Evolution of State

It's crucial to understand the distinction between the deterministic nature of state evolution and the probabilistic nature of measurement outcomes.
The time evolution of the state-vector ∣ψ(t)⟩  itself is deterministic. This means that if you precisely know the initial state ∣ψ(0)⟩ and the time-development operator U(T) (which is determined by the system's dynamics), you can precisely determine the state ∣ψ(t)⟩  at any later time 't'. There is no inherent randomness in how the state itself changes. 

However, this deterministic evolution of the state-vector does not equate to classical determinism in predicting experimental results. In classical mechanics, knowing the state (position and momentum) allows you to predict the outcome of any measurement with certainty (assuming perfect knowledge and measurement devices). In quantum mechanics, knowing the quantum state ∣Ψ(t)⟩ allows us to compute the probabilities of obtaining various outcomes when an experiment is performed. The act of measurement typically involves a probabilistic collapse of the state to one of the eigenstates of the measured observable. This probabilistic outcome upon measurement is a core difference between classical and quantum mechanics. 

Since the norm squared ⟨Ψ∣Ψ⟩ represents the total probability (which must be 1 for a normalized state), unitarity ensures that the total probability is conserved over time. This is essential for a consistent probabilistic interpretation.

This crucial result shows that the operator H must be Hermitian. Hermitian operators in quantum mechanics correspond to observable physical quantities. The operator H derived here is identified as the Hamiltonian of the system, representing its total energy. Thus, the generator of time translation in quantum mechanics is the Hamiltonian operator.

The evolution of a quantum state ∣Ψ(t)⟩ in a closed system is deterministic and governed by the time-development operator U(t), such that ∣Ψ(t)⟩=U(t)∣Ψ(0)⟩. The operator U(t) must be linear and unitary to preserve the structure of Hilbert space and the probabilistic interpretation (conservation of norm/probability).

Considering infinitesimal time evolution, we find that the generator of time translation is a Hermitian operator, which is identified as the system's Hamiltonian H. This leads directly to the time-dependent Schrödinger equation:

iℏ(∂/∂t)​∣Ψ(t)⟩=H∣Ψ(t)⟩

This equation describes how the state vector changes from one instant to the next. While the state vector evolves deterministically according to this equation, the outcome of a measurement performed on the system is generally probabilistic, dictated by the state vector via the Born rule. Planck's constant ℏ is a fundamental constant that sets the scale for quantum action and ensures the dimensional consistency of the Schrödinger equation.