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Heisenberg's Matrix Mechanics

This formulation of quantum mechanics utilizes matrices to represent wavefunctions and operators, providing a mathematical framework for quantum phenomena.
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The statement of the theorem

Let H\mathcal{H} be a finite-dimensional Hilbert space, and let A^,B^∈L(H)\hat{A}, \hat{B} \in \mathcal{L}(\mathcal{H}) be Hermitian operators representing observables. The state of the system is represented by a normalized state vector ∣ψ(t)⟩∈H|\psi(t)\rangle \in \mathcal{H}. The time evolution is governed by the Hamiltonian operator H^∈L(H)\hat{H} \in \mathcal{L}(\mathcal{H}). The dynamics are defined by the matrix Schrödinger equation:\n\nddt∣ψ(t)⟩=−i1ℏH^∣ψ(t)⟩\frac{d}{dt} |\psi(t)\rangle = -i\frac{1}{\hbar} \hat{H} |\psi(t)\rangle\n\nFurthermore, the operators satisfy the canonical commutation relations, which must be preserved in the matrix representation: \n\n[X^,P^]=X^P^−P^X^=iℏI^[\hat{X}, \hat{P}] = \hat{X}\hat{P} - \hat{P}\hat{X} = i\hbar \hat{I} \n\nwhere X^\hat{X} and P^\hat{P} are the position and momentum operators, respectively, and I^\hat{I} is the identity matrix. The expectation value of any observable A^\hat{A} is given by ⟨A^⟩=⟨ψ(t)∣A^∣ψ(t)⟩\langle \hat{A} \rangle = \langle \psi(t)| \hat{A} |\psi(t)\rangle.