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Wavefunction Representation

The wavefunction \Psi(x, t) describes the state of a quantum system, representing the probability amplitude of finding the particle at a specific position and time.
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The statement of the theorem

Let H\mathcal{H} be a separable Hilbert space representing the state space of a quantum system. The state is described by the wavefunction Ψ(x,t)∈H\Psi(\mathbf{x}, t) \in \mathcal{H}. The time evolution of Ψ\Psi is governed by the time-dependent Schrödinger equation, which mandates that the Hamiltonian operator H^:H→H\hat{H}: \mathcal{H} \to \mathcal{H} must be Hermitian, such that:\n∂∂tΨ(x,t)=−iH^ℏΨ(x,t)\frac{\partial}{\partial t} \Psi(\mathbf{x}, t) = -i \frac{\hat{H}}{\hbar} \Psi(\mathbf{x}, t)\nFurthermore, the probability density function P(x,t)P(\mathbf{x}, t) is defined by the Born rule, requiring the normalization condition:\n∫R3∣Ψ(x,t)∣2d3x=1\int_{\mathbb{R}^3} |\Psi(\mathbf{x}, t)|^2 d^3\mathbf{x} = 1