2026-08-19
The uncertainty of the internal phase as a geometric basis for the Heisenberg principle
In standard quantum mechanics, the uncertainty relation relates the width of a particle's spatial distribution to the width of its momentum distribution. It is usually derived from the non-commutativity of the position and momentum operators or from the properties of the Fourier transform. However, this derivation alone does not answer the more obvious question: what internal geometric process could underlie this uncertainty?
In the theory of Wave Electricity, the primary object is considered to be a normalized wave. A particle represents a localized and closed state of this wave, and the operator \(J\) defines its instantaneous phase. Therefore, it is natural to assume that quantum uncertainty may originate not with the particle's lack of state, but with the inability to determine the precise initial phase of its internal motion.
Let the internal parameter develop according to a known law \(a=\varpi t\), but its initial value is shifted by an unknown quantity \(\phi\). Then, instead of a single realization \(J(a)\), we should consider a family of states \(J(a+\phi)\). The velocity integral \(cJ(a+\phi)\) transforms this phase uncertainty into an uncertainty of position on the internal trajectory. With complete ignorance of the initial phase, the characteristic product of uncertainties \(\hbar/2\) arises.
But this is only the first part of the construction. A single harmonic with an unknown phase does not yet yield a general Heisenberg principle. For a universal inequality, it is necessary to move from a single internal harmonic to a localized spatial packet. Localization requires a set of spatial frequencies, and the coordinate and spectral representations of such a packet are inevitably related by a Fourier transform.
The main idea of the article is as follows: the unknown initial phase of an internal wave creates a primary uncertainty in its position; integration of the operator transforms the phase into a coordinate; wave decoupling and localization transform this internal uncertainty into a Fourier conjugacy of the coordinate and momentum.
1. Internal parameter \(a\)
The full normalized Wave Electricity operator is written as
\[\tag{1} \boxed{ J(a,b) =\j^{a}(-\j)^{b} =\ep e^{i\pi b} +\em e^{i\pi a}. } \] Here, two complementary idempotents are used: \(\ep\) and \(\em\). Parameter \(a\) describes the internal state of the particle, and parameter \(b\) describes its external motion. They are related to the internal frequency and the observed velocity by the following relationships.
\[\tag{2} a=\varpi t, \qquad \omega=\pi\varpi, \qquad b=\frac{\arcsin\beta}{\pi}, \qquad \beta=\frac{v}{c}. \] The physical phase of the internal component is not equal to the parameter \(a\) itself, but to the magnitude.
\[\tag{3} \boxed{ \theta(t)=\pi a(t)=\omega t. } \] Thus, \(a\) is a normalized phase parameter: a change in \(a\) by \(2\) corresponds to a change in the usual phase \(\theta\) by \(2\pi\). The periodic internal state returns to its original value after a complete phase cycle.
Knowing the frequency \(\omega\) and the laboratory time \(t\) does not mean that the actual phase of the internal wave is known. This would require knowing at what point in the cycle it was at \(t=0\). This is where the initial phase uncertainty appears.
2. Unknown Initial Phase
We introduce the unknown shift \(\phi\) and write the internal parameter of each individual realization as
\[\tag{4} a_{\phi}(t)=\varpi t+\phi. \] This corresponds to the usual phase
\[\tag{5} \theta_{\phi}(t) =\pi a_{\phi}(t) =\omega t+\pi\phi. \] Therefore, the correct notation for the family of realizations is
\[\tag{6} \boxed{ J_{\phi}(t)=J\!\left(a(t)+\phi\right). } \] Writing \(J(a)=J(a+\phi)\) for an arbitrary \(\phi\) would be incorrect: these are not the same instantaneous phase state. Equality holds only for a shift of an integer number of periods. The unknown \(\phi\) means that we have several possible realizations with the same frequency and the same norm, but with different initial positions of the internal wave.
In this case, the motion itself does not become random. The phase derivative remains defined:
\[\tag{7} \frac{d\theta_{\phi}}{dt}=\omega. \] Therefore, we must distinguish between an unknown initial phase and a random phase walk. In the first case, after choosing \(\phi\), the movement of each individual realThe phase transformation is entirely consistent. In the second case, time-varying noise would be added to the phase, which would describe a different physical process interaction with the environment, decoherence, or stochastic dynamics.
In this article, neither the internal frequency nor the law of motion is random. Only the initial position of the wave in its periodic cycle is unknown.
