Why can't a state with a precisely defined energy be localized in time, while a short wave event inevitably contains a certain range of energies? In standard quantum mechanics, the answer is expressed by the uncertainty principle \(\Delta E\,\Delta t\geqslant\hbar/2\). However, the formula itself does not yet reveal what geometric arrangement of the state creates the energy spectrum and why its width is related to the duration of phase consistency.

This paper proposes to relate these quantities to multilevel idempotent splitting. The splitting creates orthogonal phase channels, the random relative phase generates a spread in the phase-sensitive projection energy, and the different rotation speeds of the energy components limit the time of their co-localization. After precisely determining the energy and time widths, it will be shown that their product cannot be less than \(\hbar/2\), with the lower bound achieved only by a Gaussian wavelet with consistent relative phases.

This work relies on the unified concept of Wave Electricity, the construction of deep idempotent splitting, and the study of random and coupled phases. Here, these concepts are combined into a single, consistent construction.

The quantity \(\Delta t\) hereafter denotes neither the uncertainty of the external clock readings nor the spectrum of an individual self-adjoint time operator. It is defined as the root-mean-square duration of a normalized time packet: a pulse, a registration event, or a phase localization interval. Therefore, the relation being proven is a spectral-temporal inequality for two Fourier-conjugate representations of a single wave state.

An approximate estimate of \(\Delta E\,\Delta t\sim\hbar\) is obtained from the condition of a phase divergence of approximately one radian. The coefficient \(1/2\) requires a more rigorous definition of variances and the application of the Cauchy-Bunyakovsky inequality.

The ground state of the model is defined by a normalized operator

Parameter \(a\) describes the internal phase state, and parameter \(b\) describes the external motion:

If the initial position of the internal wave is not known in advance, a random addition \(\phi\) is introduced:

The physical random phase angle is \(\pi\phi\). For a uniform distribution over a full cycle,

The static random addition \(\phi\) does not change the intrinsic frequency, since \(d\phi/dt=0\). Therefore, the uncertainty of the initial phase alone does not create uncertainty in the energy. For the phase to manifest itself in the energy distribution, the two branches must be projected onto a common phase-sensitive channel.

Let two idempotent branches be associated with orthonormal metric states \(|+\rangle\) and \(|-\rangle\):

The external phase defines two real projections

We use them as the amplitudes of the two daughter branches, and the random internal phase as their relative phase:

The state is normalized:

With direct observationWhen the branches \(|+\rangle\) and \(|-\rangle\) are separated, their weights are \(1-\beta^2\) and \(\beta^2\). The relative phase \(\Delta\) disappears from these quantities. Therefore, random phase cannot change the energy of two already separated orthogonal channels.

The phase becomes observable when the branches are again projected onto a common basis:

The amplitudes of the two outputs are equal

Let the total energy of this splitting be \(\varepsilon\), and the projection energy be proportional to the square of its amplitude. Then

For any value of the random phase, the conservation law holds:

Therefore, the random phase does not change the total energy. It redistributes the energy between two complementary projections of a single split state.

Let's denote the amplitude of the energy deviation

Then the energy of the first output is given by

If \(\phi\) is uniformly distributed, then \(\Delta\) is also uniform over the entire cycle. The standard replacement of the random variable yields the exact energy density:

This is an arcsine distribution, not a normal one. Its mean and variance are equal.

At each deep level, we define idempotents

Let the \(r\)th level contribute energy \(\varepsilon_r\), characterized by parameter \(\beta_r\) and relative phase \(\Delta_r\). The energy of the selected phase-sensitive projection is

The total projection energy of several levels:

If the phases \(\Delta_r\) are independent and uniform, then

Therefore, the mean energy and variance are equal

The exact distribution for finite \(N\) is the convolution of arcsine distributions. Its characteristic function has the form

where \(J_0\) is the zeroth-order Bessel function. If no single level determines the entire variance, then the condition is satisfied

central preThe theorem leads to a normal distribution:

Thus, splitting creates independent channels, random phases create random phase-sensitive contributions, and the summation of a large number of such contributions creates a Gaussian energy profile.

