Schrödinger Equation of Motion

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

for any three kets then we should have

This rule determines the time-evolved kets to within a single arbitrary phase-factor to be multiplied into all of them. The evolved kets cannot be multiplied by individual phase-factors because this would invalidate the superposition relation at later times.

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.,

where is a linear operator that depends only on the times and . The arbitrary phase-factor by which all time-evolved kets may be multiplied results in being undetermined to an arbitrary multiplicative constant of modulus unity.

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

(223) |

for any ket , which immediately yields

Hence, the time evolution operator is

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

should exist. Note that this limit is simply the derivative of with respect to . Let

It is easily demonstrated from Equation (224) that is anti-Hermitian: i.e.,

(227) |

The fact that can be replaced by (where is real) implies that is undetermined to an arbitrary

(228) |

When written for general , this equation becomes

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

(230) |

where is the operator representing the momentum conjugate to . Furthermore, we have just shown that if the operator evolves the system in time from to then

(231) |

Thus, the dynamical variable corresponding to the operator stands to time as the momentum stands to the coordinate . By analogy with classical physics, this suggests that is the operator representing the total

Substituting into Equation (229) yields

(232) |

Because this must hold for any initial state , we conclude that

This equation can be integrated to give

where use has been made of Equations (224) and (225). (Here, we assume that Hamiltonian operators evaluated at different times commute with one another.) The fact that is undetermined to an arbitrary real additive constant leaves undetermined to a phase-factor. Incidentally, in the above analysis, time is