Principal component analysis (PCA) has been applied to analyze random fields in various scientific disciplines. However, the explainability of PCA remains elusive unless strong domain-specific knowledge is available. This paper provides a theoretical framework that builds a duality between the PCA eigenmodes of a random field and eigenstates of a Schrödinger equation. Based on the duality we propose the Schrödinger PCA algorithm to replace the expensive PCA solver with a more sample-efficient Schrödinger equation solver. We verify the validity of the theory and the effectiveness of the algorithm with numerical experiments.