3. Why the Internal Phase May Be Inaccessible to an Observer
Preparing a particle can fix its energy, direction of external motion, charge, or spin orientation, but does not necessarily fix the point of the internal phase cycle. To determine the absolute phase, the measuring instrument would need to have its own synchronized process with the same frequency and a known phase relationship to the wave being studied.
If such synchronization is absent, different particles in an identically prepared ensemble may have the same internal frequency \(\omega\) but different values \(\phi\). The uncertainty then relates not to the existence of the internal state, but to our knowledge of its initial point.
For a statistical description, we introduce the distribution density.
\[\tag{8} P(\phi)\geq0, \qquad \int_{0}^{2}P(\phi)\,d\phi=1. \] If no position of the internal wave is distinguished, a natural special case is a uniform distribution over the full period:
\[\tag{9} \boxed{ P(\phi)=\frac12, \qquad 0\leq\phi<2. } \] Uniformity here is a physical assumption about the complete absence of phase information. It does not follow solely from the periodicity of the operator. If the preparation process partially synchronizes the internal wave, the distribution \(P(\phi)\) may be non-uniform.
4. Operator as the Direction of Instantaneous Motion
In Wave Electricity, the operator \(J\) defines the normalized instantaneous state of motion. The velocity is determined by the formula
\[\tag{10} \boxed{ V_{\phi}(t)=cJ_{\phi}(t) =cJ\!\left(a(t)+\phi\right). } \] Since the norm of the operator is unity, the fundamental scale of the velocity is preserved:
\[\tag{11} |J_{\phi}(t)|=1, \qquad |V_{\phi}(t)|=c. \] The operator specifies the velocity or phase direction, but does not yet specify the final trajectory. To obtain the position, it must be integrated over time:
\[\tag{12} \boxed{ R_{\phi}(t) =R_{\phi}(0) +\int_{0}^{t}cJ\!\left(a(t')+\phi\right)dt'. } \] This is a fundamental transition. So far, only \(J(a+\phi)\) is considered; the uncertainty is related to the phase. After integration, different values of \(\phi\) lead to different positions of \(R_{\phi}(t)\). Thus, the phase uncertainty is spatially represented.
5. Isolation of the Internal Phase Component
For a stationary external state, we can set \(b=0\). Then the complete operator has the form
\[\tag{13} J(a+\phi,0) =\ep+\em e^{i(\omega t+\pi\phi)}. \] The constant idempotent component \(\ep\) and the rotating component \(\em e^{i\theta_{\phi}}\) belong to different phase planes. The internal periodic trajectory is obtained by projecting the complete state onto the plane in which the parameter \(a\) is located:
\[\tag{14} \Pi_{a}J(a+\phi,0) =\em e^{i\theta_{\phi}}. \] This distinction is necessary to avoid identifying the constant component of the total operator with the external translational motion of the particle. It is the rotating internal component that is integrated:
\[\tag{15} R_{\mathrm{int},\phi}(t) =R_{c} +c\int \em e^{i(\omega t+\pi\phi)}dt. \] As a result
\[\tag{16} \boxed{ R_{\mathrm{int},\phi}(t) =R_{c} +\em\frac{c}{i\omega} e^{i(\omega t+\pi\phi)}. } \] The factor \(1/i\) denotes a quarter-period rotation and arises because the coordinate is the integral of the velocity. The absolute value of the variable part of the coordinate is
\[\tag{17} \boxed{ r_{\mathrm{int}}=\frac{c}{\omega}. } \] Thus, the frequency of the internal phase motion automatically determines the spatial scale. The higher the frequency, the smaller the radius of the corresponding closed trajectory.
6. How Phase Uncertainty Becomes Coordinate Uncertainty
Let's select one observable axis in the physical representation of the internal trajectory. By denoting
\[\tag{18} \theta=\omega t+\pi\phi, \] the coordinate and velocity projections can be written as
\[\tag{19} \boxed{ x_{\phi}=r_{\mathrm{int}}\sin\theta, \qquad v_{\phi}=c\cos\theta. } \] The specific choice of sine for the coordinate and cosine for the velocity depends on the phase reference. It is not the name of the functions that is important, but their shift by a quadrature.The period. The coordinate and velocity are two quadratures of a single phase motion.
