The exact state of a fully isolated quantum mechanical system is described by its wave function, which contains all the information about the system's quantum state. The wave function is a complex-valued function that evolves over time according to the Schrödinger equation, which is the fundamental equation of quantum mechanics.
Mathematically, the wave function of a system can be denoted by the symbol Ψ (psi) and is typically represented as a vector in a Hilbert space, which is a mathematical space that describes the state of a quantum system. The wave function represents the probability amplitudes of finding the system in different possible states.
In general, the wave function can be a superposition of different basis states, which correspond to the possible outcomes of measurements of the system. For example, in the case of a qubit, the wave function can represent a superposition of the basis states |0⟩ and |1⟩.
However, when a measurement is performed on the system, the wave function collapses to one of the basis states with a probability determined by the squared magnitude of the probability amplitudes in the wave function. This collapse is a fundamental aspect of quantum mechanics and introduces probabilistic behavior into the measurement outcomes.
It's important to note that the exact state of a quantum system is not directly observable but is rather inferred through measurements. The act of measurement disturbs the system, causing the wave function to collapse into one of the possible outcomes.
The concept of entanglement is also relevant when considering the state of a fully isolated quantum system composed of multiple subsystems. In an entangled state, the quantum states of the subsystems are correlated in such a way that the system cannot be described independently of its parts.
Overall, the exact state of a fully isolated quantum mechanical system is described by its wave function, which evolves deterministically according to the Schrödinger equation, but the outcomes of measurements are probabilistic and cause the wave function to collapse to one of the possible states.