No, the state of a closed quantum system does not change with time. In quantum mechanics, the evolution of a closed system is described by the Schrödinger equation, which is a deterministic equation governing the time evolution of the wave function.
The Schrödinger equation states that the time derivative of the wave function is given by the Hamiltonian operator acting on the wave function. Mathematically, it can be written as:
iħ ∂ψ/∂t = Hψ
where ħ is the reduced Planck's constant, ψ is the wave function, t is time, and H is the Hamiltonian operator.
The Schrödinger equation indicates that the state of a closed quantum system evolves unitarily and smoothly over time. This means that the wave function retains its overall shape and magnitude as it progresses through time. The evolution is deterministic, meaning that if the initial state of the system is known, the state at any future time can be determined precisely.
The unitary evolution of a closed quantum system preserves various properties such as the norm of the wave function, energy eigenvalues, and symmetries. As a consequence, the probabilities associated with different measurement outcomes remain constant as the system evolves.
It is important to note that the notion of a closed system implies that there are no external influences or interactions with other systems. In reality, perfect isolation from the external environment is challenging to achieve. However, for the purpose of theoretical analysis and idealized scenarios, closed quantum systems are commonly considered.
In summary, in the context of a closed quantum system, the state does not change with time. Instead, it evolves deterministically according to the Schrödinger equation, preserving its overall characteristics throughout the evolution.