2026-07-13
Schrödinger equation in split operator geometry
Why is energy represented in quantum mechanics as a time-varying operator, momentum as a spatial derivative, and the external state of a particle obeying the Schrödinger equation? In the standard exposition, these rules form a ready-made mathematical apparatus. This paper considers how they can be reconciled with a more complete phase geometry of a particle.
The initial object is the operator \(J(a,b)=\j^a(-\j)^b\), which unites the internal periodicity and the external state of motion in two orthogonal idempotent planes. The internal periodicity defines the primary energy scale, its mass projection forms the rest energy, and the accumulated phase of action along the physical trajectory introduces the energy and momentum of the external motion. After passing to the nonrelativistic limit, this sequence leads to the Schrödinger equation.
The boundary of the derivation should be stated immediately. The time generator \(i\hbar\partial_t\) naturally arises from harmonic phase dynamics. The spatial phase, momentum operator, relativistic invariant, and probability mapping require additional physical rules. Therefore, this article demonstrates a consistent construction of the Schrödinger equation based on phase geometry, but does not derive all of quantum mechanics from the single operator \(J\).
1. Two planes of complete phase state
Consider two complementary idempotents:
\[\tag{1} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=\em\ep=0, \qquad \ep+\em=1. \] The associated hyperbolic unit is defined by the equality
\[\tag{2} \j=\ep-\em, \qquad \j^2=1. \] Its continuous powers create independent rotations in two complex planes:
\[\tag{3} \j^a =\ep+\em e^{i\pi a}, \qquad (-\j)^b =\ep e^{i\pi b}+\em. \] Their product forms a complete finite operator:
\[\tag{4} \boxed{ \begin{aligned} J(a,b) &=\j^a(-\j)^b \\[2pt] &=\ep e^{i\pi b} +\em e^{i\pi a}. \end{aligned} } \] Parameter \(a\) describes the internal state, and parameter \(b\) describes the external motion:
\[\tag{5} a=\varpi t, \qquad \pi\varpi=\omega_{\mathrm{int}}, \] \[\tag{6} b=\frac{\arcsin\beta}{\pi}, \qquad \beta=\frac{v}{c}, \qquad v=v(t). \] The internal phase \(\pi a\) develops in the \((\em,i\em)\) plane, and the external modulation \(\pi b\) develops in the \((\ep,i\ep)\) plane. These planes are orthogonal in the algebraic sense, since the product of their projections is zero.
The adjoint operator has the form
\[\tag{7} \overline J(a,b) =\ep e^{-i\pi b} +\em e^{-i\pi a}. \] This implies a unit finite norm:
\[\tag{8} \boxed{ J\overline J =\ep+\em =1 }. \] The equality \(J\overline J=1\) denotes the completeness of the normalized phase state. It does not assert that energy, mass, or probability are numerically equal to unity. Physical quantities arise only after mapping the operator into the corresponding space.
2. Phase Change Generator
To match the standard time phase sign, it is convenient to use the conjugate state:
\[\tag{9} \Psi_J(t) =\overline J(a(t),b(t)) =\ep e^{-i\pi b(t)} +\em e^{-i\pi a(t)}. \] Differentiating with respect to time yields
\[\tag{10} \frac{\partial\Psi_J}{\partial t} =-i\pi\dot b\,\ep e^{-i\pi b} -i\pi\dot a\,\em e^{-i\pi a}. \] After multiplying by \(i\hbar\), we obtain
\[\tag{11} i\hbar\frac{\partial\Psi_J}{\partial t} =\widehat G_J\Psi_J, \] where
\[\tag{12} \boxed{ \widehat G_J =\hbar\pi \left( \dot b\,\ep +\dot a\,\em \right) }. \] The quantity \(\widehat G_J\) is the generator of the phase operator change. It should not automatically be called the physical Hamiltonian of the particle. In particular, for uniform motion,
\[\tag{13} v=\operatorname{const} \quad\Longrightarrow\quad b=\operatorname{const} \quad\Longrightarrow\quad \dot b=0, \] although the particle's kinetic energy remains nonzero. Therefore, the term \(\hbar\pi\dot b\) shows the rate of change of the external modulation but does not replace the energy of translational motion. The physical energy of motion will be obtained separately through momentum and the relativistic invariant.