If \(\phi\) is precisely known, then at a given instant the projection \(x_{\phi}\) is also known. If the initial phase is unknown, the same formula defines a set of possible coordinates. Therefore, the spatial distribution arises without the assumption that the wave is simultaneously materially present at all points along the trajectory. Each individual realization has one phase and one coordinate, and the ensemble of realizations creates a distribution of possible outcomes.
At this level, we have obtained a statistical model of the internal coordinate. It is not yet a complete quantum description, since the universal probability rule \(|\psi|^{2}\) has not yet been established and the localization of a free wave has not yet been considered.
7. Energy and Momentum of an Internal Wave
An internal frequency is associated with an energy scale.
\[\tag{20} E_{\mathrm{int}}=\hbar\omega. \] If a fundamental wave propagates with velocity \(c\), its total momentum is determined by the ratio of energy to velocity:
\[\tag{21} \boxed{ p_{\mathrm{int}} =\frac{E_{\mathrm{int}}}{c} =\frac{\hbar\omega}{c}. } \] We write the projection of the internal momentum onto the selected axis using the same phase as the velocity:
\[\tag{22} p_{\phi}=p_{\mathrm{int}}\cos\theta. \] The coordinate amplitude is \(c/\omega\), and the momentum amplitude is \(\hbar\omega/c\). Their product is independent of frequency:
\[\tag{23} \boxed{ r_{\mathrm{int}}p_{\mathrm{int}} =\frac{c}{\omega}\frac{\hbar\omega}{c} =\hbar. } \] This equality is the central geometric scale of the construction. As the frequency changes, the radius and momentum change in opposite directions: increasing \(\omega\) decreases the characteristic length but increases the momentum in the same proportion.
8. Internal and Mechanical Momenta of an Electron
For an electron in the adopted model, a distinction must be made between the total momentum of the internal wave and the observed mechanical scale \(m_{e}c\). The rest mass is considered as a projection of the internal frequency:
\[\tag{24} m_{e}c^{2} =\alpha_{\mathrm{fs}}\hbar\omega_{\mathrm{int}}. \] From here
\[\tag{25} m_{e}c =\alpha_{\mathrm{fs}} \frac{\hbar\omega_{\mathrm{int}}}{c} =\alpha_{\mathrm{fs}}p_{\mathrm{int}}. \] Therefore, the internal momentum is greater than the mechanical scale by a factor of \(1/\alpha_{\mathrm{fs}}\):
\[\tag{26} \boxed{ p_{\mathrm{int}} =\frac{m_{e}c}{\alpha_{\mathrm{fs}}}. } \] If we use \(m_{e}c\) directly in the geometric product, we get
\[\tag{27} r_{\mathrm{int}}m_{e}c =\alpha_{\mathrm{fs}}\hbar. \] This is not a contradiction, but an indication of different levels of description. The radius \(r_{\mathrm{int}}=c/\omega_{\mathrm{int}}\) refers to the total intrinsic frequency, so its conjugate scale is \(p_{\mathrm{int}}=\hbar\omega_{\mathrm{int}}/c\). The observed mechanical momentum appears after the projection of the internal state into the external channel.
The equality \(r_{\mathrm{int}}p_{\mathrm{int}}=\hbar\) refers to the full internal wave. To identify this momentum with the particle's ordinary measured momentum, a separate description of the external projection mechanism is required.
9. Completely Unknown Phase
Consider the special case of a complete lack of information about the initial phase. Then, the angle \(\theta\) is uniformly distributed over the full cycle:
\[\tag{28} 0\leq\theta<2\pi, \qquad P(\theta)=\frac{1}{2\pi}. \] The average values of sine and cosine are zero:
\[\tag{29} \langle\sin\theta\rangle=0, \qquad \langle\cos\theta\rangle=0. \] The average squares are
\[\tag{30} \langle\sin^{2}\theta\rangle =\langle\cos^{2}\theta\rangle =\frac12. \] Therefore, for the coordinate and projection of the internal momentum, we obtain
\[\tag{31} \langle x\rangle=0, \qquad \langle p\rangle=0, \] \[\tag{32} (\Delta x)^{2}=\frac{r_{\mathrm{int}}^{2}}{2}, \qquad (\Delta p)^{2}=\frac{p_{\mathrm{int}}^{2}}{2}. \] Hence,
\[\tag{33} \Delta x=\frac{r_{\mathrm{int}}}{\sqrt2}, \qquad \Delta p=\frac{p_{\mathrm{int}}}{\sqrt2}. \] By multiplying these quantities and using the formula \(r_{\mathrm{int}}p_{\mathrm{int}}=\hbar\), we find
\[\tag{34} \boxed{ \Delta x\,\Delta p =\frac{r_{\mathrm{int}}p_{\mathrm{int}}}{2} =\frac{\hbar}{2}. } \] Thus, the characteristic quantum quantity \(\hbar/2\) arises from two quadratures of one internal wave with a uniformly unknown phase. The factor \(1/2\) appears because the average quadratureThe square of each of the two orthogonal projections of a complete circular motion is equal to half the square of its amplitude.
