The total energy of a system of N moving particles $E$ is defined as the summation of the particles kinetic energy ${\displaystyle \underset{j=1}{\overset{N}{\sum }}\frac{1}{2}{m}_j{v}_j^2}$ and the total potential energy V

So that

$V=E-{\displaystyle \underset{j=1}{\overset{N}{\sum }}\frac{1}{2}{m}_j{v}_j^2}$ (1)

Equation (1) is rewritten as

$V={\displaystyle \underset{j=1}{\overset{N}{\sum }}-\left(\frac{1}{2}{v}_j-\frac{{\tilde{\alpha }}_{j}E+{\tilde{\beta }}_j}{m_j{v}_j}\right)m_j{v}_j}$ (2)

in which ${\displaystyle \underset{j=1}{\overset{N}{\sum }}{\tilde{\alpha }}_j}=1$, ${\displaystyle \underset{j=1}{\overset{N}{\sum }}{\tilde{\beta }}_j}=0$, $m_j{v}_j\ne 0$, $j=1,\cdots ,N$

where ${\tilde{\alpha }}_j$ is an energy distribution coefficient and ${\tilde{\beta }}_j$ is a complementary energy term. The physical significance for ${\tilde{\alpha }}_j$ and ${\tilde{\beta }}_j$ is given in section 4. ${\tilde{\alpha }}_j$ and ${\tilde{\beta }}_j$ are related to the particle total energy $E_j$ and the total energy of the system of particles $E$ by

$E_j={\tilde{\alpha }}_{j}E+{\tilde{\beta }}_j$, $j=1,\cdots ,N$ (3)

${\tilde{\alpha }}_j$ and ${\tilde{\beta }}_j$ are real functions, i.e. ${\tilde{\alpha }}_j\in R$ and ${\tilde{\beta }}_j\in R$ where $R$ is the set of real numbers.

Since ${\displaystyle \underset{j=1}{\overset{N}{\sum }}{\tilde{\beta }}_j}=0$ then

$-{\tilde{\beta }}_j={\displaystyle \underset{i=1}{\overset{N-1}{\sum }}{\tilde{\beta }}_i}$, $j=1,\cdots ,N$ (4)

Equation (4) represents the complementary energy concept which interprets the impact of a system of particles on each particle in the system. The left hand side of Equation (4) gives the value of the complementary energy for particle j whereas the right hand side is the summation of the complementary energies for the rest N − 1 particles.

Substituting in Equation (2) from Equation (3) and writing

$\left(\frac{1}{2}{v}_j-\frac{{ℰ}_j}{m_j{v}_j}\right)={\bar{v}}_j$, $m_j{v}_j\ne 0$, $j=1,\cdots ,N$ (5)

in which ${\bar{v}}_j$ is a velocity parameter, yields the expressions for the total potential energy and the particle potential energy as

$V={\displaystyle \underset{j=1}{\overset{N}{\sum }}{V}_j}$, $V_j=-m_j{v}_j{\bar{v}}_j$, $j=1,\cdots ,N$ (2.1)

The above relation expresses a general format for any potential energy for any moving particle independent completely of the kind of the field it represents.

Expressing Equation (5) as

${\bar{v}}_j{v}_j=\frac{1}{2}{v}_j^2-\frac{E_j}{m_j}$, $j=1,\cdots ,N$

Gives the particle velocity quadratic equation as

$v_j^2-2{\bar{v}}_j{v}_j-\frac{2E_j}{m_j}=0$, $j=1,\cdots ,N$ (6)

The roots of Equation (6) are $v_{j,1,2}$ given by

$v_{j,1,2}=\frac{2{\bar{v}}_j\pm \sqrt{4{\bar{v}}_j^2+4\frac{2E_j}{m_j}}}{2}={\bar{v}}_j\pm \sqrt{{\bar{v}}_j^2+\frac{2E_j}{m_j}}$, $j=1,\cdots ,N$

The roots of Equation (6) result in the existence of two available velocities. Two classical interpretations are given, the first classical interpretation of the above relation is a pair particle velocities prevail along the particle path trajectory. The second classical interpretation of the above relation is one speed for transitional motion and another speed for rotational motion. The quantum interpretation of the above relation is given in detail in section 2.7.

The particle mean velocity, ${\tilde{v}}_j$, is expressed as

${\tilde{v}}_j=\frac{v_{j,1}+v_{j,2}}{2}$, $j=1,\cdots ,N$

Expressing the roots of Equation (6) as

$v_{j,1,2}={\tilde{v}}_j\pm d_j$, $j=1,\cdots ,N$ (7)

in which

$d_j=\sqrt{{\tilde{v}}_j^2+{\tilde{v}}_{\perp j}^2}$, ${\tilde{v}}_{\perp j}=\sqrt{\frac{2E_j}{m_j}}$, $j=1,\cdots ,N$ (7.1)

gives ${\bar{v}}_j={\tilde{v}}_j$ and $d_j$ to be the deviation from the mean velocity; given the term deviation velocity of particle j.

${\bar{v}}_j$ is substituted by ${\tilde{v}}_j$ in all further equations.

A proof that the value of the mean velocity is constant and is given as follows

Rewriting Equation (7) as

${\left(v_j-{\tilde{v}}_j\right)}^2={\tilde{v}}_j^2+\frac{2E_j}{m_j}$, $j=1,\cdots ,N$

$E_j\left(v_j\right)=\frac{1}{2}{m}_j{\left(v_j-{\tilde{v}}_j\right)}^2-\frac{1}{2}{m}_j{\tilde{v}}_j^2$, $j=1,\cdots ,N$ (7.2)

Equation (7.2) is deduced by expanding $E_j\left(v_j\right)$, into Taylor series around an extremal point, ${\tilde{v}}_j$, and substituting ${E^{\prime }}_j\left({\tilde{v}}_j\right)=0$ (the condition for extremum) in the second term. The terms above the second order are null. The condition for extremum ${E^{\prime }}_j\left({\tilde{v}}_j\right)=0$ indicates that if the velocity of the moving particle j reaches the value ${\tilde{v}}_j$ then it does not contribute to its energy $E_j$, i.e. the moving particle is in a collision free motion or in a perfect collision motion (no change in energy due collision) or both.

The explanation of Equation (4) whereas the summation in the right side of the equation calculates the net collective effect of N − 1 particles upon a single particle, based on the particle’s location and particle’s status whether the effect is due to direct contact as in case of collision or via remote influence through an existing field in the space. The left hand side of Equation (4) gives the value of the complementary energy for particle j. For a particle near the surface of the material the right hand side summation represents the required amount of work done by the rest N − 1 particles to pull back the particle away from the surface of the material. The complementary energy can be understood as the potential energy at the boundary of the material which prevents each particle from escaping from the boundary of the system (evaporates).

From Equation (3) in section 2.1 the total energy of particle j is written

$E_j={\tilde{\alpha }}_{j}E+{\tilde{\beta }}_j$

In special case ${\tilde{\alpha }}_j=0$ Equation (3) becomes

$E_j={\tilde{\beta }}_j$

Substituting in Equation (3) with the kinetic energy and the potential energy of particle j gets

$E_j={\tilde{\alpha }}_{j}E+{\tilde{\beta }}_j\to {\tilde{\alpha }}_{j}E=\frac{1}{2}{m}_j{v}_j^2+V_j-{\tilde{\beta }}_j$

According to the argument given above the value of the complementary energy $-{\tilde{\beta }}_j$ is independent form the potential energy value $V_j$, so in case the potential energy $V_j$ of particle j vanishes at certain positions. i.e. $V_j$ values halt contributing to the total energy of particle j, the complementary energy $-{\tilde{\beta }}_j$ still contributing to the total energy of particle j then the above equation is written

${\tilde{\alpha }}_{j}E=\frac{1}{2}{m}_j{v}_j^2-{\tilde{\beta }}_j$

where $-{\tilde{\beta }}_j$ can be considered as potential energy,

So the general format for any potential energy of a particle j given by Equation (2.1) is adequate to represent the value of the complementary energy of particle j. For a particle j its complementary energy is expressed by the equation

${\tilde{\beta }}_j=m_j{v}_{j,1,2}{\tilde{v}}_j$ (2.2)

The probabilistic formulation of the quantum theory indicates an assumption of the impossibility to measure an exact value for the velocity but you can measure a probable value some time called expected value or average value, ${\tilde{v}}_j$ represents the measured value of the velocity for a moving particle j. Another aspect of the quantum formulation is an assumption of the presence of an associated wave to any moving particle [1] in other word the associated wave adheres the moving particle composing a particle-wave object. The adherence of the wave to the moving particle enables the wave to borrow one of the available speeds of the particle to be the speed of the associated wave, consequently $v_{j,1,2}$ can represent the speed of the associated wave. ${\tilde{\beta }}_j$ is a representation of complementary energy for the particle-wave object which is a bi-natural for both a moving particle and moving wave and Equation (2.2) is a particle-wave equation. The determination of the complementary energy ${\tilde{\beta }}_j$ requires two velocities values, one for a particle and another for a wave, so ${\tilde{\beta }}_j$ is a potential energy of a combined particle-wave object. Another application for ${\tilde{\beta }}_j$ is to express the motion of a moving object in a continuous medium (fluid, gas, electric field, magnetic field, gravitational field.) by considering the average velocity is the speed of the wave caused in the medium due to the motion of the object in the medium with one of its available speeds. The potential energy replaces the complementary energy in case of single object.