3. Tangential change and conservation of norm
Since the norm of the state is constant, its derivative is zero:
\[\tag{14} \frac{d}{dt} \left( \Psi_J\overline{\Psi}_J \right) =\dot\Psi_J\overline{\Psi}_J +\Psi_J\dot{\overline{\Psi}}_J =0. \] For the complex stateThe product \(\overline{\Psi}_J\dot\Psi_J\) is generally not zero but is purely imaginary in each phase component. Therefore, the orthogonality of the radial and tangential changes is expressed through the real metric:
\[\tag{15} \boxed{ \operatorname{Re} \left( \overline{\Psi}_J\dot\Psi_J \right) =0 }. \] The change in state is directed along the tangent to a manifold of constant norm. Multiplication by \(i\) expresses the phase rotation of the tangent relative to the state and does not change its modulus. This provides the geometric basis for the time generator, but does not in itself introduce probability or spatial dynamics.
4. Intrinsic Frequency and Two Electron Scales
Let's isolate the intrinsic projection of the adjoint operator:
\[\tag{16} \em\Psi_J =\em e^{-i\pi a} =\em e^{-i\omega_{\mathrm{int}}t}. \] It satisfies the equation
\[\tag{17} i\hbar\frac{\partial}{\partial t} \left( \em\Psi_J \right) =\hbar\omega_{\mathrm{int}} \left( \em\Psi_J \right). \] Therefore, the internal periodicity corresponds to a primary energy scale.
\[\tag{18} \boxed{ E_{\mathrm{int}} =\hbar\omega_{\mathrm{int}} }. \] For the electron, it is necessary to distinguish two related geometric scales. The minor electromagnetic radius is equal to
\[\tag{19} r_e =\frac{c}{\omega_{\mathrm{int}}}. \] The average radius of the main closed contour is determined by the reduced Compton length:
\[\tag{20} \boxed{ R=\overline{\lambda}_C =\frac{\hbar}{m_ec} =\frac{c}{\Omega_C} }. \] These scales are related by the fine structure constant:
\[\tag{21} \boxed{ r_e=\alpha_{\mathrm{fs}}R }. \] Therefore, the corresponding frequencies are also related:
\[\tag{22} \boxed{ \Omega_C =\alpha_{\mathrm{fs}} \omega_{\mathrm{int}} }. \] The frequency \(\omega_{\mathrm{int}}\) refers to the primary electromagnetic scale \(r_e\), and \(\Omega_C\) refers to the observed mass periodicity and the Compton scale \(R\). A detailed geometric separation of these radii and the derivation of the magnetic moment are discussed in Part Three.
5. Intrinsic mass projection
The observed rest frequency is identified with the Compton frequency:
\[\tag{23} \boxed{ \omega_0 =\Omega_C =\alpha_{\mathrm{fs}} \omega_{\mathrm{int}} }. \] Then the rest energy of the electron is
\[\tag{24} \boxed{ E_0 =\hbar\omega_0 =\hbar\Omega_C =\alpha_{\mathrm{fs}} \hbar\omega_{\mathrm{int}} =m_ec^2 }. \] Accordingly, the mass is determined by the expression
\[\tag{25} m_e =\frac{\hbar\Omega_C}{c^2} =\frac{\alpha_{\mathrm{fs}} \hbar\omega_{\mathrm{int}}}{c^2}. \] After internal mapping, the observed mass phase takes the standard form
\[\tag{26} \psi_0(t) =e^{-i\Omega_Ct} =e^{-im_ec^2t/\hbar}. \] It satisfies the equation
\[\tag{27} i\hbar \frac{\partial\psi_0}{\partial t} =m_ec^2\psi_0. \] The coefficient \(\alpha_{\mathrm{fs}}\) here is the internal mass projection between two frequencies and two electron scales. It is not identified with the inverse Lorentz factor of internal motion. The Lorentz factor \(\gamma(\beta)\) refers to the external state of the already formed particle.