This is an important geometric result, but it cannot be immediately considered a general derivation of Heisenberg's principle. It is obtained for a special ensemble: a single frequency, constant amplitude, and a uniformly distributed initial phase.
10. Why Random Phase Is Not Enough for a General Inequality
Let the initial phase have an arbitrary distribution \(P(\phi)\). Then the mean values are determined by the formulas
\[\tag{35} \langle x\rangle =r_{\mathrm{int}} \int_{0}^{2}P(\phi) \sin(\omega t+\pi\phi)\,d\phi, \] \[\tag{36} \langle p\rangle =p_{\mathrm{int}} \int_{0}^{2}P(\phi) \cos(\omega t+\pi\phi)\,d\phi. \] The variances also depend on the shape of \(P(\phi)\). If the phase is concentrated in a very narrow interval, the projection uncertainties may differ from the values obtained for a uniform ensemble. Classical statistics of the unknown angle alone do not provide a universal lower bound for all possible distributions.
Therefore, the logically precise derivation is as follows
\[\tag{37} \boxed{ \text{uniformly unknown phase} \quad\Longrightarrow\quad \Delta x\,\Delta p=\frac{\hbar}{2} } \] only for the circular phase model considered. To obtain the general relation
\[\tag{38} \Delta x\,\Delta p\geq\frac{\hbar}{2}, \] it is necessary to take into account that a physically localized wave state cannot have arbitrary, independently specified distributions of position and momentum. These distributions are two representations of the same wave.
11. Disconnection of the Internal Phase into the Spatial Phase
In a closed particle, the phase returns to its original value after passing through the internal contour. When the wave disconnects, the cyclic return is replaced by continuation along the spatial coordinate. Then, the phase should depend not only on time but also on position:
\[\tag{39} \theta(x,t)=kx-\omega t+\phi_{0}. \] The sign before \(\omega t\) specifies the chosen propagation direction. The spatial wavenumber \(k\) shows how much the phase changes with a single displacement:
\[\tag{40} k=\frac{\partial\theta}{\partial x}. \] In the notation of the normalized parameter \(a\), the spatial harmonic can be written as
\[\tag{41} \boxed{ J_{k}(x,t) =J\!\left( \frac{kx-\omega t}{\pi}+\phi_{k} \right). } \] Here \(\phi_{k}\) is the initial phase of a specific spatial harmonic. After the break, the internal phase uncertainty does not disappear, but transforms into uncertainty in the position of the phase pattern along space.
12. Why does a spatial harmonic have the form \(e^{ikx}\)
The Fourier structure can be justified by the homogeneity of space. Let the transfer of a wave by a distance \(\xi\) not change its physical nature, but only add phase:
\[\tag{42} J_{k}(x+\xi)=J_{k}(x)e^{iF_{k}(\xi)}. \] If you first shift by \(\xi_{1}\), and then by \(\xi_{2}\), the result should be the same as shifting directly by \(\xi_{1}+\xi_{2}\). Therefore
\[\tag{43} F_{k}(\xi_{1}+\xi_{2}) =F_{k}(\xi_{1})+F_{k}(\xi_{2}). \] For a continuous function, the only linear solution is
\[\tag{44} F_{k}(\xi)=k\xi. \] Consequently, the natural phase mode of spatial transfer has the form
\[\tag{45} \boxed{ J_{k}(x)\sim e^{i(kx+\phi_{k})}. } \] Thus, the exponential \(e^{ikx}\) is not introduced arbitrarily. It is a direct consequence of the continuity and homogeneity of one-dimensional space: identical displacements must produce identical phase increments.
