Quantum Operator Execution Proof
Initializing quantum operator integration...
The Gaussian preset builds a complex wavefunction on two hundred fifty six points. Coordinates start at negative ten with spacing zero point zero seven eight one two five. The envelope is exponential of negative squared distance from the center divided by four sigma squared. Real and imaginary parts multiply cosine and sine of k times x. The source normalizes by its actual finite-grid probability sum times grid spacing. Keep center zero and k equal to three. Increase sigma from zero point eight to one point six. At the grid point x equal to zero, the normalized probability density decreases as the packet spreads. The marker is calculated from the full two hundred fifty six point sum for every parameter value. It does not substitute the infinite-domain continuum normalizer. The horizontal axis is sigma, not position. Doubling sigma gives approximately half the central density for these well-contained packets. Normalization remains the discrete sum rule. The page uses dimensionless source coordinates and its chosen unit convention; this is not a claim about a calibrated particle or guaranteed grid convergence. The graph explains the actual Gaussian preset arithmetic. Now hold sigma at one point five and k at three. Observe probability density at the fixed grid point x equal to zero. Initially the packet center is negative two, so the observation point lies away from the envelope center. The horizontal axis is the center input, not the location of a moving particle. Each point is a separately normalized source wavefunction. Move the center parameter from negative two through zero to positive two. Density at the fixed observation point rises as the Gaussian center approaches zero, then falls as it moves away. Every plotted value uses the actual finite-grid normalization sum. The grid is slightly asymmetric at its endpoints, so the calculation does not assume an exact infinite-domain symmetry. The largest density in this sweep occurs near center zero. This is parameter comparison, not time evolution or a measurement of a particle trajectory. The tool retains its position, momentum, kinetic operators and Fourier calculations; this scene focuses on how the actual center control changes density at one fixed grid point. Finally hold center zero and sigma at one point five. Inspect the actual grid point x equal to one point two five. Change k, the phase gradient parameter, while keeping the Gaussian envelope fixed. The real part is the normalized envelope times cosine of one point two five times k. Its signed value can change even when probability density does not. Increase k from zero to four. The marker follows the real-part cosine curve at that fixed grid point. The imaginary part changes with the corresponding sine. Squaring and adding the two parts cancels the phase, so the probability density stays invariant apart from floating-point rounding. This graph plots the real component, not momentum expectation or a time-dependent orbit. At k four the phase at this point is five radians. Real and imaginary components together still give the same Gaussian density. The source barrier preset uses constructed coefficients rather than solved scattering, and it is outside these scenes. The video explains the actual normalized Gaussian controls and their limits, while preserving the original interactive operators and exports.