6. Parameter b and the external relativistic state
The definition of parameter \(b\) yields two finite phase projections:
\[\tag{28} \boxed{ \sin(\pi b)=\beta, \qquad \cos(\pi b) =\sqrt{1-\beta^2} =\frac{1}{\gamma} }. \] Parameter \(b\) describes the instantaneous state of the external modulation. During uniform motion, it remains constant and therefore cannot be the accumulated spatial phase:
\[\tag{29} b=\operatorname{const} \quad\not\Longrightarrow\quad \text{phase accumulation along the path}. \] The final operator encodes the quantities \(\beta\) and \(1/\gamma\), but the origin of the full norm \(\gamma\) belongs to a deeply split space. Its state can be represented as a sequence of orthogonal channels:
\[\tag{30} \boldsymbol\Gamma_\beta =\boldsymbol\xi_0 +\beta\boldsymbol\xi_1 +\beta^2\boldsymbol\xi_2 +\cdots. \] Self-similarity of the splitting leads to the relation
\[\tag{31} \|\boldsymbol\Gamma_\beta\|^2 =1 +\beta^2 \|\boldsymbol\Gamma_\beta\|^2. \] From here
\[\tag{32} \boxed{ \|\boldsymbol\Gamma_\beta\| =\frac{1}{\sqrt{1-\beta^2}} =\gamma }. \] Thus, the unit norm of a finite operator and the length of a deep state belong to different spaces:
\[\tag{33} \boxed{ J\overline J=1, \qquad \|\boldsymbol\Gamma_\beta\|=\gamma }. \] After a separate physical mapping, the final projections are associated with energy and momentum:
\[\tag{34} \frac{E_0}{E} =\cos(\pi b) =\frac1\gamma, \qquad \frac{p_sc}{E} =\sin(\pi b) =\beta. \] Therefore,
\[\tag{35} E=\gamma E_0, \qquad p_sc=\gamma\beta E_0. \] And the relativistic invariant is obtained directly:
\[\tag{36} \boxed{ E^2-p_s^2c^2 =E_0^2 =m^2c^4 }. \] The full derivation of the deep norm is given in the article "The Origin of the Lorentz Factor from Infinite Idempotent Splitting". In this paper, formula (36) serves as a link between the geometry of the external state and nonrelativistic dynamics.
7. Phase of Action along a One-Dimensional Path
The parameter \(b\) specifies the instantaneous velocity but does not include the distance traveled. The accumulated spatiotemporal phase is introduced separately through the action along the physical trajectory.
Let the observed trajectory be written as a curve \(\mathbf r(s)\) in three-dimensional space, where \(s\) is the path length. The differential of the action is
\[\tag{37} \boxed{ d\mathcal S =p_s\,ds -E\,dt }. \] From this follow the Hamilton-Jacobi relations:
\[\tag{38} p_s =\frac{\partial\mathcal S}{\partial s}, \qquad E =-\frac{\partial\mathcal S}{\partial t}. \] The action is associated with a complex phase:
\[\tag{39} \boxed{ \Psi_{\mathcal S}(s,t) =A(s,t) \exp\left( \frac{i}{\hbar}\mathcal S(s,t) \right) }. \] For a locally harmonic state with constant amplitude
\[\tag{40} \Psi_{\mathcal S}(s,t) \sim \exp\left[ \frac{i}{\hbar} \left( \int p_s\,ds -\int E\,dt \right) \right]. \] For constant \(p_s\) and \(E\), this expression takes the usual form
\[\tag{41} \Psi_{\mathcal S}(s,t) \sim e^{i(p_ss-Et)/\hbar}. \] Unlike the parameter \(b\), the action \(\mathcal S\) accumulates along the path traveled. Therefore, the instantaneous external modulation and the spatial quantum phase are different characteristics of the state.
8. Energy and Momentum Operators
For a locally harmonic state, the time derivative is
\[\tag{42} \frac{\partial\Psi_{\mathcal S}}{\partial t} =-\frac{iE}{\hbar} \Psi_{\mathcal S}. \] This yields the standard operator representation of energy:
\[\tag{43} \boxed{ \widehat E =i\hbar\frac{\partial}{\partial t} }, \qquad \widehat E\Psi_{\mathcal S} =E\Psi_{\mathcal S}. \] And the spatial derivative along the path gives
\[\tag{44} \frac{\partial\Psi_{\mathcal S}}{\partial s} =\frac{ip_s}{\hbar} \Psi_{\mathcal S}, \] therefore
\[\tag{45} \boxed{ \widehat p_s =-i\hbar\frac{\partial}{\partial s} }, \qquad \widehat p_s\Psi_{\mathcal S} =p_s\Psi_{\mathcal S}. \] The form of the temporal operator is consistent with the harmonic change of the internal state in formula (11). The spatial operator arises after introducing the action along the physical trajectory. Therefore, the momentum operator does not follow directly from the parameter \(b\).
