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The main aim of this article is to present the Spectral theory of self-adjoint operators on Hilbert space and to describe its applications in the development of Quantum mechanics. Since in Quantum mechanics, observables correspond to self-adjoint operators, to achieve our aim, we employed the idea of symmetric operators in Hilbert space. We see that there is a bijection between symmetric extensions of an operator and isometric extensions of its Cayley transform. The resolvent (a) is invertible, which is the generalization of the theory of eigenvalues of a matrix, that is, the set of all eigenvalues of an operator is called the spectrum of the operator, and more importantly, the Hermitian operator. Finally, we showed how to determine the energy level of the atom of an element using some derived equations.

The Algebra of Self-Adjoint Operators in Spectral Theory with Application in Quantum Mechanics