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# Re: Schrodinger equation origins

RESUBMITTED (Things are getting lost):

http://www.kw.igs.net/~jackord/bp/i4.html shows how the Schrodinger eq
emerges from
Einstein: E=hf; DeBroglie: p=h/lamda; and p^2/2m + U(x) = E.

But
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
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. 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)

2) Now SIN[k(x-ut)] is not a solution of the first (time) order wave
equation: (d/dx)^2 PHI(x,t) = Const* d/dt PHI(x,t) :
Sines and Cosines repeat only after 2 differentiations. But the exponential
combination wave of real cosine and imaginary sine
exp[ik(x-ut)] IS a solution if we choose an imaginary Const in the wave
equation. (We speak here of a free particle.)

Bob Sciamanda
Physics, Edinboro Univ of PA (Em)
http://www.velocity.net/~trebor
trebor@velocity.net
----- Original Message -----
From: "Bob Sciamanda" <trebor@VELOCITY.NET>
To: <PHYS-L@lists.nau.edu>
Sent: Friday, February 20, 2004 3:22 PM
Subject: Re: Schrodinger equation origins

Perhaps
http://www.kw.igs.net/~jackord/bp/i4.html
will help.

Bob Sciamanda
Physics, Edinboro Univ of PA (Em)
http://www.velocity.net/~trebor/
trebor@velocity.net
----- Original Message -----
From: "Larry Smith" <larry.smith@SNOW.EDU>
To: <PHYS-L@lists.nau.edu>
Sent: Thursday, February 19, 2004 7:34 PM
Subject: Schrodinger equation origins

One of my intro Modern Physics texts says regarding the genesis (origin)
of
the Schrodinger wave equation "Like the classical wave equation, the
Schrodinger equation relates the time and space derivatives of the wave
function. Schrodinger's reasoning is somewhat difficult to follow and
is
not important for our purposes. In any case, the Schrodinger equation
cannot be derived, just as Newton's laws of motion cannot be derived."

Can the list give me a sense of Schrodinger's line of reasoning in
developing his wave equation?

Thanks,
Larry