19.3: Comparison with Raising Operators

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Actually these matrices are related to the raising and lowering operator for angular momentum (and simple harmonic oscillators) in quantum mechanics. For example, the \(4 \times 4\) matrices would be for a spin 3/2, with four \(S_{z}\) eigenstates.

The quantum mechanical raising and lowering matrices look like

\begin{equation}
\left(\begin{array}{llll}
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0
\end{array}\right), \quad\left(\begin{array}{llll}
0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0
\end{array}\right)
\end{equation}

They move the spin \(S_{z}\) component up (and down) by one notch, except that on applying the raising operator, the top state \(S_{z}=3 / 2\) is annihilated, similarly the lowering operator on the bottom state.

Our circular generalizations have one extra element:

\begin{equation}
\left(\begin{array}{llll}
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0
\end{array}\right), \quad\left(\begin{array}{llll}
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0
\end{array}\right)
\end{equation}

This makes the matrices circulants, and gives them a “recycling” property: the top element isn’t thrown away, it just goes to the bottom of the pile.

(And bear in mind that the standard notation for a vector has the lowest index (0 or 1) for the top element, so when we bend the ladder into a circle, the “raising” operator actually moves to the next lower number, in other words, it’s a shift to the left.)

We’ll take this shift operator P as our basic matrix:

\begin{equation}
P=\left(\begin{array}{llll}
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0
\end{array}\right), \text { from which } P^{2}=\left(\begin{array}{llll}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0
\end{array}\right)
\end{equation}

It should be evident from this that the circulant matrix having top row \(c_{0}, c_{1}, c_{2}, c_{3}\) is just the matrix \(c_{0} I+c_{1} P+c_{2} P^{2}+c_{3} P^{3}\).

This generalizes trivially to \(N \times N\) matrices.