The transition from the numerical quantities \(E,p_s\) to the operators \(\widehat E,\widehat p_s\) is a standard quantum correspondence. Phase geometry motivates its temporal structure but does not replace the independent physical postulate of the operator mapping.
9. Energy of a moving particle
From the relativistic invariant (36), the positive branch of the energy has the form
\[\tag{46} E =\sqrt{m^2c^4+p_s^2c^2}. \] In the nonrelativistic limit \(|p_s|\ll mc\), we factor the rest energy out from under the square root:
\[\tag{47} E =mc^2 \sqrt{ 1+\frac{p_s^2}{m^2c^2} }. \] Using the expansion \(\sqrt{1+x}=1+x/2-x^2/8+\ldots\), we obtain
\[\tag{48} E =mc^2 +\frac{p_s^2}{2m} -\frac{p_s^4}{8m^3c^2} +\ldots. \] If we additionally introduce the potential energy \(U(s,t)\) and restrict ourselves to the first nonrelativistic approximation, then
\[\tag{49} \boxed{ E_{\mathrm{tot}} \approx mc^2 +\frac{p_s^2}{2m} +U(s,t) }. \] The kinetic term \(p_s^2/(2m)\), necessary for the Schrödinger equation, arises as the first correction to the rest energy. The potential \(U\) is not derived from the free operator \(J\): it represents an external interaction and must be specified by a separate dynamic model.
10. Isolation of the Mass Phase
We represent the total external state as a productfast mass phase and slowly changing shell \(\chi(s,t)\):
\[\tag{50} \boxed{ \Psi_{\mathrm{full}}(s,t) =e^{-imc^2t/\hbar} \chi(s,t) }. \] Its time derivative is
\[\tag{51} i\hbar \frac{\partial\Psi_{\mathrm{full}}}{\partial t} =e^{-imc^2t/\hbar} \left( mc^2\chi +i\hbar \frac{\partial\chi}{\partial t} \right). \] According to energy (49) and operator mapping (45), the full equation in the nonrelativistic approximation is written as
\[\tag{52} i\hbar \frac{\partial\Psi_{\mathrm{full}}}{\partial t} =\left( mc^2 +\frac{\widehat p_s^{,2}}{2m} +U \right) \Psi_{\mathrm{full}}. \] The same mass energy \(mc^2\) is present on both sides and cancels out. For the outer shell, there remains
\[\tag{53} i\hbar \frac{\partial\chi}{\partial t} =\left( \frac{\widehat p_s^{,2}}{2m} +U \right) \chi. \] Since
\[\tag{54} \widehat p_s^{,2} =-\hbar^2 \frac{\partial^2}{\partial s^2}, \] we obtain the one-dimensional Schrödinger equation:
\[\tag{55} \boxed{ i\hbar \frac{\partial\chi(s,t)}{\partial t} =-\frac{\hbar^2}{2m} \frac{\partial^2\chi(s,t)}{\partial s^2} +U(s,t)\chi(s,t) }. \] In the proposed interpretation, this equation describes the slow outer shell of the state after the internal mass periodicity has been extracted. Formula (55) is obtained by combining the mass geometry, the relativistic invariant, the phase of action, and the standard operator correspondence.
11. Two-Channel Idempotent Representation
The outer shell and the observed mass periodicity can be formally combined into a two-channel state:
\[\tag{56} \boxed{ \boldsymbol\Phi(s,t) =\ep\chi(s,t) +\em\psi_0(t) }. \] The corresponding two-channel operator is written as
\[\tag{57} \widehat{\mathcal H}^{(2)} =\ep\widehat H_{\mathrm{ext}} +\em\widehat H_0, \] where
\[\tag{58} \widehat H_{\mathrm{ext}} =-\frac{\hbar^2}{2m} \frac{\partial^2}{\partial s^2} +U(s,t), \qquad \widehat H_0=mc^2. \] Then the compact notation is
\[\tag{59} \boxed{ i\hbar \frac{\partial\boldsymbol\Phi}{\partial t} =\widehat{\mathcal H}^{(2)} \boldsymbol\Phi }. \] Due to the property \(\ep\em=0\), it splits into two independent equations:
\[\tag{60} i\hbar \frac{\partial\chi}{\partial t} =\widehat H_{\mathrm{ext}}\chi, \] \[\tag{61} i\hbar \frac{\partial\psi_0}{\partial t} =mc^2\psi_0. \] The state \(\boldsymbol\Phi=\ep\chi+\em\psi_0\) is a proposed direct union of the two channels, not an identical replacement for the standard product \(\Psi_{\mathrm{full}}=\psi_0\chi\). Formula (59) shows the architecture of the model, but does not prove the existence of a new physical wave function without an additional mapping rule.