Tiny particles such as electrons, protons, atoms, etc., their velocity value can reach the speed of light or even close to the speed of light, so c the speed of light could be considered an available speed for the tiny particles.

The substitution with the value of the speed of light c in the particle-wave equation for a tiny particle j i.e. $v_{j,1}=c$ gives

${\tilde{\beta }}_j=m_j{\tilde{v}}_{j}c$

Since the speed of light represents a moving wave of frequency ν and wavelength λ satisfying the relation $c=\lambda \nu$ the above relation is rewritten as

${\tilde{\beta }}_j=m_j{\tilde{v}}_j\lambda v=hv$ (8)

Where

$h=m_j{\tilde{v}}_j\lambda$ (8.1)

$p_j=m_j{\tilde{v}}_j,p_j=\frac{h}{\lambda }$ (8.2)

In section 2.3 a proof is given indicating that ${\tilde{v}}_j$ is a constant extremal value for the energy function $E_j$, since $p_j$ and λ are both constant values this result that the value of h given by relation (8.1) is a constant value which is Plank’s constant [1] [2] .

Relation (8.2) is a reformulation of relation (8.1) indicating that the momentum of the wave-particle object is equal to the momentum on a perfect path i.e. the particle-wave object moves as same way like a free motion particle (review section 2.3) [1] [2] .

Relation (8) is an expression representing the complementary energy of a moving particle j as a multiplication of a wave frequency ν by a constant value h. Relations (8) and (8.2) are obtained by De Broglies using intuitive empirical method [1] [2] .

In this section the value of the second velocity $v_{j,2}$ is considered to be the rotating velocity for particle j. Using the particle mean velocity equation given in section 2.3 the second velocity with a magnitude $v_{j,2}=v_j$ is expressed as

$v_j=2{\tilde{v}}_j-c$ (9)

Since the angular velocity ${\omega }_j={\stackrel{\dot{}}{\varphi }}_j$ and $R_j$ is the distance from the rotating center then, Equation (9) can be rewritten as

$R_j{\omega }_j=\text{2}{\tilde{v}}_j-c\Rightarrow {\stackrel{\dot{}}{\varphi }}_j=\frac{2\pi }{R_j}\frac{{\tilde{v}}_j}{\pi }-\frac{{\lambda }_j{\nu }_1}{R_j}$

where ${\nu }_1$, is the first wave frequency obtained in the previous section

The above equation in terms of the wave variables is written as

${\stackrel{\dot{}}{\varphi }}_j=k_j{v}_{jp}-{\nu }_2$ (9.1)

The following are the conversion relations from particle variables to the wave variables:

In case $2R_j\ge {\lambda }_j$, $2R_j=n_j{\lambda }_j$, where $n_j$ is a natural number.

In case $n_j=2$, $R_j={\lambda }_j$ the magnitude of the wave vector $k$ becomes $\frac{2\pi }{{\lambda }_j}$ which is well known result obtained by quantum method [1] .

In case the rotating center inside the particle or on its boundary the rotation is considered spinning, if the rotating center outside the particle the rotation is considered revolving. Reflecting and refracting (bending) of particle-wave object are considered a kind of rotation.

Integrating Equation (9.1) gives

${\varphi }_j={\varphi }_{0j}+k{s}_j-{\nu }_{2}t$ (9.2)

Equation (9.2) is valid in the interval, $0\le s_j\le v_{pj}{\tau }_j$, $0\le t\le {\tau }_j$, and describing the phase of a plane wave propagating along the particle displacement direction s_j .

Substituting from (8.1) into the phase velocity relation gives, $v_{jp}=\frac{h}{\pi m_j{\lambda }_j}$, so a dispersion takes place, each velocity generates a corresponding wave with the same wave length λ and two frequencies ${\nu }_1,{\nu }_2$ related by the relation ${\nu }_2=\frac{2{\nu }_1}{n_j}$ the addition of the two waves creates a wave packet [1] .

Substituting in Equation (8.2) with the wave length in terms of the distance from the rotating center gives

$p_j=n_j\frac{h}{2R_j}$

From Equation (7.2), Equation (8.2), and the above relation the energy of a particle j moving with speed ${\tilde{v}}_j$ is

$E_j\left({\tilde{v}}_j\right)=-\frac{1}{2}{m}_j{\tilde{v}}_j^2=-\frac{p_j^2}{2m_j}=-n_j^2\frac{h^2}{8m_j{R}_j^2}$

The above formula is known as the energy quantization formula obtained by the methods of quantum mechanics [1] [2] .

