# The G-dynamics of the QCPB theory

Gen WANG 111720195765863, email<EMAIL_ADDRESS>
sterces ofo neila aiv siunew gang-pwc-zgy.

( School of Mathematical Sciences, Xiamen University,
Xiamen, 361005, P.R.China )

###### Abstract

In this paper, we further study the G-dynamics newly emerged in the covariant
dynamics defined by the QCPB theory. Then we want to seek the precise
formulations of the G-dynamics to calculate the practical quantum problems. We
explicitly verify how the role of G-dynamics plays in quantum mechanics, and
then we propose three new operators based on the G-dynamics: D-operator,
T-operator and G-operator, we find that the non-Hermitian operators are
inevitable to appear in the framework of the QCPB. Then the eigenvalue
equation of them are discussed accordingly by using the Schrödinger equation
and geometrinetic energy operator (GEO) which leads to the Ri-operator,
respectively, we obtain some useful results between two quantum situations.
Meanwhile, we present generalized covariant wave equation based on the
G-operator. The GEO for quantum harmonic oscillator as an application is
further calculated.

###### Contents

  1. 1 Introduction
    1. 1.1 The Schrödinger equation
  2. 2 Generalized geometric commutator and GAC
    1. 2.1 QCPB and quantum geobracket (QGB)
    2. 2.2 Quantum covariant Hamiltonian system
    3. 2.3 Covariant dynamics, generalized Heisenberg equation, G-dynamics
    4. 2.4 Imaginary geomenergy
    5. 2.5 The QCPB for quantum harmonic oscillator
    6. 2.6 Geomentum operator
    7. 2.7 The geoperator
  3. 3 G-dynamics
    1. 3.1 D-operator, T-operator and G-operator
  4. 4 Geometrinetic energy operator (GEO) and Ri-operator
    1. 4.1 The G-dynamics with Schrödinger equation
  5. 5 The cases of G-dynamics
    1. 5.1 For classical Hamiltonian operator
    2. 5.2 For Ri-operator
    3. 5.3 G-operator for Ri-operator
    4. 5.4 Generalized covariant wave equation
  6. 6 GEO for generalized quantum harmonic oscillator
  7. 7 Conclusions

##  1 Introduction

###  1.1 The Schrödinger equation

In quantum mechanics, the form of the Schrödinger equation depends on the
physical situation. The most general form is the time-dependent Schrödinger
equation, which gives a description of a system evolving with time [1, 2]

$\sqrt{-1}\hbar\frac{\partial\psi(r,t)}{\partial
t}={{\widehat{H}}^{\left(cl\right)}}\psi=\left({-\frac{\hbar^{2}}{2m}}\nabla^{2}+V\right)\psi(r,t)$
(1)

where $\partial/\partial t$ symbolizes a partial derivative with respect to
time $t$, $\psi$ is the wave function of the quantum system,

${{\widehat{H}}^{\left(cl\right)}}=-\frac{{{\hbar}^{2}}}{2m}{{\nabla}^{2}}+V$
(2)

is the Hamiltonian operator in terms of the coordinates. The time-dependent
Schrödinger equation described above predicts that wave functions can form
standing waves, called stationary states. Stationary states can also be
described by a simpler form of the Schrödinger equation, the time-independent
Schrödinger equation

${\displaystyle{{{\widehat{H}}^{\left(cl\right)}}}\psi=\left({-\frac{\hbar^{2}}{2m}}\nabla^{2}+V\right)\psi=E\psi}$

The energy and momentum operators, which are respectively shown as

${{\widehat{H}}^{\left(cl\right)}}=\sqrt{-1}\hbar\frac{\partial}{\partial
t}\,,\quad\widehat{p}^{\left(cl\right)}=-\sqrt{-1}\hbar\nabla$ (3)

The Schrödinger equation in its general form is shown as (1).

With the non-universal validity of the QPB that has been replaced by the
complete QCPB theory. In this paper, on the foundation of the Schrödinger
equation, we mainly explore the properties of the G-dynamics in the covariant
dynamics defined by the QCHS in QCPB, we obtain the more specific expressions
of the G-dynamics in terms of the coordinates to make the G-dynamics practical
in the quantum mechanics as it develops.

##  2 Generalized geometric commutator and GAC

Let $a$ and $b$ be any two operators, their commutator is formally defined by
${\displaystyle[a,b]_{cr}=ab-ba}$. Note that the operator for commutator can
be any mathematical form to be appeared for the calculation, such as a
function, vector, differential operator, partial differential operator, even a
number in a number field, and so on, it can be arbitrarily chosen according to
our needs.

Let $a$ and $b$ be any elements of any algebra, or any operators, their
generalized geometric commutator is formally defined by the following.

######  Definition 1 (GGC).

[3] A generalized geometric commutator (GGC) of arity two $a,b$ is formally
given by

$\left[a,b\right]={{\left[a,b\right]}_{cr}}+G\left(s;a,b\right)$

The geomutator is

$G\left(s;a,b\right)=a{{\left[s,b\right]}_{cr}}-b{{\left[s,a\right]}_{cr}}$

satisfying $G\left(s;a,b\right)=-G\left(s;b,a\right)$, where $s$ is a
geometric structure function given by domain.

