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Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental to the theory, representing the possible values and corresponding states of an operator when applied to a specific wavefunction.
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The statement of the theorem

Let H\mathcal{H} be a separable Hilbert space, and let H^:H→H\hat{H}: \mathcal{H} \to \mathcal{H} be a self-adjoint, bounded linear operator (the Hamiltonian). The eigenvalue problem is defined by the equation: H^∣ψ⟩=E∣ψ⟩ \hat{H} |\psi\rangle = E |\psi\rangle where ∣ψ⟩∈H|\psi\rangle \in \mathcal{H} is the eigenvector (or eigenstate), and E∈RE \in \mathbb{R} is the corresponding eigenvalue. The existence and properties of these solutions are guaranteed by the Spectral Theorem, which states that H^\hat{H} can be represented by a spectral measure μE \mu_E such that H^=∫σ(H^)EdμE(E)\hat{H} = \int_{\sigma(\hat{H})} E d\mu_E(E), where σ(H^)\sigma(\hat{H}) is the spectrum of H^\hat{H}.
Source: Wikipedia