Interfering, PlaneY (Complex) Waves
In this article (a) 3 plane Y waves are summed, (b) the probability density, P(x), is computed, (c) P 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 to the right. The formula for the iíth wave is
The Basic 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 are 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, this results in a groupís center propagating to the right. (Note that P(x) for any single plane wave is stationary and constantly equal to 12=1.)
Fig. 1 depicts the case for a single plane wave. Fig. 2 depicts, at t=0, the case where 3 plane waves interfere. Fig. 3 depicts the same case a short time later. The wavelengths, etc., can be ascertained from the Basic program in Appendix A.
P(x), Single Plane Wave
3 Interfering Waves at t=0
3 Interfering Waves at t>0
The Basic program can easily be modified to include more waves, or constituent waves of different amplitudes, etc.
Private Sub cmdProbabilities_Click()
Const Size As Long = 1000 'No. of complex planes, etc.
Const pi As Double = 3.1416
Const c As Double = 300000000# 'Speed of light.
Const h As Double = 6.63E-34 'Planck constant.
Const m As Double = 9.11E-31 'Electron mass.
Const v1 As Double = 0.01 * c 'Speeds of electrons.
Const v2 As Double = 0.011 * c
Const v3 As Double = 0.012 * c
Const k1 As Double = 2 * pi * m * v1 / h 'Wave numbers.
Const k2 As Double = 2 * pi * m * v2 / h
Const k3 As Double = 2 * pi * m * v3 / h
Const w1 As Double = h * k1 ^ 2 / (4 * pi * m) 'Angular frequencies.
Const w2 As Double = h * k2 ^ 2 / (4 * pi * m)
Const w3 As Double = h * k3 ^ 2 / (4 * pi * m)
Const lambda1 As Double = 2 * pi / k1 'Wavelengths.
Const lambda2 As Double = 2 * pi / k2
Const lambda3 As Double = 2 * pi / k3
Const Rangex As Double = 10 * lambda1 'Range of x for plotting.
Const deltax As Double = Rangex / Size 'x-increment between planes.
Const tau1 As Double = 2 * pi / w1 'Longest period.
Dim index As Long
Dim plane(Size) As Double 'x-values.
Dim RE1(Size) As Double 'Complex parts of constituent waves.
Dim RE2(Size) As Double
Dim RE3(Size) As Double
Dim IM1(Size) As Double
Dim IM2(Size) As Double
Dim IM3(Size) As Double
Dim P(Size) As Double 'Probability density.
Dim theta1 As Double 'Angles around x-axis.
Dim theta2 As Double
Dim theta3 As Double
Dim REPsi As Double 'Complex parts of net psi.
Dim IMPsi As Double
Dim Psi As Double 'Magnitude of net psi.
Dim t As Double 'Time.
'Use this value to plot wave group P(x) at time t=0.
t = 0
'Use this value to plot wave group P(x) a short time later.
't = tau1
'For each complex plane...
For index = 0 To Size - 1
'...compute the x intercept;
plane(index) = (index - Size / 2) * deltax
'compute the angle around the x-axis;
'(a) at t=0;
theta1 = plane(index) * k1
theta2 = plane(index) * k2
theta3 = plane(index) * k3
'(b) or at time t>0
'theta1 = theta1 - w1 * t
'theta2 = theta2 - w2 * t
'theta3 = theta3 - w3 * t
'then compute the real and imaginary parts of each constituent wave;
RE1(index) = Cos(theta1)
'RE2(index) = 0
RE2(index) = Cos(theta2)
'RE3(index) = 0
RE3(index) = Cos(theta3)
IM1(index) = Sin(theta1)
'IM2(index) = 0
IM2(index) = Sin(theta2)
'IM3(index) = 0
IM3(index) = Sin(theta3)
'compute the real and imaginary parts of the net psi wave;
REPsi = RE1(index) + RE2(index) + RE3(index)
IMPsi = IM1(index) + IM2(index) + IM3(index)
'compute the net psi wave amplitude;
Psi = Sqr(REPsi ^ 2 + IMPsi ^ 2)
'and compute the probability density.
P(index) = Psi ^ 2
'Finally, output P(x) for plotting.
Open "c:\WINMCAD\Physics\Spirals.PRN" For Output As #1
For index = 0 To Size - 1
Write #1, plane(index); P(index)
MsgBox ("Ready for plotting")