Some properties of the geomutator are given by

$\displaystyle
G\left(s,a+b,c+d\right)=G\left(s,a,c\right)+G\left(s,b,d\right)+G\left(s,b,c\right)+G\left(s,a,d\right)$
$\displaystyle G\left(s,a+b,c\right)=G\left(s,a,c\right)+G\left(s,b,c\right)$
$\displaystyle G\left(s,a,c+d\right)=G\left(s,a,c\right)+G\left(s,a,d\right)$

for operators $a,b,c,d$.

Obviously, in some sense, the anticommutator also needs to be generalized as
the commutator does, Analogously, denote by
${{\left\\{a,b\right\\}}_{ir}}=ab+ba$ the anticommutator.

######  Definition 2.

The geometric anticommutator (GAC) of any two elements $a$ and $b$ is defined
by

$\left\\{a,b\right\\}={{\left\\{a,b\right\\}}_{ir}}+Z\left(s,a,b\right)$

where $Z\left(s,a,b\right)=Z\left(s,b,a\right)$ is called anti-geomutator, $s$
is geometric function.

As the definition above stated,

######  Definition 3.

The anti-geomutator can be taken as

$Z\left(s,a,b\right)=\left(a:s:b\right)+{{\left\\{a,b\right\\}}_{ir}}s$

where $\left(a:s:b\right)=asb+bsa$, and $s$ is the geometric function created
by the environment.

As a result of the symmetry of anti-geomutator, geometric anticommutator then
follows the symmetry $\left\\{a,b\right\\}=\left\\{b,a\right\\}$. Actually,
the anti-geomutator can be expressed in the form

$\displaystyle Z\left(s,a,b\right)$
$\displaystyle=a{{\left\\{s,b\right\\}}_{ir}}+b{{\left\\{s,a\right\\}}_{ir}}$
$\displaystyle=\left(a:s:b\right)+{{\left\\{a,b\right\\}}_{ir}}s$

As seen, this form is very similar to the quantum geometric bracket, this is
why we need to generalize the anticommutator. There has a clear property of
the anti-geomutator given by

$\displaystyle Z\left(s,a+b,c+d\right)$
$\displaystyle=\left(a+b\right){{\left\\{s,c+d\right\\}}_{ir}}+\left(c+d\right){{\left\\{s,a+b\right\\}}_{ir}}$
$\displaystyle=Z\left(s,a,c\right)+Z\left(s,a,d\right)+Z\left(s,b,c\right)+Z\left(s,b,d\right)$

for operators $a,b,c,d$.

###  2.1 QCPB and quantum geobracket (QGB)

As [3] stated, the quantum covariant Poisson bracket (QCPB) is defined by
generalized geometric commutator (GGC) while quantum geometric bracket (QGB)
is given based on the geomutator. More precisely,

######  Definition 4 (QCPB).

[3] The QCPB is generally defined as

$\left[\hat{f},\hat{g}\right]={{\left[\hat{f},\hat{g}\right]}_{QPB}}+G\left(s,\hat{f},\hat{g}\right)$

in terms of quantum operator $\hat{f},~{}\hat{g}$, where
$G\left(s,\hat{f},\hat{g}\right)=-G\left(s,\hat{g},\hat{f}\right)$ is called
quantum geometric bracket.

It is zero if and only if $\hat{f}$ and $\hat{g}$ covariant commute, i,e.
$\left[\hat{f},\hat{g}\right]=0$. It is remarkable to see that the QCPB
representation admits a dynamical geometric bracket formula on the manifold.

######  Definition 5 (Quantum geometric bracket (QGB)).

[3] The quantum geometric bracket is

$G\left(s,\hat{f},\hat{g}\right)=\hat{f}{{\left[s,\hat{g}\right]}_{QPB}}-\hat{g}{{\left[s,\hat{f}\right]}_{QPB}}$

where $s$ represents the globally condition of space.

Let’s assert the role of the structure function $s$ that is a geometric
structure function given by domain based on the generalized geometric
commutator in definition 4, it means that structure function $s$ is only
determined by the environment, or spacetime, or manifolds, the domain, ect,
from this viewpoint, the environment joins the physical process, the influence
of the environment now based on new theory can’t be ignored, it’s naturally
necessary to be considered in a physical process.

######  Definition 6.

[3] The covariant equilibrium equation is given by
$\left[\hat{f},\hat{g}\right]=0$, i.e,

${{\left[\hat{f},\hat{g}\right]}_{QPB}}+G\left(s,\hat{f},\hat{g}\right)=0$

for operators $\hat{f},~{}\hat{g}$.

###  2.2 Quantum covariant Hamiltonian system

This section will give the covariant dynamics which contains two different
sub-dynamics: the generalized Heisenberg equation and G-dynamics [3]. It tells
that the generalized Heisenberg equation has improved the classical Heisenberg
equation by considering the quantum geometric bracket. Firstly, let’s give the
definition of quantum covariant Hamiltonian system (QCHS) based on the QCPB as
definition 4 defined.

In the beginning, let us start with the QCPB by taking $\hat{g}=\hat{H}$ into
consideration, then it formally yields below definition.

