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Linear Operators

Operators in matrix mechanics are represented by matrices that act on wavefunctions, transforming them and representing physical observables like momentum and position.
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The statement of the theorem

Let H\mathcal{H} be a complex Hilbert space, equipped with inner product ⟨⋅,⋅⟩\langle \cdot, \cdot \rangle. A linear operator A^:H→H\hat{A}: \mathcal{H} \to \mathcal{H} is defined as a mapping satisfying: 1. Linearity: A^(c1ψ1+c2ψ2)=c1A^ψ1+c2A^ψ2\hat{A}(c_1 \psi_1 + c_2 \psi_2) = c_1 \hat{A}\psi_1 + c_2 \hat{A}\psi_2 for all ψ1,ψ2∈H\psi_1, \psi_2 \in \mathcal{H} and c1,c2∈Cc_1, c_2 \in \mathbb{C}. 2. Boundedness: There exists a finite constant M=sup⁡ψ∈H,ψ≠0∣⟨A^ψ,A^ψ⟩∣⟨ψ,ψ⟩<∞M = \sup_{\psi \in \mathcal{H}, \psi \neq 0} \frac{|\langle \hat{A}\psi, \hat{A}\psi \rangle|}{\langle \psi, \psi \rangle} < \infty. Such an operator is often represented by a matrix A\mathbf{A} in a chosen basis, such that A^↔A∈L(H)\hat{A} \leftrightarrow \mathbf{A} \in \mathcal{L}(\mathcal{H}). If A^\hat{A} is self-adjoint (Hermitian), it satisfies A^=A^†\hat{A} = \hat{A}^{\dagger}, ensuring that its eigenvalues are real and that it represents a physical observable.