## PHY452H1S Basic Statistical Mechanics. Lecture 12: Helmholtz free energy. Taught by Prof. Arun Paramekanti

Posted by peeterjoot on February 28, 2013

# Disclaimer

Peeter’s lecture notes from class. May not be entirely coherent.

# Canonical partition

We found

\begin{subequations}

\end{subequations}

where the partition function \index{partition function} acts as a probability distribution so that we can define an average as

If we suppose that the energy is typically close to the average energy as in fig. 1.1.

, then we can approximate the partition function as

where we’ve used to express the number of states where the energy matches the average energy .

This gives us

or

where we define the *Helmholtz free energy* as

Equivalently, the log of the partition function provides us with the partition function

Recalling our expression for the average energy, we can now write that in terms of the free energy

# Quantum mechanical picture

Consider a subsystem as in fig. 1.2 where we have states of the form

and a total Hamiltonian operator of the form

where the total energy of the state, given energy eigenvalues and for the states and respectively, is given by the sum

Here are many body energies, so that .

We can now write the total number of states as

We’ve ignored the coupling term in eq. 1.0.10. This is actually a problem in quantum mechanics since we require this coupling to introduce state changes.

## Example: Spins

Given spin objects , , satisfying

Dropping we have

Our system has a state where . The total number of states is .

Our Hamiltonian is

This is the associated with the Zeeman effect, where states can be split by a magnetic field, as in fig. 1.3.

Our minimum and maximum energies are

\begin{subequations}

\end{subequations}

The total energy difference is

and the energy differences are

This is a measure of the average energy difference between two adjacent energy levels. In a real system we cannot assume that we have non-interacting spins. Any weak interaction will split our degenerate energy levels as in fig. 1.4.

We can now express the partition function

Our free energy is

For the expected value of the spin we find

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