######  Definition 7 (QCHS).

[3] The QCHS defined by QCPB in terms of quantum operator $\hat{f},~{}\hat{H}$
is generally given by

$\left[\hat{f},\hat{H}\right]={{\left[\hat{f},\hat{H}\right]}_{QPB}}+G\left(s,\hat{f},\hat{H}\right)$

where $G\left(s,\hat{f},\hat{H}\right)$ is quantum geobracket .

######  Definition 8.

[3] The quantum geobracket for a Hamiltonian operator $\hat{H}$ and operator
$\hat{f}$ is given by

$G\left(s,\hat{f},\hat{H}\right)=\hat{f}{{\left[s,\hat{H}\right]}_{QPB}}-\hat{H}{{\left[s,\hat{f}\right]}_{QPB}}$

and $s$ is a real structural function only associated with the structure
space.

Manifestly, this operator $\hat{f}$ covariantly commutes with $H$. By our
construction, the QCHS defined by the QCPB is a transition for the further
development of a complete and covariant theory that can naturally generalize
the Heisenberg equation. Since $\hat{f}$ is an observable, $s$ is assumed to
be real function as well. In this way, we have successfully given a complete
description of the Heisenberg equation.

######  Definition 9.

[3] The covariant equilibrium equation is given by
$\left[\hat{f},\hat{H}\right]=0$, i.e,

${{\left[\hat{f},\hat{H}\right]}_{QPB}}+G\left(s,\hat{f},\hat{H}\right)=0$

for operators $\hat{f},~{}\hat{H}$, then $\hat{f}$ is called quantum covariant
conserved quantity.

Furthermore, there is a certain case given by
$\left[\hat{H},\hat{H}\right]=0$, when $\hat{f}=\hat{H}$ based on definition
9. It is clear to see that $\hat{H}$ is a quantum covariant conserved
quantity.

###  2.3 Covariant dynamics, generalized Heisenberg equation, G-dynamics

In this section, we will briefly review the entire theoretical framework of
quantum covariant Hamiltonian system defined by the quantum covariant Poisson
bracket totally based on the paper [3]. More precisely, the time covariant
evolution of any observable $\hat{f}$ in the covariant dynamics is given by
both the generalized Heisenberg equation of motion and G-dynamics.

######  Theorem 1.

[3] The covariant dynamics, the generalized Heisenberg equation, G-dynamics
can be formally formulated as

The covariant dynamics:

$\frac{\mathcal{D}\hat{f}}{dt}=\frac{1}{\sqrt{-1}\hbar}\left[\hat{f},\hat{H}\right]$

The generalized Heisenberg equation:

$\frac{d\hat{f}}{dt}=\frac{1}{\sqrt{-1}\hbar}{{\left[\hat{f},\hat{H}\right]}_{QPB}}-\frac{1}{\sqrt{-1}\hbar}\hat{H}{{\left[s,\hat{f}\right]}_{QPB}}$

G-dynamics:

$\hat{w}=\frac{1}{\sqrt{-1}\hbar}{{\left[s,\hat{H}\right]}_{QPB}}$.

respectively, where $\frac{\mathcal{D}}{dt}=\frac{d}{dt}+\hat{w}$ is covariant
time operator, and ${{\hat{H}}}$ is the Hamiltonian and $[\cdot,\cdot]$
denotes the GGC of two operators.

###  2.4 Imaginary geomenergy

In this section, the imaginary geomenergy is defined based on the G-dynamics,
by using this new concept, we can better give a presentation for the covariant
dynamics, ect.

######  Definition 10.

[3] The imaginary geomenergy is defined by
${{E}^{\left(\operatorname{Im}\right)}}\left(\hat{w}\right)=\sqrt{-1}\hbar\hat{w}$,
where $\hat{w}$ means G-dynamics.

It is clear that the covariant dynamics leads us to a complete quantum system
in which the state of the system is represented by two separated dynamic
quantum system, the covariant dynamics can be totally pictured in the form
below

$\displaystyle\sqrt{-1}\hbar\frac{\mathcal{D}}{dt}\hat{f}$
$\displaystyle=\left[\hat{f},\hat{H}\right]={{\left[\hat{f},\hat{H}\right]}_{QPB}}+G\left(s,\hat{f},\hat{H}\right)$
$\displaystyle=\sqrt{-1}\hbar\frac{d}{dt}\hat{f}+\sqrt{-1}\hbar\hat{f}\hat{w}$

With the imaginary geomenergy defined above, then covariant dynamics is
rewritten in the form

$\sqrt{-1}\hbar\frac{\mathcal{D}}{dt}\hat{f}=\left[\hat{f},\hat{H}\right]=\sqrt{-1}\hbar\frac{d}{dt}\hat{f}+\hat{f}{{E}^{\left(\operatorname{Im}\right)}}\left(\hat{w}\right)$

As a consequence of the imaginary geomenergy, we can say that imaginary
geomenergy is a new kind of Hamiltonian operator.

