# Discrete Harmonic Oscillator matrix representation of $x$ for Quantum Simulation

(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!

## 1 Answer

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

• 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. – Keith McClary Jul 26 '20 at 3:24