# Schrödinger Representation

###### From A. Messiah’s *Quantum Mechanics* p.722, Dover (1999)

Knowing that our quantum system is in a certain dynamical state at time we wish to determine its state at a later time . The problem, therefore, is to determine as exactly as possible the operator describing the evolution in time of the dynamical states of the system in the Schrödinger “representation” (picture). This operator is completely determined once the Hamiltonian of the system is given.

In the **Schrödinger representation**, the operator is defined by the integral equation (or Fredholm equation):

an equivalent expression of this law is the Schrödinger equation:

with the initial condition:

The solution of these equations is the *central problem of the theory*, because *they express the fundamental law of evolution of the quantum system*.

Since is hermitean, is unitary:

whence

so, it is possible to write: