Wave Function

The repeated execution of the unit action $h$ can be modeled by a point rotating around the unit circle in the complex plane, represented by the complex function:

\[\psi(\theta) = e^{i \theta} = \cos\theta + i \sin\theta,\]

where $e$ is the base of the natural logarithm and $i$ is the imaginary unit ($i^2=-1$). As the phase angle $\theta$ increases from $0$ to $+2\pi$, the complex vector $\psi(\theta)$ completes a full counterclockwise revolution, representing the execution of the action $h^+$. Conversely, as $\theta$ decreases from $0$ to $-2\pi$, the vector completes a full rotation in reverse, corresponding to the execution of the counter-action $h^-$.

Quantum Mechanics Reimagined (ant = kinesin/step motor = action agent)

Any periodic physical process can therefore be mapped to motion in the complex plane using the identity,

\[z = \frac{\theta}{2\pi} = \frac{S}{h} = \frac{t}{\tau} = \frac{x}{\lambda},\]

where

• $z$ a dimensionless variable representing the number of completed action cycles;
• $\theta$ is the phase in radians, with $2\pi$ corresponding to a full cycle;
• $S$ is the signed action accumulated along a physical trajectory, measured in units of $h$;
• $t$ is the elapsed time measured in units of the action period $\tau$;
• $x$ is the length of an action path measured in units of the step length $\lambda$.

The physical quantity $S$ represents the cumulative action along the propagation path of an action signal. Expressing the phase $\theta$ in terms of $S$ and $h$ yields

\[\psi(S) = e^{i 2\pi \frac{S}{h}}.\]

This equation plays an essential role in Feynman’s path integral formulation of quantum mechanics.

The temporal evolution of the system can be expressed as

\[\psi(t) = e^{\pm i 2\pi \frac{t}{\tau}},\]

where the $\pm$ sign in the exponent represents the evolution of a pair of opposite actions. Because the time $t$ is a scalar quantity that increases monotonically, two opposing actions is represented using a pair of signed frequencies,

\[\psi(t) = e^{i2\pi f_{\pm} t} = e^{i \omega_{\pm} t},\]

where $f_{\pm} = \pm \frac{1}{\tau}$ correspond to the same action that unfolds in opposite directions, and $\omega_{\pm} = 2\pi f_{\pm}$ are the associated angular frequencies.

Space, by contrast, is bidirectional. The signs of $x$ (or of a displacement $\Delta x$) naturally represent two opposite directions. The spatial evolution of the system is thus represented by

\[\psi(x) = e^{\pm i 2\pi \frac{x}{\lambda}} = e^{\pm i 2\pi k x},\]

where the wavenumber $k= \frac{2\pi}{\lambda}$ corresponds to the spatial frequency of the periodic action.

Combining two equations yields the wave function that describes the evolution of the system over spacetime:

\[\begin{align*} \psi(x,t) &= e^{i 2\pi (\frac{t}{\tau} \pm \frac{x}{\lambda})} \\ &= e^{i2\pi(f t \pm \frac{x}{\lambda})} \\ &= e^{i( \omega t \pm k x)}. \end{align*}\]

This complex-valued function satisfies Schrödinger’s partial differential equation:

\[i\hbar \frac{\partial}{\partial t}\,\psi(x,t) = \hat{H}\,\psi(x,t),\]

where $\hbar = \frac{h}{2\pi}$ denotes the reduced action constant. The Hamiltonian operator

\[\hat{H} = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x)\]

represents the energy that drives the spacetime evolution of the system.