This is a cross platform software that visualizes the Amplitude(the square root of PDF) and Phase of a superposition wave function for a free particle in 1 dimension x
The Schrodinger Equation is:
in a 1D case, any linear composition of complex exponentials can be a solution to the SE:
while we can specify the correct relation between the wave number and the angular frequency (with the Energy associated with each state)
The Born's rule helps us to find the coefficients; considering that the complex exponentials are orthonormal, we can arbitrarily choose the coefficients so that:
we can choose the coefficients so that the state would be equally weighted:
The Amplitude
And there are N terms with n=m so:
The other N(N-1) terms are splittable in two groups in which each term in a group is corresponded to exactly one term in the other group, so that the corresponded terms are complex conjugate of each other (As we know for any (n,m) term, there is a (m,n) term):
And the Born's rule will be satisfied, while the integral of a cosine on a period is 0
The phase
This numerical calculations are done using pure C++ and the STL (cmath)
The graphical application is wrote using Qt5 and to visualize and plot the results, QCustomPlot is used
K array comma separated
x range the calculation range
number of points in x range
time variation speed
The particle mass and a push button to set the default electron mass
result type you can choose which one to plot: sqrt(PDF), PDF, PDF with complex implementation (similar to real implementation)
To cover a possibility of wrong mathematical calculations for series and wrong programming algorithms, programming implementation with complex numbers has been done and the results was similar to the implementation with the series of cosines.
As you can see in the tests below, increasing number of harmonics, will increase the period value.
and from the last image (more resolution in x) we can see that the derivative of PDF is continuous
