• Dr. Manohar B. Bhagirath
• Dr. Anilkumar Vishnupant Shelke
• Dr. Hiranand R. Khambayat

In this manuscript we depicted the Quantum mechanics is one of the most successful and intellectually profound theories in physics, providing a precise framework for understanding the behavior of matter and energy at microscopic scales. Central to its formulation is a rich mathematical structure rooted in linear algebra, functional analysis, and Hilbert space theory. This chapter presents an in-depth exploration of the mathematical foundations of quantum mechanics, emphasizing the pivotal role of complex vector spaces, linear operators, and eigenvalue problems. By examining the interplay between abstract mathematical concepts and their physical interpretations such as quantum states, observables, and wave functions we reveal how mathematical formalism enables accurate predictions and deeper insights into the nature of quantum systems. This foundational understanding not only illuminates the theoretical core of quantum mechanics but also equips readers with the essential tools to engage with more advanced quantum theories and applications.

Quantum mechanics, linear algebra, linear operators, wave functions

"Mathematical Foundations of Quantum Mechanics: From Linear Algebra to Wave function Spaces", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.12, Issue 7, page no.g267-g271, July-2025, Available :http://www.jetir.org/papers/JETIR2507639.pdf

"Mathematical Foundations of Quantum Mechanics: From Linear Algebra to Wave function Spaces", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.12, Issue 7, page no. ppg267-g271, July-2025, Available at : http://www.jetir.org/papers/JETIR2507639.pdf