###  2.5 The QCPB for quantum harmonic oscillator

In this section, we simply review some basic results given by the paper [3] in
QCPB. As quantum mechanics illustrated, the commutator relations may look
different than in the Schrödinger picture, because of the time dependence of
operators. The time evolution of those operators depends on the Hamiltonian of
the system. Considering the one-dimensional quantum harmonic oscillator,

$\hat{H}^{\left(cl\right)}={\frac{{{\hat{p}}}^{{\left(cl\right)2}}}{2m}}+{\frac{m\omega^{{2}}x^{{2}}}{2}}$
(4)

the evolution of the position and momentum operators is given by:

${d\over
dt}x(t)={\sqrt{-1}\over\hbar}[\hat{H}^{\left(cl\right)},x(t)]_{QPB}={\frac{{{\hat{p}}}^{{\left(cl\right)}}}{m}}$
(5) ${d\over
dt}{{\hat{p}}}^{{\left(cl\right)}}(t)={\sqrt{-1}\over\hbar}[\hat{H}^{\left(cl\right)},{{\hat{p}}}^{{\left(cl\right)}}(t)]_{QPB}=-m\omega^{{2}}x$

As for the application of the QPB or the commutation, there are some many
fields including the physics and mathematics, and so on.

As a certainly example, we will now start by briefly reviewing the quantum
mechanics of a one-dimensional quantum harmonic oscillator (4), and see how
the QCPB can be incorporated using GGC in the covariant quantization
procedure. With the Hamiltonian given by (4), let’s use the QCPB to
recalculate the (5), we can see how difference emerges. More specifically, the
covariant dynamics in terms of the position reads

$\frac{\mathcal{D}}{dt}x\left(t\right)=\frac{\sqrt{-1}}{\hbar}\left[\hat{H}^{\left(cl\right)},x\left(t\right)\right]=\frac{{{\hat{p}}^{\left(cl\right)}}\left(t\right)}{m}+\frac{\sqrt{-1}}{\hbar}x\left(t\right)\frac{{{\hat{p}}^{\left(cl\right)2}}s+2{{\hat{p}}^{\left(cl\right)}}s{{\hat{p}}^{\left(cl\right)}}}{2m}$

By direct computation, the G-dynamics is given by

${{\widehat{w}}^{\left(cl\right)}}=\frac{\sqrt{-1}}{\hbar}\frac{{{\hat{p}}^{\left(cl\right)2}}s+2{{\hat{p}}^{\left(cl\right)}}s{{\hat{p}}^{\left(cl\right)}}}{2m}$
(6)

And the generalized Heisenberg equation with respect to $x$ follows

$\displaystyle\frac{d}{dt}x\left(t\right)$
$\displaystyle=\frac{\sqrt{-1}}{\hbar}{{\left[\hat{H}^{\left(cl\right)},x\left(t\right)\right]}_{QPB}}+\frac{\sqrt{-1}}{\hbar}\hat{H}^{\left(cl\right)}{{\left[s,x\left(t\right)\right]}_{QPB}}$
$\displaystyle=\frac{{{\widehat{p}}^{\left(cl\right)}}\left(t\right)}{m}$

In the same way, the covariant dynamics for the classical momentum operator is

$\displaystyle\frac{\mathcal{D}}{dt}{{{\hat{p}}}^{\left(cl\right)}}\left(t\right)$
$\displaystyle=\frac{\sqrt{-1}}{\hbar}\left[\hat{H}^{\left(cl\right)},{{{\hat{p}}}^{\left(cl\right)}}\left(t\right)\right]$
$\displaystyle=-m{{\omega}^{2}}x-\hat{H}^{\left(cl\right)}\frac{ds}{dx}+{{\widehat{p}}^{\left(cl\right)}}\left(t\right){{\widehat{w}}^{\left(cl\right)}}$

Accordingly, the generalized Heisenberg equation in terms of the
${{\hat{p}}}^{{\left(cl\right)}}$ appears

$\displaystyle\frac{d}{dt}{{{\hat{p}}}^{\left(cl\right)}}\left(t\right)$
$\displaystyle=\frac{\sqrt{-1}}{\hbar}{{\left[\hat{H}^{\left(cl\right)},{{{\hat{p}}}^{\left(cl\right)}}\left(t\right)\right]}_{QPB}}+\frac{\sqrt{-1}}{\hbar}\hat{H}^{\left(cl\right)}{{\left[s,{{{\hat{p}}}^{\left(cl\right)}}\left(t\right)\right]}_{QPB}}$
$\displaystyle=-m{{\omega}^{2}}x-\hat{H}^{\left(cl\right)}\frac{ds}{dx}$

###  2.6 Geomentum operator

######  Definition 11.

[3] Let $M$ be a smooth manifold represented by structural function $s$, then
geomentum operator is defined as

$\hat{p}=-\sqrt{-1}\hbar D$

where $D=\nabla+\nabla s$. The component is
${{\hat{p}}_{j}}=-\sqrt{-1}\hbar{{D}_{j}}$ in which
${{D}_{j}}={{\partial}_{j}}+{{\partial}_{j}}s$ holds,
${{\partial}_{j}}=\frac{\partial}{\partial{{x}_{j}}}$.

Note that the geomentum operator is a revision of the classical momentum
operator given by (3).

