Physics 6311: Statistical Mechanics - Homework 10

Problem 1: Radiation of Betelgeuse (12 points)

The luminosity (total amount of energy emitted per time) of the star Betelgeuse is about $10^4$ times that of the sun. (The solar luminosity is approximately $3.828\times 10^{26}$ W.) The energy density $u(𝜖)$ of Betelgeuse’s radiation has its maximum at a photon energy $𝜖\approx 0.8$ eV.

a)

Find the surface temperature of Betelgeuse, assuming it emits blackbody radiation.

b)

Estimate the radius of Betelgeuse.

c)

Why is Betelgeuse called a red giant?

Problem 2: Blackbody radiation in one dimension (14 points)

Consider photons in a one-dimensional cavity of length $L$. The Hamiltonian is $H={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i}c|p_i|$.

a)

Calculate the density of states $g(𝜖)$.

b)

Calculate the internal energy $U$ and the specific heat $C_V$ as functions of $L$ and the temperature $T$. (Hint: ${\int }_0^{\infty }\mathit{dxx}∕(e^x-1)={\pi }^2∕6$)

c)

Calculate the entropy $S$, Helmholtz free energy $A$ and the pressure $p$.

d)

Calculate and discuss the isothermal compressibility $\kappa =-{(\mathit{\partial V}∕\mathit{\partial p})}_T∕V$.

Problem 3: Phonons in a 1D chain (14 points)

Consider a one-dimensional chain of atoms (model for a linear molecule). The vibrational part of the Hamiltonian is

where $x_i$ is the displacement of atom $i$ and $m$ is the mass of one of the atoms. Assume periodic boundary conditions.

a)

Determine the normal modes by diagonalizing $H$ (Hint: Use the Fourier transformation).

b)

Calculate energy and specific heat as functions of temperature for low temperatures.