13. A single harmonic cannot localize a particle.
Monochromatic spatial wave
\[\tag{46} \psi_{k}(x)=A_{0}e^{ikx} \] has a constant modulus throughout space:
\[\tag{47} |\psi_{k}(x)|^{2}=|A_{0}|^{2}. \] It has a precisely defined wavenumber \(k\), but lacks a distinct localization region. Changing the single common phase \(\phi_{k}\) only shifts the phase ridges and does not create a bounded packet.
To obtain a localized state, harmonics with different values of \(k\) must be summed. Their amplitudes and relative phases must be matched so that they reinforce each other in a certain region and cancel each other out further away.
14. Wave Packet as a Matched Phase System
The general one-dimensional localized state is written as a superposition of spatial harmonics:
div> \[\tag{48} \boxed{ \psi(x) =\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} A(k)e^{i[kx+\phi(k)]}\,dk. } \] The function \(A(k)\) defines the absolute value of the contribution of each harmonic, and \(\phi(k)\) defines its phase relative to the other harmonics. It is convenient to combine them into a single complex spectral amplitude:
\[\tag{49} \widetilde A(k)=A(k)e^{i\phi(k)}. \] Then
\[\tag{50} \boxed{ \psi(x) =\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} \widetilde A(k)e^{ikx}\,dk. } \] This is the inverse Fourier transform. It appears because the harmonics \(e^{ikx}\) are natural modes of spatial transport, and the linearity of the wave description allows them to be summed.
The inverse relation determines the spectrum of a state by its spatial shape:
\[\tag{51} \boxed{ \widetilde A(k) =\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} \psi(x)e^{-ikx}\,dx. } \] The coordinate function \(\psi(x)\) and the spectral function \(\widetilde A(k)\) are not two independently chosen distributions. They are two ways of describing the same state. If one of them is specified, the other is determined by the Fourier transform.
15. Overall and Relative Phases
For a correct physical interpretation, it is important to distinguish between the overall phase of the entire packet and the relative phases of its harmonics. Multiplying the state by the same phase factor
\[\tag{52} \psi(x)\longrightarrow e^{i\phi_{0}}\psi(x) \] does not change the spatial intensity:
\[\tag{53} |e^{i\phi_{0}}\psi(x)|^{2}=|\psi(x)|^{2}. \] Therefore, the absolute overall phase of the packet does not determine the observed position. Relative phases of different values of \(k\) behave differently. If we change them linearly,
\[\tag{54} \widetilde A(k) \longrightarrow \widetilde A(k)e^{-ikx_{0}}, \] then from the Fourier formula it follows
\[\tag{55} \psi(x) \longrightarrow \psi(x-x_{0}). \] Therefore, the position of the localized packet is encoded by the consistent slope of the spectral phase. It is not a single absolute phase, but a system of relative phases that determines exactly where the harmonics add together.
If the phases of different harmonics are made independently random, the stable localized maximum is destroyed. This condition should be associated not with improved localization, but with a loss of coherence.
16. From Packet Width to Spectral Width
Fourier conjugacy has a simple physical meaning. To create a narrow maximum in space, many harmonics with different \(k\) must be added together. If only a narrow range of wavenumbers is present, all components have nearly the same wavelength and form an extended packet.
Therefore, decreasing the spatial width \(\Delta x\) inevitably increases the spectral width \(\Delta k\). This limitation is not related to the accuracy of the instrument. It follows from the very impossibility of constructing a narrow wave structure from one or almost one harmonic.
For the physical interpretation of distributions, we adopt normalization.
\[\tag{56} \int_{-\infty}^{+\infty}|\psi(x)|^{2}dx=1, \qquad \int_{-\infty}^{+\infty}|\widetilde A(k)|^{2}dk=1. \] The quantity \(|\psi(x)|^{2}\) characterizes the normalized spatial density of state, and \(|\widetilde A(k)|^{2}\) characterizes the density of its wavenumbers. The mathematical equality of the normalizations follows from Parseval's theorem. The identification of these densities with the probabilities of measurement results is an additional physical rule that must be separately justified in a full theory.
17. The Relationship between Wavenumber and Momentum
For a free open wave, the temporal and spatial frequencies are related by the propagation velocity:
\[\tag{57} \omega=ck. \] Using the energy relation
\[\tag{58} E=\hbar\omega \] and the wave momentum
\[\tag{59} p=\frac{E}{c}, \] we obtain
\[\tag{60} \boxed{ p=\hbar k. } \] It is this formula that transforms the purely geometric constraint between coordinate and spatial frequency into a physical constraint between coordinate and momentum.
