Question

(a) Now change your program, so that block 1 has 375 oscillators, and block 2 has 125 oscillators. What (approximately) is the maximum value of ln(omega cap1omega cap2) for this system? max ln(omega cap1omega cap2)= How many quanta of energy are in block 1 when ln(omega cap1omega cap2) is maximum? Number of quanta of energy in block 1 = The graphs of ln(omega cap1) and ln(omega cap2) cross. What is the significance of this crossing point? The crossing point occurs at the value of q1 which produces a minimum of ln(omega cap1omega cap2) The crossing point indicates the most probable energy distribution. The crossing point has no physical significance. At the value of q1 for which ln(omega cap1omega cap2) is a maximum, look at the slopes of the other two curves (ln(omega cap1) vs. q1 and ln(omega cap2) vs. q1). Which of the following statements is true? (This will turn out later to be important.) One curve is almost flat, while the other is almost vertical. The slopes are equal in magnitude and opposite in sign. The slopes are both zero.

Answer 1

The reason this general problem is so useful in a wide range of areas of physics is in physics we love to deal with harmonic approximations of systems.

The point of solving the problem of N harmonic oscillators in this way is that they approximate (actually, correspond to) the behaviour of the particles in an ideal gas.

Here, we work out the number of states (microstates) available to the system, the entropy of the system, and derive the energy of the corresponding ideal gas as a function of temperature.

QUESTION

The energy levels of a harmonic oscillator with frequency νν are given by

En=(n+12)ℏω,n=0,1,2,…(1)

(1)En=(n+12)ℏω,n=0,1,2,…

A system of NN uncoupled and distinguishable oscillators has the total energy

E=N2ℏω+Mℏω(2)

(2)E=N2ℏω+Mℏω

where MM is a non-negative integer.

Calculate the number ΩMΩM of states for a given E.

Calculate the entropy S=kBln(ΩM)S=kBln⁡(ΩM) of the system using the Stirling formula for MM>>11 and NN>>11.

The temperature is defined as 1T=∂S∂E1T=∂S∂E. Express the total energy as a function of the temperature and discuss the function E(T)E(T).

Step-by-step explanation: