In a previous work, a new basis was proposed and an internal motion vector was constructed, describing both the constant and periodic components of the motion of a particle. It was shown that this geometric construction naturally separates translational motion and internal dynamics, and also links the natural frequency with the energy characteristics of the particle.

In this paper, the time evolution of this vector is considered. It is shown that its differentiation directly leads to the internal energy operator and allows one to obtain the time-dependent Schrödinger equation [1] without introducing additional postulates. In this case, the idempotent (em) itself acts as an algebraic projection onto the rotational component, and using the previously proposed relationship between the total internal energy and the rest energy, an expression for the particle's mass is derived through the frequency of its internal motion. Thus, the new basis unites the internal frequency, energy, mass, and time structure of quantum mechanics in a single geometric scheme.

An idempotent basis was previously introduced \[ \tag{1} \left\{ \ep,\;i\ep,\;\em,\;i\em \right\}, \] for which the following properties hold \[ \tag{2} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0. \] Such a basis allows one to represent motion as a combination of a constant component in the direction \(\ep\) and a periodic component rotating in the complex plane \((\em,i\em)\).

This paper examines the relationship between the internal motion vector and the energy, mass, and time-dependent Schrödinger equation. It will be shown that differentiation of the rotational component naturally leads to the energy operator, while the idempotent (\em\) itself acts as an algebraic projection onto the dynamic part of the vector.

It is necessary to distinguish two levels of construction. The energy scale (\hbar\omega\) directly arises from differentiation. The rest mass appears only after an additional physical assumption linking the total energy of internal motion to the expression (\gamma_{\mathrm{int}}mc^2\).

Consider the vector \[ \tag{3} J(t) = \ep + \em e^{-i\omega t}, \] where \(\omega\) is the angular frequency of the internal periodic process.

The \(\ep\) component is independent of time, while the component \[ \em e^{-i\omega t} \] rotates in the complex plane \[ \left\{ \em,\;i\em \right\}. \] Expanding the exponential gives \[ \tag{4} J(t) = \ep + \em\cos\omega t - i\em\sin\omega t. \] Thus, the end of the second component moves along the unit circle in the plane \((\em,i\em)\).

In general, the rotational component can contain an arbitrary initial phase: \[ \tag{5} J_{\varphi}(t) = \ep + \em e^{-i(\omega t+\varphi)}, \] where \(\varphi\) is the initial phase of the internal motion.

All vectors in the \(J_{\varphi}(t)\) family differ only in the choice of the time origin. Indeed, by replacing \[ \tag{6} t' = t+ \frac{\varphi}{\omega} \] we obtain \[ \tag{7} e^{-i(\omega t+\varphi)} = e^{-i\omega t'}. \] Therefore, changing the initial phase is equivalent to shifting the time origin and does not change the frequency, magnitude, or energy characteristics of the rotational component.

Therefore, without loss of generality, we can assume \[ \tag{8} \varphi=0 \] and use the form \[ \tag{9} J(t) = \ep + \em e^{-i\omega t}. \] The negative sign in the exponent corresponds to the standard time factor of stationary states in quantum mechanics and allows us to obtain the usual sign in the time-dependent Schrödinger equation.

Differentiating expression (9) with respect to time, we obtain \[ \tag{10} \frac{\partial J}{\partial t} = -i\omega\em e^{-i\omega t}. \] The constant component \(\ep\) disappears during differentiation, so the derivative isolates only the rotational part of the vector.

Multiply expression (10) by \(i\hbar\): \[ \tag{11} i\hbar \frac{\partial J}{\partial t} = \hbar\omega\em e^{-i\omega t}. \] The right-hand side has the dimension of energy and contains the standard quantum energy factor \[ \tag{12} W = \hbar\omega. \]

Thus, differentiating the vector \(J(t)\) directly leads to the energy scale of the internal periodic process. However, the expression \(\hbar\omega\) is still the total energy of the rotational component, not directly the rest energy of the particle.

Thanks to the properties of idempotent \[ \ep\em=0, \qquad \em^2=\em, \] multiplying the vector \(J(t)\) by \(\em\) extracts its rotational component: \[ \tag{13} \em J(t) = \em \left( \ep+ \em e^{-i\omega t} \right). \]

Expanding the product, we obtain \[ \tag{14} \em J(t) = \em\ep + \em^2e^{-i\omega t} = \em e^{-i\omega t}. \] Therefore, the idempotent \(\em\) itself already functions as an algebraic projection onto the dynamic sector of the vector. No additional projection is required for this.

Taking into account expression (14), the result of differentiation can be rewritten as \[ \tag{15} \frac{\partial J}{\partial t} = -i\omega\em J. \] After multiplying by \(i\hbar\), we obtain \[ \tag{16} i\hbar \frac{\partial J}{\partial t} = \hbar\omega\em J. \]

The standard time-domain Schrödinger equation [1] has View \[ \tag{17} i\hbar \frac{\partial\Psi}{\partial t} = \widehat H\Psi, \] where \(\widehat H\) is the Hamiltonian of the system.

Comparing expressions (16) and (17), we determine the Hamiltonian of the internal motion: \[ \tag{18} \widehat H_{\mathrm{int}} = \hbar\omega\em. \] Then the equation for the vector \(J(t)\) takes the form \[ \tag{19} i\hbar \frac{\partial J}{\partial t} = \widehat H_{\mathrm{int}}J. \]

Substituting Hamiltonian (18) yields \[ \tag{20} i\hbar \frac{\partial J}{\partial t} = \hbar\omega\em J. \] This equation is identical in structure to the time-dependent Schrödinger equation, but has an important feature: the Hamiltonian arises directly from the geometry of the idempotent basis.

Its action on the constant component is \[ \tag{21} \widehat H_{\mathrm{int}}\ep = \hbar\omega\em\ep = 0, \] while its action on the rotational component is \[ \tag{22} \widehat H_{\mathrm{int}} \left( \em e^{-i\omega t} \right) = \hbar\omega \em e^{-i\omega t}. \]

Thus, the vector \(J(t)\) consists of two energetically distinct sectors. The constant component \(\ep\) has zero energy with respect to the internal Hamiltonian, and the rotational component is an eigenstate with eigenvalue \[ \tag{23} W=\hbar\omega. \]

If we represent a vector in idempotent coordinates \[ \tag{24} J(t) \longleftrightarrow \begin{pmatrix} 1\\ e^{-i\omega t} \end{pmatrix}, \] then multiplication by \(\em\) corresponds to the matrix \[ \tag{25} \em \longleftrightarrow \begin{pmatrix} 0&0\\ 0&1 \end{pmatrix}. \]

Then the Hamiltonian of the internal motion has the form \[ \tag{26} \widehat H_{\mathrm{int}} = \begin{pmatrix} 0&0\\ 0&\hbar\omega \end{pmatrix}. \] The Schrödinger equation is written as \[ \tag{27} i\hbar \frac{\partial}{\partial t} \begin{pmatrix} 1\\ e^{-i\omega t} \end{pmatrix} = \begin{pmatrix} 0&0\\ 0&\hbar\omega \end{pmatrix} \begin{pmatrix} 1\\ e^{-i\omega t} \end{pmatrix}. \]

The left-hand side is equal to \[ \tag{28} i\hbar \begin{pmatrix} 0\\ -i\omega e^{-i\omega t} \end{pmatrix} = \begin{pmatrix} 0\\ \hbar\omega e^{-i\omega t} \end{pmatrix}, \] which is completely identical to the right-hand side. The matrix notation clearly shows the separation of the constant and dynamic sectors.

From expression (23), it follows that the rotational component is characterized by the total energy \[ \tag{29} W = \hbar\omega = h\nu, \] where \[ \tag{30} \omega=2\pi\nu. \] This is the standard quantum relationship between the frequency of a periodic process and its energy.

Within the model under consideration, it is assumed that this energy corresponds to the total relativistic energy of internal motion: \[ \tag{31} W = \gamma_{\mathrm{int}}mc^2, \] where \(m\) is the rest mass of the particle, and \(\gamma_{\mathrm{int}}\) is the hypothetical Lorentz factor of the internal dynamics.

Combining expressions (29) and (31), we obtain \[ \tag{32} \hbar\omega = \gamma_{\mathrm{int}}mc^2. \] Hence, the rest mass is equal to \[ \tag{33} m = \frac{\hbar\omega} {\gamma_{\mathrm{int}}c^2}. \]

Consequently, mass arises as a characteristic of an internal periodic process and is determined by its frequency, but taking into account the relationship between the total internal energy and the rest energy.

In previous work, the relationship was proposed \[ \tag{34} \gamma_{\mathrm{int}} = \frac{1}{\alpha_{\mathrm{fs}}}, \] where \(\alpha_{\mathrm{fs}}\) is the fine-structure constantы, and \(\gamma_{\mathrm{int}}\) is the internal Lorentz factor.

Then the expression for the mass takes the form \[ \tag{35} m = \frac{\alpha_{\mathrm{fs}}\hbar\omega} {c^2}. \] Accordingly, the rest energy is \[ \tag{36} mc^2 = \alpha_{\mathrm{fs}}\hbar\omega. \]

Since \[ \tag{37} W = \hbar\omega, \] we can write \[ \tag{38} mc^2 = \frac{W} {\gamma_{\mathrm{int}}} = \alpha_{\mathrm{fs}}W. \] Thus, the rest energy is the portion of the total energy of internal motion determined by the factor \(\alpha_{\mathrm{fs}}\).

The resulting relationship between frequency and mass is \[ \tag{39} \boxed{ m = \frac{\hbar\omega} {c^2 \gamma_{\mathrm{int}}} = \frac{\hbar\omega} {c^2} \alpha_{\mathrm{fs}} }. \]

The inverse expression allows us to determine the intrinsic frequency in terms of mass: \[ \tag{40} \omega = \frac{mc^2} {\hbar} \gamma_{\mathrm{int}} = \frac{mc^2} {\hbar \alpha_{\mathrm{fs}}}. \] For the usual frequency, we get \[ \tag{41} \nu = \frac{mc^2}{h} \gamma_{\mathrm{int}}. \]

Since the Hamiltonian of the internal motion is \[ \widehat H_{\mathrm{int}} = \hbar\omega\em, \] we can define the total energy mass operator: \[ \tag{42} \widehat M_{\mathrm{int}} = \frac{\widehat H_{\mathrm{int}}} {c^2}. \] Then \[ \tag{43} \widehat M_{\mathrm{int}} = \frac{\hbar\omega} {c^2}\em. \]

However, this operator corresponds to the total internal energy \(W\), and not directly to the rest mass. To obtain the rest mass operator, the internal Lorentz factor must be taken into account: \[ \tag{44} \widehat M_0 = \frac{1}{\gamma_{\mathrm{int}}} \widehat M_{\mathrm{int}}. \] From here \[ \tag{45} \widehat M_0 = \frac{\hbar\omega} {\gamma_{\mathrm{int}}c^2}\em. \]

Taking into account the connection \[ \gamma_{\mathrm{int}} = \frac{1}{\alpha_{\mathrm{fs}}}, \] we get \[ \tag{46} \widehat M_0 = \frac{\alpha_{\mathrm{fs}}\hbar\omega} {c^2}\em. \] Its eigenvalue on the rotational component is equal to the rest mass: \[ \tag{47} \widehat M_0 \left( \em e^{-i\omega t} \right) = m\em e^{-i\omega t}. \]

The result obtained allows the following consistent interpretation. The vector \[ J(t) = \ep + \em e^{-i\omega t} \] contains constant and rotational components. Differentiation eliminates the constant component and isolates the internal periodic process.

Multiplying the derivative by \(i\hbar\) transforms the angular frequency into an energy scale: \[ \tag{48} i\hbar \frac{\partial J}{\partial t} = \hbar\omega\em J. \] Therefore, the quantity \(\hbar\omega\) arises directly from the time evolution of the rotational component.

