Creation and Annihilation Operators - Matrix Representation

Matrix Representation

The matrix counterparts of the creation and annihilation operators obtained from the quantum harmonic oscillator model are

$a^{\dagger}=\left(\begin{array}{cccccc} 0 & 0 & 0 & \dots & \dots\\ \sqrt{1} & 0 & 0 & \dots & \dots\\ 0 & \sqrt{2} & 0 & \dots & \dots\\ 0 & 0 & \sqrt{3} & \dots & \dots\\ \vdots & \vdots & \vdots\\ 0 & 0 & 0 & \sqrt{n} & 0\dots\\ \vdots & \vdots & \vdots & \vdots & \vdots\end{array}\right)$

$a=\begin{pmatrix} 0 & \sqrt{1} & 0 & 0 & \dots & 0 & \dots \\ 0 & 0 & \sqrt{2} & 0 & \dots & 0 & \dots \\ 0 & 0 & 0 & \sqrt{3} & \dots & 0 & \dots \\ 0 & 0 & 0 & 0 & \ddots & \vdots & \dots \\ \vdots & \vdots & \vdots & \vdots & \ddots & \sqrt{n} & \dots \\ 0 & 0 & 0 & 0 & \dots & 0 & \ddots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix}$

Substituting backwards, the laddering operators are recovered. They can be obtained via the relationships and . The wavefunctions are those of the quantum harmonic oscillator, and are sometimes called the "number basis".

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