Re: Schrodinger's Equation

I think the conceptual difficulty is due to the very different
viewpoints. In classical mechanics we are representing the problem in
position space, but in quantum mechanics the representation is in
energy or momentum space. To make this clearer, consider a vibrating
string. In classical mechanics we talk of the individual atoms of the
string and how they are connected elastically to their neighbors. We
understand the motion of the string in terms of the motions of the
individual atoms. An alternative representation is as the collective
motion of the string as waves, where the motions of the individual
atoms are not explicitly represented. This wave can be described in
terms of its wavelength (i.e. momentum) and/or energy (amplitude).
Both representations are correct. (Whether each is complete is an
other question...)

The collective motion of the vibrating string makes as much sense as
the collective motion of a pendulum, even though the individual atoms
are not making the same movements at the same time, they are still
part of the same motion and are all interconnected. Anything that
alters the vibration of the string alters the collective motion
exactly analogous to a pendulum being disturbed. An just as a
pendulum can be abstracted to a point mass at the end of a massless
rod, the wave on string can be abstracted as a single pendulum.

I recall reading several years ago about a calculation for electrons
in highly excited states in an atom. If the wave functions for many
excited energy levels are combined in the appropriate way, they got a
wave function that showed a localized electron orbiting the atom. I
think this is a key point. If you study only one energy level you
only see the wave nature of the particles. If you want to see the
particle nature you need to superimpose many wave funtions
corresponding to different energy/momentum states.

Again back to the analogy with the vibrating string: A localized pulse
on the string (similar to a localized electron) can be represented by
a superposition of many vibration modes. When you do this, the pulse
will propagate back and forth along the string (assuming it has fixed
ends) just like you'd expect an electron to in a closed box. A
quantum mechanical calculation with many superimposed modes will show
the same behavior. It is the superposition of many modes (eigen
states) that gives the classical particle-like behavior.

Rich L.

.

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