For a massive particle, the group velocity of the packet can be less than \(c\). Therefore, the equality \(\omega=ck\) cannot be applied to its external nonrelativistic dispersion without clarification. However, the correspondence \(p=\hbar k\) remains as the relationship between the accumulated spatial phase and the canonical momentum. More generally, it follows from the action:
\[\tag{61} dS=p\,dx-E\,dt, \qquad \theta=\frac{S}{\hbar}. \] From here
\[\tag{62} k=\frac{\partial\theta}{\partial x} =\frac{p}{\hbar}, \qquad \omega=-\frac{\partial\theta}{\partial t} =\frac{E}{\hbar}. \] 18. Mathematical Derivation of the General Inequality
Now we can move from a particular uniform phase ensemble to an arbitrary normalized localized packet. For simplicity, we choose the coordinate and wavenumber origins so that
\[\tag{63} \langle x\rangle=0, \qquad \langle k\rangle=0. \] Then the variance of the coordinate is
\[\tag{64} (\Delta x)^{2} =\int_{-\infty}^{+\infty} x^{2}|\psi(x)|^{2}dx. \] Multiplication of the spectrum by \(k\) corresponds to differentiation of the coordinate function. Therefore, by Parseval's theorem
\[\tag{65} (\Delta k)^{2} =\int_{-\infty}^{+\infty} k^{2}|\widetilde A(k)|^{2}dk =\int_{-\infty}^{+\infty} \left|\frac{d\psi}{dx}\right|^{2}dx. \] Consider a non-negative value for an arbitrary real \(\lambda\):
\[\tag{66} 0\leq \int_{-\infty}^{+\infty} \left| x\psi+\lambda\frac{d\psi}{dx} \right|^{2}dx. \] After expanding the square, two positive terms and a mixed term appear. For a localized function vanishing at infinity, integration by parts yields
\[\tag{67} \int_{-\infty}^{+\infty} x\frac{d|\psi|^{2}}{dx}dx=-1. \] Therefore, expression (66) takes the form
\[\tag{68} 0\leq (\Delta x)^{2} -\lambda +\lambda^{2}(\Delta k)^{2}. \] The right-hand side is a square trinomial with respect to \(\lambda\). It must remain non-negative for any value of \(\lambda\), therefore, its discriminant cannot be positive:
\[\tag{69} 1-4(\Delta x)^{2}(\Delta k)^{2}\leq0. \] From this, we obtain a purely wave relation.
\[\tag{70} \boxed{ \Delta x\,\Delta k\geq\frac12. } \] Finally, taking into account \(p=\hbar k\), we have
\[\tag{71} \boxed{ \Delta x\,\Delta p\geq\frac{\hbar}{2}. } \] Now this is no longer a result for a single selected initial phase distribution, but a general inequality for any normalized localized wave state with finite variances of position and momentum.
19. Why the Lower Bound Is Not an Instrumental Error
If uncertainty were only a consequence of inaccurate measurement of predetermined classical quantities, it would be possible to simultaneously reduce position and momentum errors by improving the instrument. Fourier conjugacy is structured differently: changing the state itself, necessary to decrease \(\Delta x\), broadens the spectrum of \(k\) and thereby increases \(\Delta p\).
Therefore, the unknown initial phase in the proposed picture plays the role of a geometric source of statistical scatter, but the universal boundary arises from the structure of the localized wave. These two levels complement each other:
\[\tag{72} \boxed{ \begin{aligned} \text{unknown internal phase} &\longrightarrow \text{uncertainty of an individual realization},\\ \text{Fourier structure of the wave packet} &\longrightarrow \text{universal lower bound}. \end{aligned} } \] 20. State of Minimum Uncertainty
Equality in formula (70) is achieved when the integrand in formula (66) vanishes:
\[\tag{73} x\psi+\lambda\frac{d\psi}{dx}=0. \] The solution to this equation has a Gaussian form:
\[\tag{74} \psi(x) =C\exp\!\left(-\frac{x^{2}}{4\sigma_{x}^{2}}\right). \] For a Gaussian packet
\[\tag{75} \Delta x=\sigma_{x}, \qquad \Delta k=\frac{1}{2\sigma_{x}}, \] therefore
\[\tag{76} \Delta x\,\Delta p=\frac{\hbar}{2}. \] Here, it is necessary to distinguish between two cases in which the same value of \(\hbar/2\) appears. The uniform distribution of one internal phase yields it from two orthogonal projections of the circular motion. A Gaussian packet yields the same boundary as the optimal ratio between the spatial width and the spectral width. These distributions are not identical, but both indicate a common phase-spatial scale.
