Consider a system in a state that evolves in time. At time , the state of the system is represented by the ket . The label is needed to distinguish this ket from any other ket ( , say) that is evolving in time. The label is needed to distinguish the different states of the system at different times.
The final state of the system at time is completely determined by its initial state at time plus the time interval (assuming that the system is left undisturbed during this time interval). However, the final state only determines the direction of the final state ket. Even if we adopt the convention that all state kets have unit norms, the final ket is still not completely determined, because it can be multiplied by an arbitrary phase-factor. However, we expect that if a superposition relation holds for certain states at time then the same relation should hold between the corresponding time-evolved states at time , assuming that the system is left undisturbed between times and . In other words, if
According to Equations (220) and (221), the final ket depends linearly on the initial ket . Thus, the final ket can be regarded as the result of some linear operator acting on the initial ket: i.e.,
Because we have adopted a convention in which the norm of any state ket is unity, it make sense to define the time evolution operator in such a manner that it preserves the length of any ket upon which it acts (i.e., if a ket is properly normalized at time then it will remain normalized at all subsequent times ). This is always possible, because the length of a ket possesses no physical significance. Thus, we require that
Up to now, the time evolution operator looks very much like the spatial displacement operator introduced in the previous section. However, there are some important differences between time evolution and spatial displacement. In general, we do expect the expectation value of some observable to evolve with time, even if the system is left in a state of undisturbed motion (after all, time evolution has no meaning unless something observable changes with time). The triple product can evolve either because the ket evolves and the operator stays constant, the ket stays constant and the operator evolves, or both the ket and the operator evolve. Because we are already committed to evolving state kets, according to Equation (222), let us assume that the time evolution operator can be chosen in such a manner that the operators representing the dynamical variables of the system do not evolve in time (unless they contain some specific time dependence).
We expect, from physical continuity, that if then for any ket . Thus, the limit
Equation (229) gives the general law for the time evolution of a state ket in a scheme in which the operators representing the dynamical variables remain fixed. This equation is denoted the Schrödinger equation of motion. It involves a Hermitian operator which is, presumably, a characteristic of the dynamical system under investigation.
We saw, in Section 2.8, that if the operator displaces the system along the -axis from to then
Substituting into Equation (229) yields