######  Theorem 2 (Geometric canonical quantization rules).

[3] Geometric equal-time canonical commutation relation is

$\left[\hat{{{x}_{i}}},\hat{{{p}_{j}}}\right]=\sqrt{-1}\hbar{{D}_{j}}{{x}_{i}}$

where
$\left[\cdot,~{}\cdot\right]={{\left[\cdot,~{}\cdot\right]}_{QPB}}+G\left(s,\cdot,~{}\cdot\right)$
is QCPB.

Geometric canonical commutation relation can be expressed in a specific form

$\left[\hat{{{x}_{i}}},\hat{{{p}_{j}}}\right]=\sqrt{-1}\hbar\left({{\delta}_{ij}}+{{x}_{i}}\frac{\partial}{\partial{{x}_{j}}}s\right)$

In other words, it also can be rewritten as
$\left[{{x}_{i}},\hat{{{p}_{j}}}\right]=\sqrt{-1}\hbar{{\theta}_{ij}}$, where
${{\theta}_{ij}}={{\delta}_{ij}}+{{x}_{i}}{{\partial}_{j}}s$, and
${{\partial}_{j}}=\frac{\partial}{\partial{{x}_{j}}}$.

###  2.7 The geoperator

######  Definition 12 (Geoperator).

[4] Let $\hat{X}$ be a Hermitian operator, then geoperator can be defined as

${{\hat{X}}^{\left(s\right)}}=\hat{X}+u$

where $u=\hat{X}s=I\left(\hat{X},s\right)$ is coupling interaction between the
observable $\hat{X}$ and the environment $s$.

Obviously, the geometric operator is a extension of the Hermitian operator.
Note that the interaction term $u=\hat{X}s=I\left(\hat{X},s\right)\neq 0$
actually is a precise function with respect to the spacetime, as we state, the
structure function $s$ generated by the space or manifolds as an environment
variable only associate with the space or the manifolds, and it’s independent
to the wave function.

Note that the definition of the geoperator is reasonable, in particular, the
evidence comes from the coupling interaction between the observable $\hat{X}$
and the environment variable $s$ that is described by
$u=\hat{X}s=I\left(\hat{X},s\right)$ in which the environment variable $s$
satisfies the G-dynamics in the theorem 1, it also can be seen from (6) in the
one-dimensional quantum harmonic oscillator. In fact, geomentum operator is
one of the geoperator if the Hermitian operator
$\hat{X}=\widehat{p}^{\left(cl\right)}=-\sqrt{-1}\hbar\nabla$.

##  3 G-dynamics

In this section, G-dynamics as a new dynamical form appears in the quantum
mechanics, we mainly focus on the G-dynamics expressed by theorem 1, that is,
$\hat{w}=\frac{1}{\sqrt{-1}\hbar}{{\left[s,\hat{H}\right]}_{QPB}}$. As [3]
already stated, the G-dynamics is only induced by the structure function $s$,
and meanwhile, it’s independent to the other observables. There is a clear
fact of the G-dynamics taken as a different form by choosing different
Hamiltonian operators, in one word, various Hamiltonian operators correspond
to the multiple G-dynamics. Therefore, for a given wave function $\psi$, the
eigenvalue equation of operator is accordingly given by
$\widehat{w}\psi={{w}}\psi$, where ${{w}}$ is the eigenvalue. By using
definition 10, it gets corresponding imaginary geomenergy given by definition
10, that is, ${{E}^{\left(\operatorname{Im}\right)}}=\sqrt{-1}\hbar\hat{w}$,
thusly, it then has energy spectrum given by
${{E}^{\left(\operatorname{Im}\right)}}\psi=\sqrt{-1}{{E}^{\left(g\right)}}\psi$,
where ${E}^{\left(g\right)}=\hbar w$. Obviously, we can see that the
G-dynamics is completely determined by the structure function $s$, it implies
that the G-dynamics represents the properties of the spatial manifolds, in
other words, the spatial manifolds has abundant activities. As a result of
this point, we need to seek more clues to unlock this quantum characters. The
classical Hamiltonian operator (3) or (2) is the first to bear the brunt to be
considered in above formula.

###  3.1 D-operator, T-operator and G-operator

In this section, the discussions on the most general form will be done, we use
the G-dynamics $\widehat{w}$ to define the T-operator, and then the imaginary
geomenergy is used with the classical Hamiltonian operator (3) to construct
the G-operator as a non-Hermitian operator to deeply study some properties of
the G-dynamics.

To start with the definition of the D-operator and the T-operator,

######  Definition 13.

The D-operator is given by
$\widehat{{{D}_{t}}}={{\partial}_{t}}\pm\widehat{w}$, and then T-operator is
defined as $\widehat{{{N}_{t}}}=\sqrt{-1}\widehat{{{D}_{t}}}$, where
$\widehat{w}$ is the G-dynamics.

Note that D-operator $\widehat{{{D}_{t}}}={{\partial}_{t}}\pm\widehat{w}$
means that there are two kinds of D-operators. More precisely, we take one of
them into consideration as reality needed. Its eigenvalue equation of
D-operator is given by $\widehat{{{D}_{t}}}\psi={{D}_{t}}\psi$.

As a result of the definition 13, that naturally leads to bond the imaginary
geomenergy and the classical Hamiltonian operator (3) together to construct a
new energy operator, accordingly, we can derive following G-operator.