Schrodinger’s formulation describes the motion of moving particles using a wave equation which is a partial differential equation its solution is a wave function . The square value of the wave function results the corresponding probability value for the parameters of the moving particle such as speed, energy, and displacement . In order to switch the deterministic relations to be expressed in probabilistic format the following procedure adopted to achieve successful convergence process.

From section 2.3 the particle mean velocity, ${\tilde{v}}_j$, is expressed as

${\tilde{v}}_j=\frac{v_{j,1}+v_{j,2}}{2}=\frac{1}{2}{v}_{j,1}+\frac{1}{2}{v}_{j,2}$, $j=1,\cdots ,N$

The particle mean velocity relation can be interpreted as follows the probability of the first speed $v_{j,1}$ is $\frac{1}{2}$ and the probability of the second speed $v_{j,2}$ is $\frac{1}{2}$ then ${\tilde{v}}_j$ is the value of expected speed and that is the quantum interpretation of the particle mean velocity relation. Instead of using the equal fixed probabilities value $\frac{1}{2}$ for each speed which is a special case, a variable functions can replace the equal fixed probabilities value $\frac{1}{2}$. There are two available sets of functions can fully fill the replacing purpose explained previously. The first set is trigonometric functions $\text{sin}\theta ,\cos \theta$ where ${\sin }^2\theta +{\cos }^2\theta =1$. The second set is hyperbolic functions

$-{\sinh }^2\theta ,{\cosh }^2\theta$ where $-{\sinh }^2\theta +{\cosh }^2\theta =1$ in this article the trigonometric functions used to represent the required probabilities.

From section 2.3

$v_{j,1,2}={\tilde{v}}_j\pm d_j$, $j=1,\cdots ,N$ (7)

in which

$d_j=\sqrt{{\tilde{v}}_j^2+{\tilde{v}}_{\perp j}^2}$, ${\tilde{v}}_{\perp j}=\pm \sqrt{\frac{2E_j}{m_j}}$, $j=1,\cdots ,N$ (7.1)

Figure 1 is the geometrical representation of Equation (7.1) is a right triangle its sides ${\tilde{v}}_j$, ${\tilde{v}}_{\perp j}$, and $d_j$. It will be shown the fundamental contribution of the three sides of the triangle as well as the angel ${\tilde{\theta }}_j$ in determining and finding the appropriate structure of the wave function and the wave vector as a solution for Schrodinger’s equation.

From Figure 1 the geometrical representation of Equation (7.1) $\sin {\theta }_j$ is written as follows:

$\pm \sin {\theta }_j=\pm \frac{\sqrt{\frac{2E_j}{m_j}}}{d_j}$

Substituting ${\theta }_j=k_{0j}{x}_j$ where $x_j$ is the displacement of particle j. Specifying the value of $k_{0j}$ requires three steps, first step is expanding $\sin k_j{x}_j$ in power series and second step is equating the expansion with the above relation and third step is adjusting the $\sin k_{0j}{x}_j$ ratio to match the value of $k_{0j}$. The previous steps will be implemented as follows:

$\sin k_{0j}{x}_j=k_{0j}{x}_j-\frac{{\left(k_{0j}{x}_j\right)}^3}{3!}+\frac{{\left(k_{0j}{x}_j\right)}^5}{5!}-\frac{{\left(k_{0j}{x}_j\right)}^7}{7!}+\frac{{\left(k_{0j}{x}_j\right)}^9}{9!}-\cdots$

$\frac{\sqrt{\frac{2E_j}{m_j}}}{d_j}=\frac{1}{\hslash }\sqrt{2m_j{E}_j}\left(\frac{\hslash }{m_j{d}_j}\right)=k_{0j}\left(x_j-\frac{k_{0j}^2{\left(x_j\right)}^3}{3!}+\frac{k_{0j}^4{\left(x_j\right)}^5}{5!}-\frac{k_{0j}^6{\left(x_j\right)}^7}{7!}+\frac{k_{0j}^8{\left(x_j\right)}^9}{9!}-\cdots \right)$

Comparing the left side with right side of the above equation results.

$k_{0j}=\frac{1}{\hslash }\sqrt{2m_j{E}_j}$

Substituting with $k_{0j}$ value into $\sin k_{0j}{x}_j$ gives.

