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*From*: Bob Sciamanda <trebor@VELOCITY.NET>*Date*: Sun, 22 Feb 2004 23:13:18 -0500

A further point:

Note that the necessity of using i and a complex PHI in the Shrodinger

(free particle) wave equation,

(d/dx)^2 PHI(x,t) = i*Const* d/dt PHI(x,t),

really means that two real wave functions are needed to describe the quantum

particle, and that they obey a set of two coupled wave equations (containing

no i's):

Define PHI(x,t) = U(x,t) + iV(x,t), where U and V are real.

Sub this into the above wave equation and get the (completely real) coupled

equations:

(d/dx)^2 U(x,t) = -Const*d/dt V(x,t) and

(d/dx)^2 V(x,t) = Const*d/dt U(x,t)

Note how neatly these are satisfied by a COSINE/SINE for U/V respectively.

Bob Sciamanda

Physics, Edinboro Univ of PA (Em)

http://www.velocity.net/~trebor

trebor@velocity.net

*************************************

----- Original Message -----

Some addendum thoughts:

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

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