######  Definition 14.

The G-operator can be defined as
${{\widehat{H}}^{\left(gr\right)}}=\hbar\widehat{{{N}_{t}}}$.

Thusly, then the G-operator in details is given by

$\displaystyle{{\widehat{H}}^{\left(gr\right)}}$
$\displaystyle={{\widehat{H}}^{\left(cl\right)}}+{{E}^{\left(\operatorname{Im}\right)}}={{\widehat{H}}^{\left(cl\right)}}+\sqrt{-1}\hbar\widehat{w}$
$\displaystyle=\sqrt{-1}\hbar\left({{\partial}_{t}}+\widehat{w}\right)$
$\displaystyle=\sqrt{-1}\hbar\widehat{{{D}_{t}}}$

That T-operator is then expressed as
${{\widehat{H}}^{\left(gr\right)}}/\hbar=\widehat{{{N}_{t}}}$. Thusly, the
eigenvalue equation $\widehat{{{N}_{t}}}\psi={{N}_{t}}\psi$ follows. As we
stated, G-dynamics $\widehat{w}$ is a Hermitian operator in the most of time,
it implies that the the G-operator is a non-Hermitian operator. As a result of
this point, we say that the eigenvalue of the G-operator must be in a complex
form.

######  Theorem 3.

The eigenvalue of G-operator is ${{E}^{\left(gr\right)}}=\hbar{{N}_{t}}$,
where the eigenvalue of T-operator is

$\displaystyle{{N}_{t}}$
$\displaystyle={{E}^{\left(gr\right)}}/\hbar={{E}^{\left(cl\right)}}/\hbar+\sqrt{-1}{{E}^{\left(g\right)}}/\hbar$

where ${{E}^{\left(g\right)}}=\hbar{{w}}$.

###### Proof.

According to the definition 14 of G-operator, the eigenvalue equation can be
more specifically expressed below

$\displaystyle{{\widehat{H}}^{\left(gr\right)}}\psi$
$\displaystyle={{\widehat{H}}^{\left(cl\right)}}\psi+{{E}^{\left(\operatorname{Im}\right)}}\psi={{\widehat{H}}^{\left(cl\right)}}\psi+\sqrt{-1}\hbar\widehat{w}\psi$
(7)
$\displaystyle=\sqrt{-1}\hbar\left({{\partial}_{t}}+\widehat{w}\right)\psi=\sqrt{-1}\hbar\widehat{{{D}_{t}}}\psi$
$\displaystyle=\sqrt{-1}\hbar{{D}_{t}}\psi$
$\displaystyle=\sqrt{-1}\hbar\left({{\partial}_{t}}\psi+{{w}}\psi\right)$
$\displaystyle=\left({{E}^{\left(cl\right)}}+\sqrt{-1}\hbar{{w}}\right)\psi$
$\displaystyle={{E}^{\left(gr\right)}}\psi$

Hence, we obtain the eigenvalue of the G-operator that is given by
${{E}^{\left(gr\right)}}=\sqrt{-1}\hbar{{D}_{t}}$, it’s rewritten as
${{E}^{\left(gr\right)}}={{E}^{\left(cl\right)}}+\sqrt{-1}{{E}^{\left(g\right)}}$
by denoting ${{E}^{\left(g\right)}}=\hbar{{w}}$, and the eigenvalue of the
D-operator $\widehat{{{D}_{t}}}$ follows

${{D}_{t}}={{E}^{\left(gr\right)}}/\sqrt{-1}\hbar={{E}^{\left(cl\right)}}/\sqrt{-1}\hbar+{{w}}$

Similarly, the eigenvalue of the T-operator $\widehat{{{N}_{t}}}$ also emerges
as follows ${{N}_{t}}={{E}^{\left(gr\right)}}/\hbar$, where
${{E}^{\left(g\right)}}/\hbar={{w}}$, therefore, the eigenvalue of the
operator $\widehat{{{N}_{t}}}$ can be rewritten as

$\displaystyle{{N}_{t}}$
$\displaystyle={{E}^{\left(gr\right)}}/\hbar={{E}^{\left(cl\right)}}/\hbar+\sqrt{-1}{{E}^{\left(g\right)}}/\hbar$

Note that the eigenvalue of the G-operator can be rewritten in a form
${{E}^{\left(gr\right)}}=\hbar{{N}_{t}}$. Therefore, we complete the proof as
desired. ∎

Notice that the eigenvalue of G-operator and T-operator are complex form,
these are the features of the non-Hermitian operator. As G-dynamics stated,
it’s generated by the structure function, and it forms the G-operator, it
reveals that there exists a complete Schrödinger equation such that the
structure function is naturally involved in the equation.

##  4 Geometrinetic energy operator (GEO) and Ri-operator

As a result of the G-operator, we use geomentum operator 11 to reconstruct the
new Hamiltonian operator. In order to smoothly build such complete new
Hamiltonian operator, we firstly need to define a corresponding kinetic energy
operator associated with the structure function $s$, therefore, let’s rewrite
geomentum operator as ${{\hat{P}}^{\left(ri\right)}}=-\sqrt{-1}\hbar D$ from
definition 11, then we give the definition below based on the geomentum
operator.