A wave function ${\psi }_j$ for a particle j is written as:

${\psi }_j=\sin k_{0j}{x}_j$

The same wave function ${\psi }_j$ for a particle j can be obtained also by solving the Schrodinger’s amplitude equation given as [1] [3] :

$\frac{{\text{d}}^2{\psi }_j}{\text{d}{x}_j^2}+\frac{2m_j}{{\hslash }^2}{E}_j{\psi }_j=0$ or $\frac{{\text{d}}^2{\psi }_j}{\text{d}{x}_j^2}+k_{0j}^2{\psi }_j=0$ where $k_{0j}=\frac{1}{\hslash }\sqrt{2m_j{E}_j}$

The wave function ${\psi }_j$ is the solution of the above Schrodinger’s equation after applying the boundary conditions, to describe the motion of a particle j enclosed in a potential well where the potential energy vanishes inside the well, Also the wave function ${\psi }_j$ is the solution of the problem of passage of particles through the potential barrier. Tunnel effect [1] [3] .

Substituting in $\cos {\theta }_j$ with ${\theta }_j=k_{1j}{x}_j$ and From Figure1 the geometrical representation of Equation (7.1) $\cos {\theta }_j$ is written as follows

$\cos {\theta }_j=\cos k_{1j}{x}_j=\frac{{\tilde{v}}_j}{d_j}$

From section (2.3) the relation between the total energy of particle j $E_j$ and its potential energy $V_j$ is given as

$E_j\left(v_j\right)=\frac{1}{2}{m}_j{\left(v_j-{\tilde{v}}_j\right)}^2-\frac{1}{2}{m}_j{\tilde{v}}_j^2=-\frac{1}{2}{m}_j{\tilde{v}}_j^2+V_j$, $V_j=\frac{1}{2}{m}_j{\left(v_j-{\tilde{v}}_j\right)}^2$, $j=1,\cdots ,N$ (7.2)

The above formula is valid for values of $v_j$ and ${\tilde{v}}_j^2$ satisfying the relation $\frac{1}{2}{m}_j\left(v_j^2+{\tilde{v}}_j^2\right)=0$.

${\tilde{v}}_j=\frac{v_{j,1}+v_{j,2}}{2}\to v_{j,1}^2+{\tilde{v}}_j^2=v_{j,1}^2+\frac{{\left(v_{j,1}+v_{j,2}\right)}^2}{4}=0$ which leads to the quadratic equation $5v_{j,1}^2+2v_{j,2}{v}_{j,1}+v_{j,2}^2=0$ the same for $v_{j,2}$ we get the quadratic equation $5v_{j,2}^2+2v_{j,1}{v}_{j,2}+v_{j,1}^2=0$.

From the expansion of $\cos k_{1j}{x}_j$ in power series and the above formula and adjusting the $\cos k_{1j}{x}_j$ ratio obtained from Figure 1 the geometrical representation of Equation (7.1) to match $k_{1j}$ value we get

$\begin{array}{c}\frac{{\tilde{v}}_j}{d_j}={\tilde{v}}_j\frac{\hslash }{\hslash d_j}=\pm \frac{1}{\hslash }\sqrt{2m_j\left(V_j-E_j\right)}\left(\frac{\hslash }{m_j{d}_j}\right)\\ =\pm k_{1j}\left(\frac{1}{k_{1j}}-\frac{k_{1j}{\left(x_j\right)}^2}{2!}+\frac{k_{1j}^3{\left(x_j\right)}^4}{4!}-\frac{k_{1j}^5{\left(x_j\right)}^6}{6!}+\frac{k_{1j}^7{\left(x_j\right)}^8}{8!}-\cdots \right)\end{array}$

Comparing the left side with right side of the above equation results

$\pm k_{1j}=\pm \frac{1}{\hslash }\sqrt{2m_j\left(V_j-E_j\right)}$

A wave function ${\psi }_j$ for a particle j is written as

${\psi }_j=\cos k_{1j}{x}_j$

The same wave function ${\psi }_j$ for a particle j can be obtained also by solving the Schrodinger’s amplitude equation given as [1] .

$\frac{{\text{d}}^2{\psi }_j}{\text{d}{x}_j^2}+k_{1j}^2{\psi }_j=0$ where $\pm k_{1j}=\pm \frac{1}{\hslash }\sqrt{2m_j\left(V_j-E_j\right)}$

The wave function ${\psi }_j$ is the solution of the above Schrodinger’s equation after applying the boundary conditions, to describe the motion of a particle j in a region where the potential energy has a value $V_j$ [1] .

In this section proofs are given for some wave functions to be solutions for time dependent Schrodinger’s equation based on the results obtained in the previous section.

Some relations required to derive the proposed solutions for time dependent Schrodinger’s equation. The purpose of these relations is to transform the particle parameters to the wave parameters to be adequate to Schrodinger’s equation.

