert islands have satellite internet access, I'll have all the tables in the world at my disposal.

It's important knowing how to get there... but, for most practical uses, it's enough to know the result.

[Alex K]

Well, if you insist on doing it the hard way...

[Jackalope]

Here's something that kept me awake the other night.

You are no doubt familiar with some of the various experiments with quantum entanglement.  The experiments and their results are strange enough to be sure, but my question isn't really about that.

How are individual entangled (or non-entangled, for that matter) particles observed?  (Think back to the 1998 Caltech experiment where an individual photon was "teleported" about a meter.  What means is used to make the observations?

Often , a particular property of the particle is measured. For example, the polarization of a photon. The photon itself can be measured using a photo multiplier, and the polarization is decided by putting a polarizing surface or something in front of the photomultipliers. E.g. the decay of spin 0 positronium into two photons can be assumed to produce a quantum superposition of two photons with entangled opposite polarizations going off in opposite directions. One then catches them as far apart as possible with detectors  set up on opposite sides of the source. Does that answer the question a bit?

More or less, yeah, but I still wonder how an individual photon is isolated for observation like that. I mean that's kind of hard to wrap my brain around.

....wishing I had followed through on my intent to study physics beyond the two years I took in high school.

[Pyrrho]

[popeyespappy]

But since you bring it up, what are the misconceptions that people have about the Big Bang?

You mean besides for goddidit about 6000 years ago?