######  Definition 15 (Geometrinetic energy operator (GEO)).

Geometrinetic energy operator (GEO) is defined as

${{\hat{T}}^{\left(ri\right)}}=\frac{{{\hat{P}}^{\left(ri\right)2}}}{2m}=-\frac{{{\hbar}^{2}}}{2m}\left(\Delta+2\nabla
s\cdot\nabla\right)+{{T}^{\left(c\right)}}$

where ${{T}^{\left(c\right)}}=-\frac{{{\hbar}^{2}}}{2m}\left(\Delta s+\nabla
s\cdot\nabla s\right)$ is called structural $c$-energy.

More precisely, the derivation of the geometrinetic energy operator can be
given by logically computation. Since the square of generalized gradient
operator that is calculated as

${{D}^{2}}\psi=\Delta\psi+2\nabla s\cdot\nabla\psi+\psi\left(\Delta s+\nabla
s\cdot\nabla s\right)$

Accordingly, it yields an second order operator

${{D}^{2}}=\Delta+2\nabla s\cdot\nabla+\left(\Delta s+\nabla s\cdot\nabla
s\right)$ (8)

Then the eigenvalue equation of the geometrinetic energy operator
${{\hat{T}}^{\left(ri\right)}}$ with respect to the wave function $\psi$
follows

${{\hat{T}}^{\left(ri\right)}}\psi=\frac{{{\hat{P}}^{\left(ri\right)2}}}{2m}\psi=-\frac{{{\hbar}^{2}}}{2m}\left(\Delta\psi+2\nabla
s\cdot\nabla\psi\right)+\psi{{T}^{\left(c\right)}}$

######  Definition 16.

Ri-operator is defined as

${{\hat{H}}^{\left(ri\right)}}={{\widehat{T}}^{\left(ri\right)}}+V$

where ${{\hat{T}}^{\left(ri\right)}}$ is the geometrinetic energy operator.

As a result, Ri-operator can be rewritten in the form

${{\hat{H}}^{\left(ri\right)}}={{\hat{H}}^{\left(cl\right)}}-\frac{{{\hbar}^{2}}}{m}\nabla
s\cdot\nabla+{{T}^{\left(c\right)}}$ (9)

by using the classical Hamiltonian operator (2).

###  4.1 The G-dynamics with Schrödinger equation

This section, we will mainly discuss the eigenvalue equation of G-dynamics
based on the Schrödinger equation (1), it nicely proves that G-operator and
T-operator are natural results on G-dynamics. Rewriting the G-dynamics that is
originally proposed as

${{E}^{\left(\operatorname{Im}\right)}}/\sqrt{-1}\hbar=\widehat{w}=\frac{1}{\sqrt{-1}\hbar}{{\left[s,\widehat{H}\right]}_{QPB}}$

where the Hamiltonian operator $\widehat{H}$ in it is completely determined by
the physical truth.

######  Theorem 4.

The eigenvalue of the G-dynamics ${{\widehat{w}}^{\left(cl\right)}}$ based on
Schrödinger equation (1) in time parameter or Hamiltonian operator (3) is
given by ${{w}^{\left(q\right)}}=-{{\partial}_{t}}s$.

###### Proof.

Let’s take $H={{\widehat{H}}^{\left(cl\right)}}$, then, more precisely,

${{\widehat{w}}^{\left(cl\right)}}\psi=\frac{1}{\sqrt{-1}\hbar}{{\left[s,{{\widehat{H}}^{\left(cl\right)}}\right]}_{QPB}}\psi=\frac{1}{\sqrt{-1}\hbar}\left(s{{\widehat{H}}^{\left(cl\right)}}\psi-{{\widehat{H}}^{\left(cl\right)}}\left(s\psi\right)\right)$

Plugging the Schrödinger equation (1) into above eigenvalue equation, we
obtain

$\sqrt{-1}\hbar{{\partial}_{t}}\psi={{\widehat{H}}^{\left(cl\right)}}\psi,~{}~{}\sqrt{-1}\hbar{{\partial}_{t}}\left(s\psi\right)={{\widehat{H}}^{\left(cl\right)}}\left(s\psi\right)$

and the eigenvalue equation of G-dynamics ${{\widehat{w}}^{\left(cl\right)}}$
as an operator is

$\displaystyle{{\widehat{w}}^{\left(cl\right)}}\psi$
$\displaystyle=\frac{1}{\sqrt{-1}\hbar}{{\left[s,{{\widehat{H}}^{\left(cl\right)}}\right]}_{QPB}}\psi$
(10)
$\displaystyle=\frac{1}{\sqrt{-1}\hbar}\left(s\sqrt{-1}\hbar{{\partial}_{t}}\psi-\sqrt{-1}\hbar{{\partial}_{t}}\left(s\psi\right)\right)$
$\displaystyle=\left(s{{\partial}_{t}}\psi-{{\partial}_{t}}\left(s\psi\right)\right)$
$\displaystyle=-\psi{{\partial}_{t}}s$

We denote ${{\widehat{w}}^{\left(cl\right)}}\psi={{w}^{\left(q\right)}}\psi$
here. Thusly, we certainly get the specific eigenvalue for
${{\widehat{w}}^{\left(cl\right)}}$ given by
${{w}^{\left(q\right)}}=-{{\partial}_{t}}s$. ∎

Note that theorem 4 has indicated a fact that for any function form given for
the G-dynamics $\widehat{w}$ in such condition, the eigenvalue is always
dependent on the structure function.