12. Conservation of Probability and the Registration Rule
A standard probability density is introduced for the outer shell
\[\tag{62} \rho(s,t) =\chi^*(s,t)\chi(s,t). \] For a real potential \(U(s,t)\), the Schrödinger equation and its adjoint have the form
\[\tag{63} i\hbar\partial_t\chi =-\frac{\hbar^2}{2m} \partial_s^2\chi +U\chi, \] \[\tag{64} -i\hbar\partial_t\chi^* =-\frac{\hbar^2}{2m} \partial_s^2\chi^* +U\chi^*. \] Multiplying the first equality by \(\chi^*\), the second by \(\chi\), and subtracting them, we obtain the continuity equation:
\[\tag{65} \boxed{ \frac{\partial\rho}{\partial t} +\frac{\partial j_s}{\partial s} =0 }. \] Here \(j_s\) is the probability flux density:
\[\tag{66} j_s =\frac{\hbar}{2mi} \left( \chi^* \frac{\partial\chi}{\partial s} -\chi \frac{\partial\chi^*}{\partial s} \right). \] If there is no flux through the boundaries of the region, the cumulative probability is preserved:
\[\tag{67} \frac{d}{dt} \int|\chi(s,t)|^2\,ds =0. \] The geometric condition \(J\overline J=1\) and the quantum condition (67) belong to different levels of description. The former characterizes the finite phase norm, the latter the norm of the distributed external amplitude.
The identification of \(\rho=|\chi|^2\) with the detection probability is the Born rule. In the present construction, it is adopted as an additional physical law and is not derived solely from the unit norm of the operator \(J\).
13. From a One-Dimensional Path to a Three-Dimensional Amplitude
Formula (55) uses a single coordinate \(s\), measured along the physical trajectory \(\mathbf r(s)\). This corresponds to the statement that each local motion of a point occurs alongь one tangent:
\[\tag{68} \widehat{\boldsymbol\tau}(s) =\frac{d\mathbf r}{ds}, \qquad \left| \widehat{\boldsymbol\tau} \right|=1. \] For a spatial function, the derivative along the chosen path is
\[\tag{69} \frac{d}{ds} =\widehat{\boldsymbol\tau} \boldsymbol\cdot\nabla. \] The second derivative contains not only the second derivative in the tangent direction, but also the change in the tangent itself:
\[\tag{70} \frac{d^2\chi}{ds^2} =\sum_{i,j} \tau_i\tau_j \frac{\partial^2\chi} {\partial x_i\partial x_j} +\frac{d\widehat{\boldsymbol\tau}}{ds} \boldsymbol\cdot\nabla\chi. \] Therefore, for a single curve in the general case
\[\tag{71} \boxed{ \frac{d^2}{ds^2} \ne\nabla^2 }. \] To move to a three-dimensional equation, it is necessary to introduce the amplitude \(\chi(\mathbf r,t)\), which covers the family of admissible one-dimensional extensions. If the local propagation is isotropic and all spatial directions are treated equally, their combined second differential contribution is represented by the Laplace operator:
\[\tag{72} \nabla^2 =\frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2} +\frac{\partial^2}{\partial z^2}. \] After such spatial generalization, the external equation takes the form
\[\tag{73} \boxed{ i\hbar \frac{\partial\chi(\mathbf r,t)}{\partial t} =-\frac{\hbar^2}{2m} \nabla^2\chi(\mathbf r,t) +U(\mathbf r,t) \chi(\mathbf r,t) }. \] An individual recorded trajectory remains a one-dimensional curve in three-dimensional space. The three-dimensional amplitude, however, describes not the simultaneous motion of a point along three independent coordinates, but a family of admissible spatial extensions of the state.
