Recent content by gravityandlev

Hi pzona and Ygggdrasil, I saw that you stumbled across my blog entry. I'm certainly glad you liked it! pzona, you're right to think that entropy can be defined in terms of quantum numbers. In a way, that's the most correct way to define entropy. If N is the number of distinct sets of...

The phrasing of the question actually seems a little wrong to me. Generally what we call the "group velocity" actually IS the velocity a particle would move at. The expression v = \hbar k/m is only true in the non-relativistic limit (take your expression above for v_g and consider the...

Sorry. I gave a quick, unhelpful reply, and it had a typo in it to boot. I meant that you could start with your relation between frequency \omega and wavevector k (which we usually call the "dispersion relation"): \frac{\omega ^2}{c^2}=k^2 + \frac{m^2 c^2}{\hbar ^2} and do an implicit...

Try using implicit differentiation on your dispersion relation: \frac{2 \omega}{c^2} d\omega = 2 k \dk \Rightarrow \frac{d\omega} {dk} \frac{\omega}{k} = c^2 d\omega/dk is the group velocity and \omega/k is the phase velocity. The product of the two is c^2 .

Thanks, Bob. I'll look it up. Come to think of it, I have heard of this concept in the context of electromagnetic fields. I guess I just didn't think to apply it to mass energy as well. I wonder why it is not commonly taught?

center of mass and "center of energy" Hi everyone, I recently came across a little thought experiment in a book that seemed to imply that the center of mass of an isolated system is not a strictly conserved quantity. Rather, it is the "center of energy" that remains fixed. The idea of a...

I wrote a post on this subject fairly recently that explains minimization of free energy slowly, and in detail, using a specific example. Maybe you'll find it useful: http://gravityandlevity.wordpress.com/2009/05/02/when-nature-plays-skee-ball-the-meaning-of-free-energy/

Oops! Here it is, sans formatting errors: The probability that the first digit is a 1 is P_1 = 0.6 . The probability the the last two digits are 00 is P_{00} = (0.4)^2 . The probability that the string either begins with a 1 or ends with 00 is given by P_1 + P_{00} - P_1 P_{00} = 0.664 ...

In my opinion, entropy is one of those things that is easy to think about on the small scale and hard to think about on the large scale. Unfortunately, it is almost always applied on large scales! Its definition is indeed made in terms of probability, but when you deal with lots of things it's...

The other posts give a pretty good start. Here's a complete solution, for posterity's sake: The probability that the first digit is a 1 is $latex P_1 = 0.6$. The probability the the last two digits are 00 is $latex P_{00} = (0.4)^2$. The probability that the string either begins with a 1 or...

You can actually view the law "nothing goes faster than the speed of light" as a consequence of the laws of electricity and magnetism: http://gravityandlevity.wordpress.com/2009/04/08/why-cant-i-go-faster-than-the-speed-of-light-hints-from-electrodynamics/ Although it's probably more correct to...