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Old May 19th 2016, 07:33 AM   #1
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Question about wave function spreading and shrinking

My (limited) understanding of QM has been that, if you measure
the position of a free electron (no forces on it) with an idealized
precision (assumed zero error), then immediately after that
measurement, the (position) wave-function psi(x) is a Dirac delta
function, and the momentum wave-function phi(p) is a plane wave,
indicating that the momentum has no definite value at that instant,
but rather is in a superposition of all possible momenta, all weighted
equally, and expressed as a plane wave.

As time then progresses without any additional measurements being
made, the Schrodinger equation for the position wave function psi(x)
says (I think) that the delta function spreads out spatially as time increases.
And I've always thought that the momentum wave function phi(p) starts
to shrink as time progresses, i.e., the momentum plane wave becomes
a wave packet whose width continuously decreases.

I had always suspected that, in the limit as time goes to infinity, that the
two wave function widths approach equality (in some sense) ... i.e.,
that the two quantities become "equally uncertain" at some "medium level"
of uncertainty. That would be "nature's equilibrium" for (undisturbed) quantum
objects.

But if I haven't misunderstood Bohm (in his "Quantum Physics" book),
he seems to say that, if (as opposed to the scenario above) you initially
have the position wave function phi(x) as a plane wave, then the
momentum has a definite value, and is conserved as time progresses
(without any further measurements). I.e., he seems to be saying that
the momentum always remains at that initial definite value, and the position
always remains uniformly spread out. That seems to be inconsistent with
my understanding of the first scenario.

Was my original understanding incorrect, or am I misinterpreting Bohm?


-- Mike Fontenot
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