14. What follows from standard physics and what is a hypothesis?
The standard physical relations used in the construction include:
1. relationship between energy and frequency \(E=\hbar\omega\);
2. action \(d\mathcal S=p_s\,ds-E\,dt\);
3. operator correspondence \(\widehat E=i\hbar\partial_t\), \(\widehat p_s=-i\hbar\partial_s\);
4. relativistic invariant \(E^2-p_s^2c^2=m^2c^4\);
5. its nonrelativistic expansion;
6. continuity equation for a real potential;
7. Born rule \(P\propto|\chi|^2\).
2. action \(d\mathcal S=p_s\,ds-E\,dt\);
3. operator correspondence \(\widehat E=i\hbar\partial_t\), \(\widehat p_s=-i\hbar\partial_s\);
4. relativistic invariant \(E^2-p_s^2c^2=m^2c^4\);
5. its nonrelativistic expansion;
6. continuity equation for a real potential;
7. Born rule \(P\propto|\chi|^2\).
From the algebra of a finite operator it follows:
1. expansion \(J(a,b)=\ep e^{i\pi b}+\em e^{i\pi a}\);
2. orthogonality of phase channels;
3. unit finite norm \(J\overline J=1\);
4. independence of internal phase \(a\) and external modulation \(b\);
5. phase change generator \(\widehat G_J\).
2. orthogonality of phase channels;
3. unit finite norm \(J\overline J=1\);
4. independence of internal phase \(a\) and external modulation \(b\);
5. phase change generator \(\widehat G_J\).
The physical hypotheses of Wave Electricity are:
1. Mapping of the internal phase onto the primary energy process;
2. Mass projection \(\Omega_C=\alpha_{\mathrm{fs}}\omega_{\mathrm{int}}\);
3. Deep self-similar splitting creating the norm \(\gamma\);
4. Mapping of finite projections \(\beta,1/\gamma\) onto momentum and energy;
5. Two-channel state \(\boldsymbol\Phi=\ep\chi+\em\psi_0\);
6. Representation of the three-dimensional amplitude as a family of admissible one-dimensional extensions.
2. Mass projection \(\Omega_C=\alpha_{\mathrm{fs}}\omega_{\mathrm{int}}\);
3. Deep self-similar splitting creating the norm \(\gamma\);
4. Mapping of finite projections \(\beta,1/\gamma\) onto momentum and energy;
5. Two-channel state \(\boldsymbol\Phi=\ep\chi+\em\psi_0\);
6. Representation of the three-dimensional amplitude as a family of admissible one-dimensional extensions.
The Schrödinger equation does not arise from a single algebraic operation. It is obtained at the intersection of intrinsic mass geometry, relativistic kinematics, phase of action, and the standard quantum operator mapping.
Conclusion
The full operator \(J(a,b)\) defines two independent sides of the final phase state. The intrinsic phase determines the primary frequency \(\omega_{\mathrm{int}}\), and the intrinsic mass projection transforms it into the Compton frequency \(\Omega_C\) and the rest energy \(mc^2\). The parameter \(b\) encodes the instantaneous external state of motion, but does not replace the accumulated phase of the action.
Deep splitting forms the norm \(\gamma\), and the physical mapping of the projections \(1/\gamma\) and \(\beta\) creates energy, momentum, and a relativistic invariant. Action along a trajectory introduces a spatiotemporal phase. After the standard operator correspondence and mass phase extraction, the outer nonrelativistic shell satisfies the Schrödinger equation.
\[\tag{74} \boxed{ \begin{aligned} J_{\mathrm{int}} &\longrightarrow \omega_{\mathrm{int}} \longrightarrow \Omega_C \longrightarrow mc^2, \\[3pt] \boldsymbol\Gamma_\beta &\longrightarrow \gamma, \qquad J(a,b) \longrightarrow \frac1\gamma,\beta \longrightarrow E,p_s, \\[3pt] d\mathcal S &=p_s\,ds-E\,dt \longrightarrow \widehat E,\widehat p_s \longrightarrow i\hbar\partial_t\chi =\widehat H_{\mathrm{ext}}\chi. \end{aligned} } \] TakiThus, phase geometry does not replace established quantum dynamics, but rather explains the internal state of a particle that precedes its external nonrelativistic description. The Schrödinger equation acts as an external equation for the slow amplitude of an already formed massive particle, while the internal phase periodicity and deep geometry of motion remain hidden levels of the complete state.
Materials used
- Wikipedia. Schrödinger Equation.