Recent content by Kaguro

Thank you! I'll begin my study here.

Given the unperturbed Hamiltonian ##H^0## and a small perturbating potential V. We have solved the original problem and have gotten a set of eigenvectors and eigenvalues of ##H^0##, and, say, two are degenerate: $$ H^0 \ket a = E^0 \ket a$$ $$ H^0 \ket b = E^0 \ket b$$ Let's make them...

Okay, I think I got it: It doesn't matter if you consider a small volume of phase as a single microstate. So ([q0,q0+dq],[p0,p0+dp]) is a small patch and belongs to a single microstate. It doesn't matter exactly how small the dq and dp are. Planck's constant doesn't appear in Classical...

Hello all. I am studying stat mech from Pathria's book. It says a system is completely described by all positions and momenta of all the N particles. This maybe represented by a single point in 6N-D gamma space. So, each point is a (micro)state. Now if we restrict the system (N,V,E to E+ΔE)...

Earlier, ##W=\frac{N!}{n1! n2! n3!...}g_1^{n1} g_2^{n2}... ## Now we have divided by N! the get: ##W=\frac{1}{n1! n2! n3!...}g_1^{n1} g_2^{n2}... ##How can I interpret this one? Doesn't this just mean, out of N particles, we have arranged n1 in g1 states with energy E1. And their order...

In classical statistics, we derived the partition function of an ideal gas. Then using the MB statistics and the definition of the partition function, we wrote: $$S = k_BlnZ_N + \beta k_B E$$, where ##Z_N## is the N-particle partition function. Here ##Z_N=Z^N## This led to the Gibb's paradox...

My book defines four-vectors as quantities with four components that transform via LTs under a change of frame. Yes! :biggrin: So the invariant quantity associated with position four vector is spacetime interval, The invariant quantity associated with velocity four vector is speed of light...

Okay! So I could also say this in this way: Since (ct,x,y,z) is a four vector so I can say that the coordinates of the clock is also a four vector and its norm should remain invariant under different frames. And this is exactly the invariance of the spacetime interval.

Okay.. Can you recommend me a good place/book which illustrates the use of 4-vectors to derive everything we have derived using LTs? Focusing heavily on applications of four vectors.

I have a related question, if someone asks me to prove length contraction using four vectors, would that be similar to what I originally asked in this thread?

Wouldn't that be just plain derivation using Lorentz Transformations ?

Thank you! I got the idea.

Deriving time dilation was easy: Imagine two events in frame O' at the same location. ##ds^2 = -c^2 dt'^2## The same viewed in O frame is: ##ds^2 = dx^2+dy^2 + dz^2 - c^2 dt^2## ##\Rightarrow dx^2+dy^2 + dz^2 - c^2 dt^2 = -c^2 dt'^2## ##\Rightarrow (\frac{dx}{dt})^2+(\frac{dy}{dt})^2+...

To all future people who want to know: I found out how can we show equivalence. We need to define mu only once, but neither in these places. We need to define mu in the fundamental thermodynamic equation: First consider Grand Canonical Ensemble, that is, allow particle number to change...

I am trying to learn statistical physics. While learning MB statistics, my textbook defined chemical potential as ##\mu = (\frac{\partial F}{\partial N})_{V,T}##. That's nice. Later when I started on Quantum statistics, my textbook described all three distribution functions via: ##n_i =...