We have the deviation $d_j$ is given as follows

$d_j=\sqrt{{\tilde{v}}_j^2+{\tilde{v}}_{\perp j}^2}$, ${\tilde{v}}_{\perp j}=\pm \sqrt{\frac{2E_j}{m_j}}$, $j=1,\cdots ,N$ (7.1)

$d_j^2={\tilde{v}}_j^2+\frac{2E_j}{m_j}\to m_j^2{d}_j^2=m_j^2{\tilde{v}}_j^2+2m_j{E}_j$

From the previous section we have

$k_{0j}=\frac{1}{\hslash }\sqrt{2m_j{E}_j}\to {\hslash }^2{k}_{0j}^2=2m_j{E}_j$

$\pm k_{1j}=\pm \frac{1}{\hslash }\sqrt{2m_j\left(V_j-E_j\right)}\to {\hslash }^2{k}_{1j}^2=2m_j\left(V_j-E_j\right)$

Using the relation (7.2) and the above relation we get

$E_j\left(v_j\right)=-\frac{1}{2}{m}_j{\tilde{v}}_j^2+V_j\to m_j^2{\tilde{v}}_j^2=2m_j\left(V_j-E_j\right)={\hslash }^2{k}_{1j}^2$

From the above relations $m_j^2{d}_j^2$ is rewritten as follows

$m_j^2{d}_j^2={\hslash }^2{k}_{0j}^2+{\hslash }^2{k}_{1j}^2$

$$Now we write the functions $\sin {\theta }_j$ and $\cos {\theta }_j$ in terms of $k_{0j}$and $k_{1j}$ from the previous section and the substitution with value of $m_j^2{d}_j^2$ given above we get

${\sin }^2{\theta }_{1j}=\frac{k_{0j}^2{\hslash }^2}{m_j^2{d}_j^2}=\frac{k_{0j}^2{\hslash }^2}{{\hslash }^2{k}_{0j}^2+{\hslash }^2{k}_{1j}^2}=\frac{k_{0j}^2}{k_{0j}^2+k_{1j}^2},$

${\cos }^2{\theta }_{2j}=\frac{k_{1j}^2{\hslash }^2}{m_j^2{d}_j^2}=\frac{k_{1j}^2{\hslash }^2}{{\hslash }^2{k}_{0j}^2+{\hslash }^2{k}_{1j}^2}=\frac{k_{1j}^2}{k_{0j}^2+k_{1j}^2}$

Using the equality ${\sin }^2\theta +{\cos }^2\theta =1$ and multiplying by i we get

$i\cos {\theta }_{1j}=\sin {\theta }_{1j}\sqrt{1-\frac{1}{{\sin }^2{\theta }_{1j}}}$, $i\sin {\theta }_{2j}=\cos {\theta }_{2j}\sqrt{1-\frac{1}{{\cos }^2{\theta }_{2j}}}$

$i\cos {\theta }_{1j}=\sin {\theta }_{1j}\sqrt{\frac{k_{0j}^2-\left(k_{0j}^2+k_{1j}^2\right)}{k_{0j}^2}}=\frac{i{k}_{1j}}{k_{0j}}\sin {\theta }_{1j}$

$i\sin {\theta }_{2j}=\cos {\theta }_{2j}\sqrt{\frac{k_{1j}^2-\left(k_{0j}^2+k_{1j}^2\right)}{k_{1j}^2}}=\frac{i{k}_{0j}}{k_{1j}}\cos {\theta }_{2j}$

One dimension functions to be solutions for time dependent Schrodinger’s equation

Is a solution for time dependent Schrodinger’s equation given as [1]

$-i\hslash \frac{\partial {\psi }_j\left(x,t\right)}{\partial t}=\frac{{\hslash }^2}{2m_j}\frac{{\partial }^2{\psi }_j\left(x,t\right)}{\partial x^2}-V_j\left(x\right){\psi }_j\left(x,t\right)$

Proof

The first term on the right hand side

$\frac{{\hslash }^2}{2m_j}\frac{{\partial }^2{\psi }_j\left(x,t\right)}{\partial x^2}=-\frac{{\hslash }^2}{2m_j}{k}_{0j}^2\sin {\theta }_{1j}=-E_j\sin {\theta }_{1j}$

$\text{R}\text{.H}\text{.S}=-E_j\sin {\theta }_{1j}-V_j\left(x\right)\sin {\theta }_{1j}=-\left(E_j+V_j\right)\sin {\theta }_{1j}=-k_{2j}^2\sin {\theta }_{1j}$

