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# Bernoulli Effect Observation

I just noticed the Bernoulli effect in action in an everyday situation I
haven't seen described before. I was filling a cup from a slow faucet.
When the cup was well below the faucet, the water splashed and air bubbles
were forced under the surface as it filled the cup. When the cup was close
to the faucet, it flowed in smoothly.

Well, as I brought the cup upward through the transition, I notice some of
the bubbles remained. They were held directly under the stream of water
entering the cup. The stream pushed them down while buoyancy pushed them
up. Bernoulli kept them from going off to either side and returning to the
surface.

Sort of the reverse of the "ping pong ball above a stream of air" demo.

Tim

Instructor of Physics
Barton County Community College
Great Bend, KS
620-792-9320
"The only thing necessary for the triumph of evil is for good men to do
nothing." Edmund Burke

-----Original Message-----
From: John Denker [mailto:jsd@AV8N.COM]
Sent: Monday, February 23, 2004 11:20 AM
To: PHYS-L@lists.nau.edu
Subject: Re: Schrodinger equation origins

Quoting Bob Sciamanda <trebor@VELOCITY.NET>:

1)Why did S use a first order time derivative and not a 2nd order time
derivative as in the "standard" wave equation: (d/dx)^2 PHI(x,t) =
Const*(d/dt)^2 PHI(x,t)? and

2) Why is the imaginary (i) necessary?

1) The wave function PHI(x,t) is to be a complete description of the
particle's state at any time t. This means that the governing
differential
equation must be able to develop PHI(x,t) solely from a knowledge of of
PHI(x,0), where t=0 is any convenient "starting" time. This requires the
governing differential (Wave) equation to be first order in time
derivatives. A second order time derivative in the wave equation would
require a knowledge of both PHI(x,0) and (d/dt) PHI(x,0) as initial
conditions, and PHI(x,0) would not alone be a complete state description.
(In the same way, the second order N2: F=m*(d/dt)^2 x(t) requires a
knowledge of both the position x and the velocity dx/dt as initial
conditions to specify and develop a particle state - given the
environment,
F)

I find that argument unconvincing for several reasons.
(I also have doubts about the historicity of that line
of reasoning, but let's not go there but rather stick
to the technical issues.)

First of all, why do we think we are obliged to "develop PHI(x,t)
solely from a knowledge of of PHI(x,0)"?? I don't recall that
requirement being graven on any stone tablets.

In more detail: To be specific, let's consider the case of the
Schrödinger equation for the motion of an alpha particle. We
know that the massive scalar Klein-Gordon equation is a *better*
description of the physics (better than the Schrödinger equation),
and is second-order in time.

As a second, independent argument: Since the wavefunction is
complex, to develop PHI(x,t) from initial conditions involves
knowing the RealPart *and* the ImaginaryPart of PHI(x,0) which
is more that we can ever know. Classical mechanics only tells
us |PHI|^2 and QM burdens us with the Heisenberg uncertainty
principle. So the whole argument about developing PHI from
given initial conditions is a non-starter.

To pile on additional evidence, we can turn the "first order"
argument on its head: because the Schrödinger equation is first
order in time and second order in space, it *cannot* withstand
scrutiny of its short-time behavior. Zitterbewegung and all that.

In summary: It was a very bold stroke for Schrödinger to propose
a wave equation that was first order in time. We should not try
to "explain" it with Kiplingesque just-so stories; that would be
unfaithful to the physics and to the history.
http://www.boop.org/jan/justso/camel.htm

As for question (2), if you've got a parabolic differential
equation (first order in time, second order in space) you need
a factor of i or you haven't got a wave equation at all, but
rather a diffusion equation. To say the same thing, the
Schrödinger equation is in some sense second-order in time
already, since it involves two coupled differential equations
(one for the RP and one for the IP). Without the factor of i,
the two parts decouple and the resulting equation cannot
describe wavelike behavior.