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🎭 Background🪶

The inverse Schrödinger problem is a fundamental problem in quantum mechanics that uses partial observations to determine the potential \(V(x)\) that generates a given set of eigenfunctions and eigenvalues satisfying the Schrödinger equation. Unlike the forward problem, where the potential is known and eigenstates are computed, this inverse problem is generally non-unique and highly sensitive to measurement noise.

Project 2 investigates this setting for the 1-D time-independent Schrödinger equation (TISE) of a harmonic oscillator using a physics-informed neural architecture. ✨ In particular, this architecture learns a potential parameterization \(V_\theta(x)\) whose induced Hamiltonian \(\hat{H}_\theta\) simultaneously satisfied observed, noisy spectral constraints and the TISE.

Learned variables include:

  • \(N=3\) orthonormalized eigenfunctions \(\hat{\psi}_n^\theta\) where \(n=\{0, 1, ..., N-1\}\).
  • Eigenvalues \(E_n^\theta\) are predicted for the energy eigenvalues. These are constrained to be ordered \(E_0 < E_1 < E_2\).
  • The neural network architecture \(V_\theta : \mathbb{R} \rightarrow \mathbb{R}\) approximates the unknown potential \(V_\theta(x)\).

📒 Core Terminology🪶

  • Identifiability: The extent to which information contained in the observed densities and energies constrains the underlying learned potential.
  • Perfect identifiabillity corresponds to unique recovery, while poor identifiability permits multiple distinct potentials to explain the same observations.
  • Physics Residual: Used in the loss function to ensure the learned wavefunctions and learned potential are physically consistent.
  • 1-D TISE (Time-Independent Schrödinger Equation): The physical constraint $ \hat{H}\psi_n = E_n\psi_n $ where $$ \hat{H}=-\frac{\hbar2}{2m}\frac{d2}{dx^2}+V(x)$$ is used to define the physics residual for the inverse problem.
  • POD (Proper Orthogonal Decomposition): A data-driven method (equivalent to PCA) used to find the spatial features (modes) in the learned wavefunctions along which the variance in the data varies the most.
  • Mode Mixing: A failure mode in which a learned basis state contains contributions from multiple physical eigenstates, reducing interpretability and obscuring state-to-state correspondence.

Summary Tables:🪶

Table 1: Variable Relations🪶

Variable Definition Diagnostic Use
\(\psi_n(x)\) Analytic ground truth wavefunction for the \(n\)-th eigenstate of the 1-D TISE A validation of model accuracy.
\(\psi_n^\theta(x)\) Learned wavefunction for the \(n\)-th eigenstate of the 1-D TISE Represents the model's learned approximation of the corresponding physical eigenstate.
\(\hat{\psi}_n^\theta(x)\) Orthonormalized learned wavefunction for the \(n\)-th eigenstate of the 1-D TISE Used to define the physics residual and perform POD diagnostics.
\(U_k\) (POD modes) The SVD spatial basis. Specifically the \(k\)-th POD mode, which is the \(k\)-th eigenvector of the covariance matrix of the orthonormalized learned wavefunctions. Reveals the dominant geometric directions of variance within the learned eigenstate manifold.
\(V_{nk}\) (Composition) Weights connecting the \(n\)-th true eigenstate to the \(k\)-th POD mode. Used to quantify the contribution of each POD mode to the true eigenstate. Identifies mode mixing and alignment quality

Table 2: Matrix Description of \(H_\theta\)🪶

The Schrödinger operator appears in both continuous and discretized forms throughout Project 2. The following table summarizes the relationship between the continuous Hamiltonian and its finite-difference approximation.

Variable Definition Description
\(\hat{H}_\theta\) \(-\frac{1}{2}\partial_{xx}+V_\theta(x)\) A continuous operator representing the learned Hamiltonian in the parameterized space.
\(H_\theta\) $ -\frac{1}{2}D_{xx}+\text{diag}(V_\theta(x)) $ The learned Hamiltonian matrix. Represents the discretized learned Hamiltonian operator on the spatial collocation grid.

Note: Used to compute the physics residual and assess model accuracy.
\([D_{xx}\psi^\theta_{n,i}]\) $\frac{\hat{\psi}^\theta_{n,i-1} - 2\hat{\psi}^\theta_{n,i} + \hat{\psi}^\theta_{n,i+1}}{\Delta x^2} $ The central difference matrix. Used to approximate the second derivative in the Hamiltonian.
\(\text{diag}(V_\theta(x))\) a diagonal matrix where the \(i\)-th diagonal element is the NN-predicted potential \(V_\theta(x_i)\). Summing with the first term in \(H_\theta\) gives is a discretized approximation of \(\hat{H}_\theta\).