Massachusetts Institute of Technology This is to certify that Takayuki Mashima has successfully completed Quantum Mechanics: A First Course Issued: June 10, 2026
Barton Zwiebach

Professor of Physics

Massachusetts Institute of Technology

Dimitris Bertsimas

Vice Provost for Open Learning

Associate Dean, Online Education and Artificial Intelligence

Boeing Professor of Operations Research

Massachusetts Institute of Technology

Valid Certificate ID: de835987-8738-4f32-9fcc-bf2e42f97435

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