21. A Possible Interpretation of the Measurement
In the proposed picture, an individual particle has a specific phase of its internal wave, but this phase is not synchronized with the measuring instrument. Interaction with the instrument establishes a specific phase relationship and translates the internal state into an observable coordinate or momentum projection.
When repeating the experiment, the external preparation conditions may be the same, while the initial phases of individual realizations differ. Therefore, the results form a statistical distribution. This explanation allows us to link the probabilistic description to a real internal process, without claiming that the particle has no phase at all before the measurement.lo state.
However, this interpretation alone does not yet imply the Born rule.
\[\tag{77} P(x)=|\psi(x)|^{2}. \] For a complete theory, it is necessary to show why the intensity of the normalized wave amplitude determines the frequency of registration results and how a specific interaction with the device selects one outcome. In this article, the rule \(|\psi|^{2}\) is used as a physical correspondence between the wave state and measurement statistics.
22. What has been obtained and what remains a hypothesis.
It follows directly from the adopted geometry: the operator \(J\) defines the phase direction; the velocity is of the form \(V=cJ\); the coordinate is obtained by integration; The internal harmonic creates the scale \(r=c/\omega\); the unknown initial phase means an unknown point on the periodic trajectory.
After energy correspondence, we obtain: \(E=\hbar\omega\), \(p_{\mathrm{int}}=\hbar\omega/c\), and \(r_{\mathrm{int}}p_{\mathrm{int}}=\hbar\). For a uniform distribution of the initial phase, two orthogonal projections give \(\Delta x\,\Delta p=\hbar/2\).
From the general mathematics of localized waves it follows: the homogeneity of space distinguishes harmonics \(e^{ikx}\); Localization requires their superposition; the coordinate and spectral representations are related by the Fourier transform; therefore, \(\Delta x\,\Delta k\geq1/2\), and for \(p=\hbar k\), the standard Heisenberg inequality is obtained.
Additional physical hypotheses remain: the complete origin of the Born rule from the operator \(J\); the mechanism for converting the internal phase into a single measurement result; an exact mapping of the total internal momentum onto the external mechanical momentum; restrictions on physically admissible phase distributions; a description of composite and entangled states.
23. Final sequence
\[\tag{78} \boxed{ \begin{gathered} a_{\phi}(t)=\varpi t+\phi \longrightarrow J_{\phi}(t)=J(a+\phi),\\ V_{\phi}=cJ_{\phi} \longrightarrow R_{\phi}=\displaystyle\int V_{\phi}dt,\\ r_{\mathrm{int}}=\dfrac{c}{\omega}, \qquad p_{\mathrm{int}}=\dfrac{\hbar\omega}{c} \longrightarrow r_{\mathrm{int}}p_{\mathrm{int}}=\hbar,\\ \text{uniformly unknown phase} \longrightarrow \Delta x\,\Delta p=\dfrac{\hbar}{2},\\ \text{spatial opening} \longrightarrow J_{k}(x)\sim e^{ikx},\\ \text{localization} \longrightarrow \psi(x)=\dfrac{1}{\sqrt{2\pi}} \displaystyle\int\widetilde A(k)e^{ikx}dk,\\ x\leftrightarrow k, \qquad p=\hbar k \longrightarrow \Delta x\,\Delta p\geq\dfrac{\hbar}{2}. \end{gathered} } \] Thus, the uncertainty begins with the phase, but does not end with it. The unknown value \(\phi\) defines an unknown point of continuous internal motion. Integrating \(cJ(a+\phi)\) transforms the phase difference into a spatial one. When a wave must be localized in open space, a single harmonic is no longer sufficient: a spectrum of wave numbers and a consistent system of relative phases emerge.
In this sequence, Heisenberg's principle receives a two-level explanation. At the internal geometric level, the quantity \(\hbar/2\) emerges from the uniformly unknown phase and two quadratures of the normalized wave. At the general spatial level, the inequality arises from the impossibility of creating an arbitrarily narrow localized packet without broadening its pulse spectrum.
Quantum uncertainty in the proposed model does not mean the absence of internal motion, but the impossibility of simultaneously fixing a phase cycle point and maintaining a narrow spectrum of spatial extensions of the same wave.