Re: Schrodinger equation origins

[Bob Sciamanda <trebor@VELOCITY.NET>]

My replies are embedded below:

----- Original Message -----
From: "John Denker" <jsd@AV8N.COM>
To: <PHYS-L@lists.nau.edu>
Sent: Monday, February 23, 2004 12:20 PM
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.

Correct me if I'm wrong, but is not H(PHI) = i*Const*d/dt(PHI) arguably at
the heart of all of QM?

Within any useful mathematical model, the specification of the state of a
system (including its environment) is here naturally taken to mean that
information which alone enables the model to develop the time evolvement of
that system.  What else might it mean?
> 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.

The Klein-Gordon equation was an attempt to "better" the Schroedinger
equation by making it Lorentz invariant (ie. relativistically correct).  To
do this one has to treat time and space coordinates equally.  The KG
equation accomplishes this by using second order derivates in both space and
time.  The result has several failings, not the least of which is a non
positive-definite probability density.

Dirac,  heeding H(PHI) = i*Const *d/dt(PHI) and its import to state
definition and development,  took the other option and forced his equation
into an ALL FIRST derivative model.  This necessitated four coupled complex
equations involving four "component" scalar wave functions. {Writable as a
single four vector equation, involving matrices.)  The result was
spectacularly successful.
> 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.

The point is not what you can or cannot know in practice - we are concerned
with the completeness and integrity of the mathematical model, in principle.
> 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.

I think you refer to relativistic limitations.
The  SE is a nonrelativistic equation.  It is, in fact the non-relativistic
limit of the Dirac equation (except spin must be added ad hoc to SE).
> 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.

Agreed - I said as much in 2)

Bob Sciamanda
Physics, Edinboro Univ of PA (Em)
http://www.velocity.net/~trebor/
trebor@velocity.net