The Harmonic Oscillator Basis
The harmonic oscillator (HO) basis is, in some sense, the fundamental basis for studying vibrations. It gives the solution to the Schrödinger equation for the Hamiltonian
\[H = -\frac{1}{2 m} \frac{d^2}{d x^2} + m \omega^2 x^2\]where $m$ is the mass of the system and $\omega$ is the frequency of the vibration. For this, $x$ should range from $[-\infty, \infty]$. Unsurprisingly, for a vibration that’s perfectly harmonic, the HO basis gives the simplest possible representation.
For nearly harmonic vibrations, the HO basis is often still good.
Properties of the Basis
The HO wavefunctions are
\[\phi_n(x) = N_n H_n(x) e^{-(x^2/2)}\]where $N_n$ is a normalization factor and $H_n$ is the $n^{th}$ Hermite polynomial.
This basis has two important operators for which it gives simple representations.
We have the coordinate operator
\[\langle i|\hat{x}|j\rangle = \begin{cases} c_{i+1} & j=i+1 \\ c_{i} & j=i-1 \\ 0 & \text{else} \end{cases}\]and the derivative operator
\[\langle i|\frac{d}{d x}|j\rangle = \begin{cases} k_{i+1} & j=i+1 \\ -k_{i} & j=i-1 \\ 0 & \text{else} \end{cases}\]where the $c_i$ and $k_i$ are simple coefficients that only depend on the value of $i$.
It’s worth noting that these representations are simple, because for most values of $i$ and $j$, these matrix elements will be zero. Specifically, only when $i = j \pm 1$ will this be non-zero, which means our matrix representation will have zeroes everywhere except for the first upper and lower sub-diagonals.
For completeness, the coefficients are
\[\begin{align} c_i = \frac{a}{\sqrt{2}}\sqrt{i} \\ k_i = \frac{1}{a \sqrt{2}}\sqrt{i} \end{align}\]where $a$ takes care of the units and is $1$ if we’ve put our coordinate in dimensionless units.
Hamiltonian Forms
Taken together, this means our harmonic oscillator basis is good for problems where we have a Cartesian-like (i.e. non-curvilinear) coordinate and a potential that can be described by
\[V(x)=\sum_{n=0} v_n x^n\]I haven’t justified why this last property is true, recalling from the General Overview that
\[X^n_{i,j} = (X X ... X)_{i, j}\]and we already know how to get the representation of $\hat{x}$.
Therefore, given a Hamiltonian like
\[H = \frac{1}{2 m} \frac{d^2}{d x^2} + \sum_{n=0} v_n x^n\]we know how to get its matrix representation.
Sample Applications
On our McCode Academy Exercises Page, you will find an exercise to help you implement this Basis Set representation. Check it out
Got questions? Ask them on the McCoy Group Stack Overflow