Thermodynamics and Statistical Mechanics I - Home Exercise 4

Thermodynamics and Statistical Mechanics I - Home Exercise 4
1. Classical spins
~ attached to a reservoir at temperConsider a system of N spins in a magnetic field H
ature τ . Each spin has a magnetic moment m
~ that can continuously rotate, pointing
in any direction (this is referred to as ”classical spin”).
(a) What are the possible energies of a single spin?
(b) What is the partition function of the system?
(c) Find the average magnetization M in the z direction. Express it using Langevin’s
function: L(x) = coth(x) − x1 .
(d) Expand this result for small x, and in addition find the limit of Langevin’s function
for x → ∞. Use this information to plot a qualitative graph of the magnetization
as a function of x.
(e) Show that Curie’s law - M ∝
1
T
applies at high temperatures. Find proportionality
constant.
2. General spins in magnetic field Consider a system of N fixed spins with magnetic
moments in the z direction: µz = Am, where m = −J, −J + 1, ..., J − 1, J and J is
given. A magnetic field H is applied at the z direction at temperature τ .
(a) Find the partition function. Transform to a unitless parameter x (what should
you pick as x?).
(b) Find the magnetization M in the z direction.
(c) Write the magnetization using the Brillouin function defined by
1
1
1
1
BJ (x) = 1 +
coth 1 +
x −
coth
x
2J
2J
2J
2J
(d) Expand BJ (x) for small x. Find an expression for the first order magnetization
at high temperatures and identify Curie’s law - M ∝
constant.
1
1
.
T
Find proportionality
(e) What happens when J = 1/2? How does one adjust the result to get the known
result where µz = ±1?
3. Quantum harmonic oscillator
A quantum harmonic oscillator has energy levels of the form En = ~ω(N + 12 ).
(a) Find the partition function of a single oscillator.
(b) Find the partition function of N fixed oscillators.
(c) Find the free energy.
(d) What is the average (internal) energy of the system?
(e) What is the average (internal) energy at high temperatures? Low temperatures?
What does it mean?
4. Classical vs. quantum harmonic oscillator
(a) Calculate the partition function of a classical harmonic oscillator (meaning x ∈
R3 ). The problem is 3-dimensional and the Hamiltonian of the oscillator is H =
p
~2
2m
+ mω 2~r2 .
(b) Find the internal energy.
(c) What is the connection between the partition functions of the classical and the
quantum harmonic oscillators? Hint: try a high temperature expansion.
2