# Abridged harmonic oscillator notes

[This is an abbreviation of more extensive PDF notes associated with the latter part of this post.]

# Motivation and summary of harmonic oscillator background

After having had some trouble on a non-1D harmonic oscillator lattice problem on the exam, I attempted such a problem with enough time available to consider it properly. I found it helpful to consider first just two masses interacting harmonically in 3D, each displaced from an equilibrium position.

The Lagrangian that described this most naturally was found to be

This was solved in absolute and displacement coordinates, and then I moved on to consider a linear expansion of the harmonic potential about the equilibrium point, a problem closer to the exam problem (albeit still considering only two masses). The equilibrium points were described with vectors as in fig. 2.1, where .

Using such coordinates, and generalizing, it was found that the complete Lagrangian, to second order about the equilibrium positions, is

Evaluating the Euler-Lagrange equations, the equations of motion for the displacements were found to be

or

Observe that on the RHS above we have a projection operator, so we could also write

We see that the equations of motion for the displacements of a system of harmonic oscillators has a rather pleasant expression in terms of projection operators, where we have projections onto the unit vectors between each pair of equilibrium position.

# A number of harmonically coupled masses

Now let’s consider masses at lattice points indexed by a lattice vector , as illustrated in fig. 2.2.

With a coupling constant of between lattice points indexed and (located at and respectively), and direction cosines for the equilibrium direction vector between those points given by

the Lagrangian is

Evaluating the Euler-Lagrange equations for the mass at index we have

and

where . Equating both, we have in vector form

or

This is an intuitively pleasing result. We have displacement and the direction of the lattice separations in the mix, but not the magnitude of the lattice separation itself.

# Two atom basis, 2D diamond lattice

As a concrete application of the previously calculated equilibrium harmonic oscillator result, let’s consider a two atom basis diamond lattice where the horizontal length is and vertical height is .

Indexing for the primitive unit cells is illustrated in fig. 2.3.

Let’s write

For mass assume a trial solution of the form

The equations of motion for the two particles are

Insertion of the trial solution gives

Regrouping, and using the matrix form for the projection operators, this is

As a single matrix equation, this is

Observe that this is an eigenvalue problem for matrix

and eigenvalues .

To be explicit lets put the and functions in explicit matrix form. The orthogonal projectors have a simple form

For the and projection operators, we can use half angle formulations

After some manipulation, and the following helper functions

the block matrices of eq. 2.0.17.17 take the form

A final bit of simplification for possible, noting that , and , so

and

It isn’t particularly illuminating to expand out the determinant for such a system, even though it can be done symbolically without too much programming. However, what is easy after formulating the matrix for this system, is actually solving it. This is done, and animated, in twoAtomBasisRectangularLatticeDispersionRelation.cdf