Spectral Tomography of the Quantum Environment: Bayesian Reconstruction of the Hamiltonian from Discrete Energy Levels

The Heisenberg Uncertainty Principle imposes a fundamental limit on the simultaneous knowledge of kinematic variables. However, it does not preclude the complete determination of the static forces governing a quantum system. In this work, we demonstrate that the full potential landscape V (x) of a bound quantum system can be uniquely reconstructed solely from its discrete energy spectrum (Eng). Using a Global Markov Chain Monte Carlo (MCMC) inversion, we perform a Bayesian tomography of the Hamiltonian structure. We find that the potential is recoverable with high precision (< 1% uncertainty) in the physical region probed by the wavefunction, while rigorously quantifying the information horizon in classically forbidden regions. This implies that while the instantaneous state of a particle remains probabilistic, the structural laws of its environment are deterministic and fully recoverable from spectral data.