Massachusetts Institute of Technology
Department of Physics
Physics 8.04 Thu Oct 12 12:31:02 EDT 1995
So far we have built up a quantum kinematic theory based on the
Principle of Superposition and the interpretation of the square of
quantum amplitudes in a superposition as probabilities and the first
de Broglie hypothesis (
). Upon this base we built the
mathematical framework of operators, and as verification of
our kinematic framework, we have seen that the Heisenberg uncertainty
may be proven directly from within this framework.
We then used the principle of quantum determinism and the second de
Broglie hypothesis (
) to lead us to posit the time dependent
Schrödinger equation (TDSE) to describe the dynamic evolution of
states. In its most general form, the TDSE reads,
where 
is the Hamiltonian (energy) operator.
For the rest of this course we will concern ourselves nearly exclusively with
simple position-momentum systems of a single particle. We will also
work primarily within the position representation. In these circumstances,
the Hamiltonian operator in the position representation reads 
in the three dimensions, so that the TDSE becomes
In this course we will for the most part focus on system in one dimension, in which case the TDSE is just
All of what we are about to discuss is true for both one and three
dimensional systems. To keep our results general, we will stick to
the more general form (2) for now. The student should
be able to recover the developments for the one dimensional case by
simply removing the vector ``
'' symbols and replacing 
with 
in all of the equations below. The rest of this
note is prepared with that understanding.
The purpose of this set of notes is to explore the general features of
the Schrödinger equation (2). In particular, as a
check on our hypothesized dynamics, we will verify two things. We
will first verify that
(2) is consistent with the interpretation of
as a probability by showing that under (2)
remains normalized, that probability is conserved.
We will then show that
(2) is consistent with the correspondence principle
by proving Ehrenfest's theorem, the statement that well-localized
wave packets obey Newtonian dynamics,
and 
. We will prove, in fact, that Ehrenfest's
theorem is completely general and holds not only for well-localized
wavepackets but for all quantum states.
