PHYSICS 313
Problem set 2 2003
Given Sept. 22 2003
Due Sept 29 2003

Problem 1:
The probability that $N=N_T+N_H$ coin tosses with a fair coin gives rise to exactly $N_H$ heads is

$$p(N_H)=(\frac{1}{2})^H\frac{N!}{N_H!(N-N_H)!}$$

a: Calculate the probability that $20$ tosses give rise to exactly 10 heads using the exact formula above.
b: The crude version of Stirlings formula is

$$\ln n!=n\ln n -n$$

Calculate the probability of exactly 10 heads from 20 tosses using the above formula.
c: A more accurate version of Stirlings formula is

$$n!=n^ne^{-n}\sqrt{2\pi n}$$

Use the more accurate formula to calculate the probability of 10 heads from 20 tosses of a fair coin.
Problem 2:
An Einstein solid contains 1 mol of atoms. Calculate the entropy of a state with

$$[1]\; q=0.5,\;\;[2]\;q=1\;\;[3] q=3$$

vibrational quanta/oscillator (there are 3 Einstein oscillators/atom).
Problem 3:
The Einstein temperature of a solid is defined as

$$T_E=\frac{\hbar\omega}{k_B}$$

Assume that the Einstein temperature of a solid is $100$ K. Calculate the vibrational internal energy at

$$[1]\; 50 K\;\;[2]\; 100 K\;\;[3]\;\; 200 K$$