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(The paper I'm referring to in this question is "Quantum simulations of one dimensional quantum systems")

I've been trying to understand the paper above, specifically on constructing a matrix representation of the position operator, $\hat{x}$, in discrete real space (Equation (11)).

In analogy with the CV QHO, we define a discrete QHO by the Hamiltonian $$H^{\text{d}}=\frac{1}{2}((x^{\text{d}})^2+(p^{\text{d}})^2). \tag{10}$$ The Hilbert space dimension is $N$, where $N\geq 2$ is even for simplicity. $x^{\text{d}}$ is the discrete "position" operator given by the $N\times N$ diagonal matrix $$x^{\text{d}} = \sqrt{\frac{2\pi}{N}}\frac{1}{2} \begin{pmatrix} -N & 0 & \dots & 0 \\ 0 & -(N+2) & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & (N-2) \end{pmatrix}, \tag{11}$$

I'm quite a bit lost on how this matrix is derived. Since we are in the basis of real space, I expect that the matrix should be diagonal (as it is). My guess is that the basis of real space that we are in is really the basis of Hermite Polynomials: the diagonal entries are the entries that would satisfy something along the lines of:

$$ \hat{H} H_n(x) = a_{nn}H_n(x)$$

where $a_{nn}$ is the diagonal entry in the $n$th row and column, and $H_n(x)$ is the $n$th Hermite polynomial.

I'm not entirely sure if this is proper thinking, so any insight would be greatly appreciated!

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I think this is it: It seems like we can treat $X$ as the Fourier transform of $p$ to explain the factors.

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  • $\begingroup$ I think (11) is a definition (not derived). (15) appears to express approximate eigenvalues of $H^d$ as sums of eigenvalues of $x^d$ (the $\vert j \rangle$), with Hermite function coefficients. $\endgroup$ – Keith McClary Jul 26 '20 at 3:24

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