Vector wave function

In pilitron physics, the vector wave function is a function that explains how, at certain speeds, a particle can only move by constantly slowing down and speeding up, or, in certain directions, how it must constantly turn, because its speed or direction is impossible in quantised space-time of the pilitron theory.

Speed
Pilitron physics measured speed in Planck lengths per Planck time. Because at a given Planck time a particle either stays in place or moves by 1 Planck length, the top part of the fraction stays at 1, whereas the bottom one changes. So 1/2 would means 1 Planck length for every 2 Planck times. So therefore we say that the delay is 2. The speed of light (c) is exactly 1, as there is no delay.

To maintain a constant speed, a particle must therefore travel at a speed with constant delay, like 1/1, 1/2, 1/3, etc. but if it travels at, for example, 2/3, it would have to constantly slow down and speed up to maintain that speed.

The bottom part of the speed fraction s is the delay, and is noted D. The top part is called the magic factor, noted F. The speed s can be achieved when the particle moves for F Planck times, and stops for D-F Planck lengths. This can therefore be though of as a wave with the wavelength D.

Direction
Direction can be given as a vector across 3 dimensions, giving the ratio of speed across those 3 dimensions:


 * $$V = \left(1, \frac{1}{3}, \frac{1}{5} \right)$$

The Main Function
An object's vector wave function is measured in kappa moments (denoted ), and is related to speed, direction and active energy jumping between rings. It is defined as:


 * $$w = \frac{S\sqrt{V_x^2 + V_y^2 + V_z^2}}{E}$$

So, if one knows the vector wave function of a given object, you can calculate its energy as:


 * $$E = \frac{S\sqrt{V_x^2 + V_y^2 + V_z^2}}{w}$$

The speed can be given as:


 * $$S = \frac{Ew}{\sqrt{V_x^2 + V_y^2 + V_z^2}}$$

And the direction can also be found from:


 * $$\sqrt{V_x^2 + V_y^2 + V_z^2} = \frac{Ew}{S}$$

This means that the vector wave function of a photon with a frequency of 1 can be given as:


 * $$\Lambda \equiv \frac{c}{h} = 1\kappa$$

Which shows that:


 * $$h = c = \Lambda \,\!$$

A photon whose electromagnetic wave has the frequency f has a vector wave function of:


 * $$w = \frac{c}{hf} = \frac{1}{f} \kappa$$

When an object is moving in a perfectly straight line with a constant vector, then its vector wave function can be simplified to:


 * $$w = \frac{S}{E}$$

If the object is moving in a straight line with acceleration a, initial speed S, then over time t, its vector wave function can be given as:


 * $$w = \frac{S + at}{E}$$

If the object is moving at a constant speed, but with a direction changing by angle  across time t, then its vector wave function can be given as:


 * $$w = \frac{S\sqrt{\left(V_x + t \times \sin \theta \right)^2 + \left(V_y + t \times \sin \left(\theta + \mu \right) \right)^2 + \left(V_z + t \times \sin \left(\theta + \pi \right) \right)^2}}{E}$$

where is equal to /2. If acceleration and change of direction are taken into account, then the vector wave function can be fully defined as:


 * $$w = \frac{\left(S + at \right)\sqrt{\left(V_x + t \times \sin \theta \right)^2 + \left(V_y + t \times \sin \left(\theta + \mu \right) \right)^2 + \left(V_z + t \times \sin \left(\theta + \pi \right) \right)^2}}{E}$$

Please note that the angle is measured in radians.