The left hand side

$-i\hslash \frac{\partial {\psi }_j\left(x,t\right)}{\partial t}=-\hslash \frac{k_{0j}}{i\hslash k_{1j}}{k}_{2j}^{2}i\cos {\theta }_{1j}=-\hslash \frac{k_{0j}}{i\hslash k_{1j}}{k}_{2j}^2\frac{i{k}_{1j}}{k_{0j}}\sin {\theta }_{1j}=-k_{2j}^2\sin {\theta }_{1j}=\text{R}\text{.H}\text{.S}$

Let the frequency $v=\frac{k_{0j}}{ih{k}_{1j}}{k}_{2j}^2$ in case the potential energy $V_j=0$ the frequency becomes $v=\frac{k_{0j}}{ih{k}_{1j}}{k}_{2j}^2=\frac{\frac{1}{\hslash }\sqrt{2m_j{E}_j}}{i\hslash \frac{1}{\hslash }\sqrt{-2m_j{E}_j}}{E}_j=\frac{E_j}{\hslash }\to E_j=\hslash v$ comparing this result with De Broglie’s relations obtained in section (2.5) ${\tilde{\beta }}_j=hv$ using the relation $E_j={\tilde{\alpha }}_{j}E+{\tilde{\beta }}_j$ we get

${\tilde{\alpha }}_{j}E=\hslash v-hv=v\left(\hslash -h\right)=hv\left(\frac{1}{2\pi }-1\right)$

is a solution for time dependent Schrodinger’s equation.

Proof

The first term on the right hand side

$\frac{{\hslash }^2}{2m_j}\frac{{\partial }^2{\psi }_j\left(x,t\right)}{\partial x^2}=-\frac{{\hslash }^2}{2m_j}{k}_{1j}^2\cos {\theta }_{2j}=-\left(V_j-E_j\right)\cos {\theta }_{2j}$

$\text{R}\text{.H}\text{.S}=-\left(V_j-E_j\right)\cos {\theta }_{2j}-V_j\cos {\theta }_{2j}=\left(E_j-2V_j\right)\cos {\theta }_{2j}=k_{3j}^2\cos {\theta }_{2j}$

The left hand side

$-i\hslash \frac{\partial {\psi }_j\left(x,t\right)}{\partial t}=\hslash \frac{k_{1j}}{i\hslash k_{0j}}{k}_{3j}^{2}i\sin {\theta }_{2j}=\hslash \frac{k_{1j}}{i\hslash k_{0j}}{k}_{3j}^2\frac{i{k}_{0j}}{k_{1j}}\cos {\theta }_{2j}=k_{3j}^2\cos {\theta }_{2j}=\text{R}\text{.H}\text{.S}$

Proof

The first term on the right hand side

$\frac{{\hslash }^2}{2m_j}\frac{{\partial }^2{\psi }_j\left(x,t\right)}{\partial x^2}=\frac{{\hslash }^2}{2m_j}{k}_{0j}^2{\text{e}}^{{\theta }_{3j}}=E_j{\text{e}}^{{\theta }_{3j}}$

$\text{R}\text{.H}\text{.S}=E_j{\text{e}}^{{\theta }_{3j}}-V_j{\text{e}}^{{\theta }_{3j}}=-\left(V_j-E_j\right){\text{e}}^{{\theta }_{3j}}=-\frac{{\hslash }^2}{2m_j}{k}_{1j}^2{\text{e}}^{{\theta }_{3j}}$

The left hand side

$-i\hslash \frac{\partial {\psi }_j\left(x,t\right)}{\partial t}=-i\hslash \frac{\hslash }{2m_j}\frac{k_{1j}^2}{i}{\text{e}}^{{\theta }_{3j}}=-\frac{{\hslash }^2}{2m_j}{k}_{1j}^2{\text{e}}^{{\theta }_{3j}}=\text{R}\text{.H}\text{.S}$

$$Proof

The first term on the right hand side

$\frac{{\hslash }^2}{2m_j}\frac{{\partial }^2{\psi }_j\left(x,t\right)}{\partial x^2}=\frac{{\hslash }^2}{2m_j}{k}_{4j}^2{\text{e}}^{{\theta }_{4j}}=\left(V_j-E_j\right){\text{e}}^{{\theta }_{4j}}$

$\text{R}\text{.H}\text{.S}=\left(V_j-E_j\right){\text{e}}^{{\theta }_{4j}}-V_j{\text{e}}^{{\theta }_{4j}}=-E_j{\text{e}}^{{\theta }_{4j}}=-\frac{{\hslash }^2}{2m_j}{k}_{0j}^2{\text{e}}^{{\theta }_{4j}}$