The idempotent \(\em\) is simultaneously an element of the basis and a projector onto the dynamical sector. This property allows us to write the Hamiltonian without introducing an additional operator: \[ \tag{49} \widehat H_{\mathrm{int}} = \hbar\omega\em. \]

Mass, however, is not a direct result of differentiation alone. The total internal energy follows directly from the derivative. \[ W=\hbar\omega. \] The transition to mass requires an additional physical relation. \[ W=\gamma_{\mathrm{int}}mc^2. \] It is this assumption that links the internal frequency to the observed rest mass.

Within this model, frequency is the primary characteristic of internal motion, the total energy is defined as \(\hbar\omega\), and the rest mass is a projection or fraction of this energy: \[ \tag{50} mc^2 = \frac{\hbar\omega} {\gamma_{\mathrm{int}}}. \]

The solution considered can be generalized by introducing constant complex amplitudes: \[ \tag{51} \Psi(t) = A\ep + B\em e^{-i\omega t}, \] where \(A\) and \(B\) are independent of time.

Differentiation yields \[ \tag{52} i\hbar \frac{\partial\Psi}{\partial t} = \hbar\omega B\em e^{-i\omega t}. \] On the other hand, \[ \tag{53} \em\Psi = B\em e^{-i\omega t}. \] Therefore, \[ \tag{54} i\hbar \frac{\partial\Psi}{\partial t} = \hbar\omega\em\Psi. \]

Therefore, the equation \[ \tag{55} i\hbar \frac{\partial\Psi}{\partial t} = \widehat H_{\mathrm{int}}\Psi, \qquad \widehat H_{\mathrm{int}} = \hbar\omega\em, \] is satisfied not only for the normalized vector \(J(t)\), but also for the entire family of states consisting of a constant \(\ep\)-component and a harmonic \(\em\)-component.

The resulting equation describes onlyto the time evolution of the internal periodic component. It currently lacks spatial derivatives, potential energy, and interaction with external fields. Therefore, the expression \[ i\hbar \frac{\partial J}{\partial t} = \hbar\omega\em J \] should be viewed as a time-dependent equation for a free internal state with a fixed frequency.

To construct a more general equation, it will be necessary to define a spatial operator in the same idempotent basis and establish a relationship between frequency, momentum, and external interactions. Only then will it be possible to compare the resulting construction with the full space-time Schrödinger equation.

Nevertheless, even at this stage it is clear that the standard time structure of quantum mechanics arises directly from the harmonic dependence of the rotational component of the new basis.

Differentiation of the vector \[ J(t) = \ep + \em e^{-i\omega t} \] separates its rotational component and leads to the expression \[ i\hbar \frac{\partial J}{\partial t} = \hbar\omega\em J. \] Since the idempotent \(\em\) annihilates the constant component \(\ep\) and preserves the rotational component, it itself acts as a projector onto the dynamic sector.

On this basis, the Hamiltonian of the internal motion is determined without introducing additional operators: \[ \widehat H_{\mathrm{int}} = \hbar\omega\em. \] The resulting equation \[ i\hbar \frac{\partial J}{\partial t} = \widehat H_{\mathrm{int}}J \] coincides in form with the time-dependent Schrödinger equation. The energy eigenvalue of the rotational component is equal to \(\hbar\omega\).

The rest mass arises from the additional assumption of the relationship between the total energy of internal motion and relativistic energy: \[ \hbar\omega = \gamma_{\mathrm{int}}mc^2. \] Hence \[ m = \frac{\hbar\omega} {\gamma_{\mathrm{int}}c^2}. \] Under the condition \(\gamma_{\mathrm{int}}=1/\alpha_{\mathrm{fs}}\), this expression takes the form \[ m = \frac{\alpha_{\mathrm{fs}}\hbar\omega} {c^2}. \]

Thus, in the proposed model, the internal frequency determines the total energy of a periodic process, and the rest mass is a derived characteristic related to this energy via the internal Lorentz factor. The new idempotent basis naturally separates the constant and dynamic sectors and leads to the time structure of the Schrödinger equation.