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# Disclaimer

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

# Interacting spin

For these notes

This lecture requires concepts from phy456 [1].

We’ll look at pairs of spins as a toy model of interacting spins as depicted in fig. 1.1.

Fig 1.1: Pairs of interacting spins

**Example: **

Simple atomic system, with the nucleus and the electron can interact with each other (hyper-fine interaction).

Consider two interacting spin operators each with components , ,

We rewrite the dot product term of the Hamiltonian in terms of just the squares of the spin operators

The squares , , can be thought of as “length”s of the respective angular momentum vectors.

We write

for the total angular momentum. We recall that we have

where , and implies that .

(singlet).

. Triplet: .

state.

For

energies

For

energies

state

For

energies

For

energies

These are illustrated schematically in fig. 1.2.

Fig 1.2: Energy levels for two interacting spins as a function of magnetic field

Our single pair partition function is

So for pairs our partition function is

Our free energy

Our magnetization is

The moment per particle, after cancellation, is

**Low temperatures, small ()**

The term will dominate.

Fig 1.3: magnetic moment

The specific heat has a similar behavior

Considering a single spin system, we have energies as illustrated in fig. 1.4.

Fig 1.4: Single particle spin energies as a function of magnetic field

At zero temperatures we have a finite non-zero magnetization as illustrated in fig. 1.5, but as we heat the system up, the state of the system will randomly switch between the 1, and 2 states. The partition function democratically averages over all such possible states.

Fig 1.6: Single spin magnetization

Once the system heats up, the spins are democratically populated within the entire set of possible states.

We contrast this to this interacting spins problem which has a magnetization of the form fig. 1.6.

Fig 1.6: Interacting spin magnetization

For the single particle specific heat we have specific heat of the form fig. 1.7.

Fig 1.7: Single particle specific heat

We’ll see the same kind of specific heat distribution with temperature for the interacting spins problem, but the peak will be found at an energy that’s given by the difference in energies of the two states as illustrated in fig. 1.8.

# References

[1] Peeter Joot. *Quantum Mechanics II.*, chapter: Two spin systems, angular momentum, and Clebsch-Gordon convention. URL http://sites.google.com/site/peeterjoot2/math2011/phy456.pdf.