Question on ladder operators

Question

Suppose we have a finite , discrete set of orthonormal states $|k\rangle $

We can construct raising and lowering operators intuitively, for example $$a_+ =\sum_{k=1}^nC_{k+1}|k+1\rangle \langle k|$$

However most textbooks begin by defining ladders in terms of the linear combinations of hermitian operators.

How do we get from the above construction to showing that such ladder operators have the form $$a_\pm = \frac{\alpha\pm i\beta}{\sqrt{2}}$$ where $\alpha,\beta$ are hermitian?

Answer 1

Since the vector space is finite dimensional, say of dimension $n$, then $\hat{a}_-$ is represented by a $n\times n$ matrix $A$ with respect to the basis that OP mentions. As valdo suggests, define $\hat{a}_+$ via the Hermitian conjugate matrix $A^\dagger$. Now use that any matrix $A$ can uniquely be written as

$$ A~=~H+iK,$$

where $H$ and $K$ are Hermitian. It is not hard to see that

$$H~=~\frac{A+A^\dagger}{2}, \qquad K~=~\frac{A-A^\dagger}{2i}, \qquad A^\dagger~=~H-iK. $$