Quantum phase is not directly observable and is usually determined by interferometric methods. We present a method to map complete electron wave functions, including internal quantum phase information, from measured single-state probability densities. We harness the mathematical discovery of drum-like manifolds bearing different shapes but identical resonances, and construct quantum isospectral nanostructures with matching electronic structure but divergent physical structure. Quantum measurement (scanning tunneling microscopy) of these "quantum drums"-degenerate two-dimensional electron states on the copper(111) surface confined by individually positioned carbon monoxide molecules-reveals that isospectrality provides an extra topological degree of freedom enabling robust quantum state transplantation and phase extraction.
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