The Schrödinger equation uses a wave function \(\psi:\mathbb{R}^d\to\mathbb{R}^m\) to characterize the quantum particle. The steady-state equation with constant energy \(E\) reads: $$-\frac{\boldsymbol{\Sigma}_m(\boldsymbol{x})}{2}\boldsymbol{\Delta\psi}(\boldsymbol{x})+\mathbb{V}(\boldsymbol{x})\boldsymbol{\psi}(\boldsymbol{x})=\mathbb{E}\boldsymbol{\psi}(\boldsymbol{x})$$ where \(\boldsymbol{\Delta\psi}\) is the Laplacian operator, \(\mathbb{V}(\boldsymbol{x})\) potential field and \(\boldsymbol{\Sigma}_m\in R^{d\times d}\) a positive definite matrix representing the inverse mass matrix of the particle.

• Kinetic energy of the particle: \(-\frac{\boldsymbol{\Sigma}_m(\boldsymbol{x})}{2}\boldsymbol{\Delta\psi}(\boldsymbol{x})\)
• Potential energy of the particle \(\mathbb{V}(\boldsymbol{x})\boldsymbol{\psi}(\boldsymbol{x})\)

The left hand side of the steady-state Schrödinger equation is the application of an Hermitian operator \( \hat{\mathbb{H}}\) on \(\boldsymbol{\psi}(\boldsymbol{x})\), i.e. \(\hat{\mathbb{H}}\boldsymbol{\psi}=\mathbb{E}\boldsymbol{\psi}\).

Therefore, the Hermitian operator \(\hat{\mathbb{H}}\) provides a complete set of real \(\textbf{eigenstates}\) \(\boldsymbol{\tilde\psi}_i(\boldsymbol{x})\) according to the following equations: $$\hat{\mathbb{H}}\boldsymbol{\tilde\psi}_i(\boldsymbol{x})=\mathbb{E}_i\boldsymbol{\tilde\psi}_i \int_{\mathbb{R}^d}\langle\boldsymbol{\tilde\psi}_i(\boldsymbol{x})\vert\boldsymbol{\tilde\psi}_j(\boldsymbol{x})\rangle d\boldsymbol{x}=\delta_{ij}$$ with \(\mathbb{M}=\mathbb{I}\).

Each eigenstate is associated to an \(\textbf{eigen-energy }\)\(E_i\).

The standard equation (in physics) of a one-dimensional harmonic oscillator is written as follows: $$-\frac{1}{2m}\Delta \psi(x) +\frac{1}{2}mx^2\omega^2\psi(x)=E\psi(x)$$ with \(\hat{H}(\cdot)=-\frac{1}{2m}\Delta(\cdot) +\frac{1}{2}mx^2\omega^2(\cdot)\).

First, we generate a set of potentials (given by a polynomial \(V(x) = \sum_{i=0}^{k-1} \alpha_i x^i)\) and solve the Schrödinger equation associated with the potential using numerical methods to provide the wave functions and their energies.

The potentials \(V(x)\) we discretise on a segment \((x_\text{min}, x_{max})\) of the length \(L = x_{max} - x_{min}\) into \(N\) points.

We construct a dataset, which is composed of the potentials as features and wave functions as labels. We will denote the \((n_\text{s} \times N)\) array containing \(n_\text{s}\) wave functions \(\psi^i\), \(i=0,\ldots,n_\text{s}\) by \(\boldsymbol{\psi}\), while the \((n_\text{s} \times N)\) array containing \(n_\text{s}\) potentials \(V^i\), \(i=0,\ldots,n_\text{s}\) by \(\boldsymbol{V}\). After the standard splitting of the dataset into a training and evaluation subsets, we train the neural network model such that the loss function is minimised, which we chose to be a mean square error $$\text{MSE}(\psi,\tilde \psi) = \frac{1}{n_s} \sum_{i=0}^{n_s-1} \frac{1}{N} \sum_{j=0}^{N-1}|\psi_i^j-\tilde \psi_i^j|^2,$$ between wave functions \(\psi\) associated to the potentials \(V\) and the predictions \(\tilde\psi\) that given the model:   \(\tilde \psi = \text{MLP}(V) .\)

After the training, we evaluate the thus obtained model on the validation data, which we have put away for this exact occasion.