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1)Why did S use a first order time derivative and not a 2nd order timedifferential
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
equation must be able to develop PHI(x,t) solely from a knowledge of ofenvironment,
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
F)