The left hand side

$-i\hslash \frac{\partial {\psi }_j\left(x,t\right)}{\partial t}=-i\hslash \frac{\hslash }{2m_j}\frac{k_{0j}^2}{i}{\text{e}}^{{\theta }_{4j}}=-\frac{{\hslash }^2}{2m_j}{k}_{0j}^2{\text{e}}^{{\theta }_{4j}}=\text{R}\text{.H}\text{.S}$

Three dimensions functions to be solutions for time dependent Schrodinger’s equation

$k_{x0j}=c_1{k}_{0j},k_{y0j}=c_2{k}_{0j},k_{z0j}=c_3{k}_{0j}$ where $c_1^2+c_2^2+c_3^2=1$

Suggested values for $c_1,c_2,c_3$ $c_1=\frac{1}{\sqrt{3}},c_2=\frac{1}{\sqrt{3}},c_3=\frac{1}{\sqrt{3}}$ or $c_1=\frac{1}{\sqrt{2}},c_2=\frac{1}{2},c_3=\frac{1}{2}$ or $c_1=\sqrt{\frac{1}{5}},c_2=\sqrt{\frac{2}{5}},c_3=\sqrt{\frac{2}{5}}$

$\sqrt{k_{x0j}^2+k_{y0j}^2+k_{z0j}^2}=\sqrt{k_{0j}^2\left(c_1^2+c_2^2+c_3^2\right)}=k_{0j}\to k_{0j}^2=k_{x0j}^2+k_{y0j}^2+k_{z0j}^2$

is a solution for time dependent Schrodinger’s equation given as [1] [4] .

$-i\hslash \frac{\partial {\psi }_j}{\partial t}=\frac{{\hslash }^2}{2m_j}\left(\frac{{\partial }^2{\psi }_j}{\partial x^2}+\frac{{\partial }^2{\psi }_j}{\partial y^2}+\frac{{\partial }^2{\psi }_j}{\partial z^2}\right)-V_j{\psi }_j$

Proof

The first term on the right hand side

$\begin{array}{c}\frac{{\hslash }^2}{2m_j}\left(\frac{{\partial }^2{\psi }_j}{\partial x^2}+\frac{{\partial }^2{\psi }_j}{\partial y^2}+\frac{{\partial }^2{\psi }_j}{\partial z^2}\right)=-\frac{{\hslash }^2}{2m_j}\left(k_{x0j}^2+k_{y0j}^2+k_{z0j}^2\right)\sin {\theta }_{1j}\\ =-\frac{{\hslash }^2}{2m_j}{k}_{0j}^2\sin {\theta }_{1j}=-E_j\sin {\theta }_{1j}\end{array}$

Continue the proof the same steps as the proof in one dimension function.

The argument ${\theta }_{1j}$ can be rewritten

${\theta }_{1j}=k_{x0j}x+k_{y0j}y+k_{z0j}z+\frac{k_{0j}}{ih{k}_{1j}}{k}_{2j}^{2}t+{\theta }_{0j}=k_{0j}{r}_j\cos {\phi }_j+\frac{k_{0j}}{ih{k}_{1j}}{k}_{2j}^{2}t+{\theta }_{0j}$

where $r_j$ is the radius vector with components $r_j\left(x,y,z\right)$ and $k_{0j}$ is the wave vector with components $k_{0j}\left(k_{x0j},k_{y0j},k_{z0j}\right)$ their dot product is given by $k_{0j}{r}_j\cos {\phi }_j$, ${\phi }_j$ is the angel between the two vectors $r_j$ and $k_{0j}$. $\cos {\phi }_j$ is a wave function could be used to interpret the spinning motion of the particle.

The following functions are solutions for time dependent Schrodinger’s equation the proof steps as the same proofs given above ${\psi }_j=\cos {\theta }_{2j}$ where ${\theta }_{2j}=k_{x1j}x+k_{y1j}y+k_{z1j}z+\frac{k_{1j}}{ih{k}_{0j}}{k}_{3j}^{2}t+{\theta }_{0j}$, $k_{3j}=\sqrt{E_j-2V_j}$

${\psi }_j={\text{e}}^{{\theta }_{3j}}$ where ${\theta }_{3j}=k_{x0j}x+k_{y0j}y+k_{z0j}z+\frac{\hslash }{2m_j}\frac{k_{1j}^2}{i}t+{\theta }_{0j}$

${\psi }_j={\text{e}}^{{\theta }_{4j}}$ where ${\theta }_{4j}=k_{x1j}x+k_{y1j}y+k_{z1j}z+\frac{\hslash }{2m_j}\frac{k_{0j}^2}{i}t+{\theta }_{0j}$