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Interfering, Plane Y (Complex) Waves

Abstract. Three plane Y waves are summed. The probability density, P(x), is then computed. Finally The probability is plotted vs. x.

The spinning spiral model is used for each plane wave. All of the waves have a magnitude of unity. And each wave travels in the positive x direction. The formula for the iíth wave is

. (1)

The Python program that computes P(x) is provided in Appendix A. In general the objective is to illustrate, in a straightforward and non-mathematical way, how the 3 waves (a) produce P wave groups, and (b) how the groups propagate in the positive x-direction with the passage of time.

By way of review, the paradigm for the spinning spiral model is as follows. (1) A set of complex planes is conceptually constructed at regularly spaced points along the x-axis. Each plane is perpendicular to the x-axis, with (a) its Real axis parallel to the z-axis, and (b) its Imaginary axis parallel to the y-axis. (2) Looking in the positive x-direction, the tip of Y, for any simple plane wave, describes a counterclockwise spiral around the x-axis, and it spins clockwise around the x-axis with increasing time. It is assumed that, at time t=0, the 3 constituent waves are all in phase at the Origin.

Since the waves have different wavelengths, at t=0 they drift out of phase as one moves away from the Origin. This produces the Probability groups. And, since they spin at different rates, a groupís center propagates to the right (positive x direction). 

Fig. 1 depicts, at t=0, the case where 3 plane waves interfere. Fig. 2 depicts the same case a short time later. The wavelengths, etc., can be ascertained from the program in Appendix A.

Figure 1

3 Interfering Waves at t=0

 

Figure 2

3 Interfering Waves at t>0

 

The program can easily be modified to include more waves, or constituent waves of different amplitudes, etc.

Appendix A

"""This program is for the article, Interfering Psi Complex Waves."""
import math
Steps=1000
c=3e8
h=6.63e-34
m=9.11e-31
Particle1Speed=.01*c
Particle2Speed=.011*c
Particle3Speed=.012*c
Particle1WaveNumber=2*math.pi*m*Particle1Speed/h
Particle2WaveNumber=2*math.pi*m*Particle2Speed/h
Particle3WaveNumber=2*math.pi*m*Particle3Speed/h
Particle1Frequency=h*Particle1WaveNumber**2/(4*math.pi*m)
Particle2Frequency=h*Particle2WaveNumber**2/(4*math.pi*m)
Particle3Frequency=h*Particle3WaveNumber**2/(4*math.pi*m)
Particle1WaveLength=2*math.pi/Particle1WaveNumber
Particle2WaveLength=2*math.pi/Particle2WaveNumber
Particle3WaveLength=2*math.pi/Particle3WaveNumber
Rangex=10*Particle1WaveLength
deltax=Rangex/Steps
tau1=2*math.pi/Particle1Frequency
plane=[]
RealPartOfPsi1=[]
RealPartOfPsi2=[]
RealPartOfPsi3=[]
ImaginaryPartOfPsi1=[]
ImaginaryPartOfPsi2=[]
ImaginaryPartOfPsi3=[]
Probability=[]
#Choose t=0 for Probability at t=0; t=tau1 for Probability at t=tau1. 
#t=0.
t=tau1
for index in range(Steps-1):
plane.append((index-Steps/2)*deltax)
theta1=plane[index]*Particle1WaveNumber
theta2=plane[index]*Particle2WaveNumber
theta3=plane[index]*Particle3WaveNumber
theta1=theta1-Particle1Frequency*t
theta2=theta2-Particle2Frequency*t
theta3=theta3-Particle3Frequency*t
RealPartOfPsi1.append(math.cos(theta1))
RealPartOfPsi2.append(math.cos(theta2))
RealPartOfPsi3.append(math.cos(theta3))
ImaginaryPartOfPsi1.append(math.sin(theta1))
ImaginaryPartOfPsi2.append(math.sin(theta2))
ImaginaryPartOfPsi3.append(math.sin(theta3))
RealPartOfPsi=RealPartOfPsi1[index]+RealPartOfPsi2[index]+RealPartOfPsi3[index]
ImaginaryPartOfPsi=ImaginaryPartOfPsi1[index]+ImaginaryPartOfPsi2[index]+ImaginaryPartOfPsi3[index]
Psi=math.sqrt(RealPartOfPsi**2+ImaginaryPartOfPsi**2)
Probability.append(Psi**2)
f=open('c:/Python33/Website/PLOTVALS.dat','w')
f.close()
f=open('c:/Python33/Website/PLOTVALS.dat','a')
for index in range(Steps-1):
#if(index%10000==0):
f.write(str(plane[index]))
f.write(',')
f.write(str(Probability[index]))
f.write('\n')
f.close()
print('Ready for plotting')