Why is this called a `Harmonic Oscillator Chain'?

Question

Consider the following general setup:

Assume have a chain of atoms (of mass $m=1$) in one dimension interacting with their nearest neighbor through a interaction potential $U$, and which are in an external potential $V$. Assume that the interparticle potential is such that the particles do not cross each other and so their ordering on the line is maintained. Let the position and momenta be $(q,p)$, the Hamiltonian for the system is

$$H=\sum_{i=1}^N \frac{p^2_i}{2}+V(q_i)+\sum_{i=1}^{N-1} U(q_i-q_{i+1}).$$

My question is, why when $V=0$ and $U(r)=\frac{r^2}{2}$ do people say that the atom chain is a

$$\textit{harmonic oscillator chain?}$$

Answer 1

The principal mental picture of a harmonic oscillator is two masses connected by a spring. For $m=1,$ the Hamiltonian of a single harmonic oscillator is $$H_{12}=\frac{p_1^2}2+\frac{p_2^2}2+\frac12kd_{12}^2,$$ where $k$ is a constant specifying the stiffness of the spring and $d_{12}$ is the displacement of the masses from the equilibrium position. (You should make it clear to yourself that this Hamiltonian just restates Hooke's law. Hooke's law states that the force generated by a spring is $F=-kd$ on each body and therefore there is a potential energy of $U=\frac12kd^2.$)

Now set $k=1$ and $d_{12}=q_2-q_1$ (the equilibrium position is considered to be zero separation between the masses). Then extend the above Hamiltonian to many particles by adding, for each particle $i$ in order, 1) a kinetic term $p_i^2/2$ 2) a potential term (i.e. add a spring) connecting to the $i-1$-th particle $\frac12kd_{(i-1)i}^2=\frac12(q_i-q_{i-1})^2.$ Then you arrive at $$H=\sum_{i=1}^N\frac{p_i^2}2+\sum_{i=1}^{N-1}\frac12(q_{i+1}-q_i)^2.$$ By construction, this $H$ represents a line of particles connected with springs between adjacent particles. I.e. it defines a harmonic oscillator chain.

Answer 2

• "Chain" because it is a chain of particles
• "Oscillator" because it will oscillate or have wavelike solutions.
• "Harmonic" refers to a single frequency oscillation (loosely linked to what we can find in music under the term of "harmonic"). It was first studied by Huygens who saw a link between the single frequency oscillation with time for a pendulum at small amplitude: $$m\dfrac{d^2\theta}{dt^2}\simeq \dfrac{k}{2} \theta^2\Rightarrow\theta(t)\sim\sin(wt)$$ and the acoustic wave associated to an harmonic in music which is also sinusoidal (and not a sum of sinusoidal functions).

In your case, the chain itself is not harmonic in Huygens sense (even if the potential is quadratic) because the solution will involve all normal modes: $$q_j(t) = \sum_{k=1}^Na_k\cos(\Omega_kt+c_k)$$ As you see it is a sum of sinusoidal functions. However, it has some nice spectral properties. For example the different modes (each term of the sum labeled by a $k$) don't interact between them. In any case, the term harmonic has remained commonly used when the interaction is quadratic and thus, the equation of motion linear.