The density \(P(E)\) does not yet contain complete information about the wave state. A complex energy amplitude must be entered. \[\tag{26} \widetilde\psi(E)=\sqrt{P(E)}\,e^{i\chi(E)}, \qquad \int_{-\infty}^{\infty}|\widetilde\psi(E)|^2dE=1. \] \[\tag{27} \omega(E)=\frac{E}{\hbar}, \qquad e^{-i\omega t}=e^{-iEt/\hbar}. \] \[\tag{28} \boxed{ \psi(t)= \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} \widetilde\psi(E)e^{-iEt/\hbar}\,dE. } \] \[\tag{29} \widetilde\psi(E)= \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} \psi(t)e^{iEt/\hbar}\,dt. \] \[\tag{30} \int_{-\infty}^{\infty}|\psi(t)|^2dt=1. \] \[\tag{31} \langle t\rangle =\int_{-\infty}^{\infty}t|\psi(t)|^2dt, \qquad (\Delta t)^2 =\int_{-\infty}^{\infty} (t-\langle t\rangle)^2|\psi(t)|^2dt. \] \[\tag{32} \langle E\rangle =\int_{-\infty}^{\infty}E|\widetilde\psi(E)|^2dE, \qquad (\Delta E)^2 =\int_{-\infty}^{\infty} (E-\langle E\rangle)^2|\widetilde\psi(E)|^2dE. \] \[\tag{33} \langle t\rangle=0, \qquad \langle E\rangle=0. \] \[\tag{34} (\Delta E)^2 =\hbar^2 \int_{-\infty}^{\infty} \left|\frac{d\psi}{dt}\right|^2dt. \] \[\tag{35} 0= \int_{-\infty}^{\infty} \frac{d}{dt}\left[t|\psi(t)|^2\right]dt. \] \[\tag{36} 1+2\operatorname{Re} \int_{-\infty}^{\infty} t\psi^*(t)\frac{d\psi}{dt}\,dt=0. \] \[\tag{37} \left| \int_{-\infty}^{\infty} t\psi^*(t)\frac{d\psi}{dt}\,dt \right| \geqslant\frac12. \] \[\tag{38} \left( \int t^2|\psi|^2dt \right) \left( \int\left|\frac{d\psi}{dt}\right|^2dt \right) \geqslant \left| \int t\psi^*\frac{d\psi}{dt}\,dt \right|^2 \geqslant\frac14. \] \[\tag{39} (\Delta t)^2\frac{(\Delta E)^2}{\hbar^2} \geqslant\frac14. \] \[\tag{40} \boxed{ \Delta E\,\Delta t \geqslant\frac{\hbar}{2}. } \] \[\tag{41} \frac{d\psi}{dt}=-\lambda t\psi, \qquad \lambda>0. \] \[\tag{42} \psi(t)= \frac{1}{(2\pi\Delta t^2)^{1/4}} \exp\!\left[ -\frac{(t-t_0)^2}{4\Delta t^2} \right] e^{-iE_0t/\hbar}e^{i\phi_0}. \] \[\tag{43} |\widetilde\psi(E)|^2 =\frac{1}{\sqrt{2\pi}\,\Delta E} \exp\!\left[ -\frac{(E-E_0)^2}{2(\Delta E)^2} \right]. \] \[\tag{44} \boxed{ \Delta E\,\Delta t=\frac{\hbar}{2}. } \] \[\tag{45} \widetilde\psi(E)=\sqrt{P(E)}e^{i\chi(E)}. \] \[\tag{46} \widetilde\psi(E)\longrightarrow e^{i\phi_0}\widetilde\psi(E). \] \[\tag{47} \chi(E)=\chi_0+\frac{Et_0}{\hbar} \] \[\tag{48} (\Delta t)^2 =\hbar^2 \int_{-\infty}^{\infty} \left(\frac{d\sqrt{P}}{dE}\right)^2dE +\hbar^2\operatorname{Var}_{P} \!\left(\frac{d\chi}{dE}\right). \] \[\tag{49} (\Delta E)^2(\Delta t)^2 =\frac{\hbar^2}{4} +\hbar^2(\Delta E)^2 \operatorname{Var}_{P} \!\left(\frac{d\chi}{dE}\right). \] \[\tag{50} \operatorname{Var}_{P}\!\left(\frac{d\chi}{dE}\right)>0 \quad\Longrightarrow\quad \Delta E\,\Delta t>\frac{\hbar}{2}. \] \[\tag{51} \Delta\omega=\frac{\Delta E}{\hbar}. \] \[\tag{52} \Delta\Theta(t)\sim\Delta\omega\,t =\frac{\Delta E}{\hbar}t. \]