- The state of a quantum mechanical system is represented by a unit vector, called a state vector, in a complex separable Hilbert space.
- An observable is represented by a Hermitian linear operator in that space.
- When a system is in a state |ψ>, a measurement of an observable A produces an eigenvalue a with probability
- There is a distinguished observable H, known as the Halmiltonian, corresponding to the energy of the system. The time evolution of the state vector |ψ(t)> is given by Schrodinger's equation:
- i (h/2π) d/dt |ψ(t)> = H |ψ(t)>
In this framework, Heisenberg's uncertainty principle becomes a theorem about noncommuting operators. Furthermore, both continuous and discrete observables may be accomodated; in the former case, the Hilbert space is a space of square-integrable wavefunctions.