######  Theorem 5.

The heat equation for the structure function $s$ based on the Schrödinger
equation (1) is given by

${{\partial}_{t}}s=\frac{\sqrt{-1}\hbar}{m}\left(\Delta s/2+\nabla
s\cdot\nabla\ln\psi\right)$

###### Proof.

In the one hand, theorem 4 has evidently given. In another hand, we take the
(2) into the G-dynamics $\widehat{w}$, and it deduces a result

$\displaystyle{{\hat{w}}^{\left(cl\right)}}\psi$
$\displaystyle=\frac{1}{\sqrt{-1}\hbar}{{\left[s,{{\widehat{H}}^{\left(cl\right)}}\right]}_{QPB}}\psi$
$\displaystyle=\frac{1}{\sqrt{-1}\hbar}{{\left[s,-\frac{{{\hbar}^{2}}}{2m}\Delta+V\right]}_{QPB}}\psi$
$\displaystyle=-\frac{\hbar}{2\sqrt{-1}m}{{\left[s,\Delta\right]}_{QPB}}\psi$
$\displaystyle=\frac{\hbar}{2\sqrt{-1}m}\left(2\nabla s\cdot\nabla+\Delta
s\right)\psi$

According to (10), it yields

${{\hat{w}}^{\left(cl\right)}}\psi=-\psi{{\partial}_{t}}s=\frac{\hbar}{2\sqrt{-1}m}\left(\Delta
s+2\nabla s\cdot\nabla\right)\psi$

It’s definite to form a heat equation associated with the structure function

$\displaystyle{{\partial}_{t}}s$
$\displaystyle=\frac{\sqrt{-1}\hbar}{2m\psi}\left(\psi\Delta s+2\nabla
s\cdot\nabla\psi\right)=\frac{\sqrt{-1}\hbar}{m}\left(\Delta s/2+\nabla
s\cdot\nabla\ln\psi\right)$

Then, the proof is completed. ∎

######  Corollary 1.

The G-dynamics in terms of the Hamiltonian operator (2) in coordinates can be
expressed as

${{\widehat{w}}^{\left(cl\right)}}=\frac{\hbar}{2\sqrt{-1}m}\left(\Delta
s+2\nabla s\cdot\nabla\right)$ (11)

As we can see, the G-dynamics on such expression given by the (11) is totally
determined by the structure function. Geometrinetic energy operator by using
the G-dynamics ${{\widehat{w}}^{\left(cl\right)}}$ can be rewritten as

${{\widehat{T}}^{\left(ri\right)}}={{\widehat{T}}^{\left(cl\right)}}-{{E}^{\left(\operatorname{Im}\right)}}\left({{{\hat{w}}}^{\left(cl\right)}}\right)-{E}^{\left(s\right)}/2$

where the imaginary geomenergy is
${{E}^{\left(\operatorname{Im}\right)}}\left({{{\hat{w}}}^{\left(cl\right)}}\right)=\sqrt{-1}\hbar{{\hat{w}}^{\left(cl\right)}}$,
and ${{\widehat{T}}^{\left(cl\right)}}=-\frac{{{\hbar}^{2}}}{2m}\Delta$, and
${{E}^{\left(s\right)}}=\frac{{{\hbar}^{2}}}{m}\nabla s\cdot\nabla s$. As a
consequence, based on the theorem 5, we can get a corollary shown as follows.

######  Corollary 2.

If wave function $\psi$ satisfies both (1) and heat equation 5, then

$\left\\{\begin{matrix}{{\partial}_{t}}s=\frac{\sqrt{-1}\hbar}{m}\left(\Delta
s/2+\nabla s\cdot\nabla\ln\psi\right)\\\
\sqrt{-1}\hbar{{\partial}_{t}}\psi=-\frac{{{\hbar}^{2}}}{2m}\Delta\psi+V\psi\\\
\end{matrix}\right.$

Take a deep analysis to above two equations, obviously, it yields

$\sqrt{-1}\hbar\psi{{\partial}_{t}}s=-\frac{{{\hbar}^{2}}}{2m}\left(\psi\Delta
s+2\nabla s\cdot\nabla\psi\right)$

It follows the bonding equation

$\sqrt{-1}\hbar\left({{\partial}_{t}}\psi+\psi{{\partial}_{t}}s\right)=-\frac{{{\hbar}^{2}}}{2m}\left(\Delta\psi+2\nabla
s\cdot\nabla\psi+\psi\Delta s\right)+V\psi$

Definitely, it can be rewritten as

$\displaystyle\sqrt{-1}\hbar\left({{\partial}_{t}}+{{\partial}_{t}}s\right)\psi$
$\displaystyle=-\frac{{{\hbar}^{2}}}{2m}\left(\Delta+2\nabla
s\cdot\nabla+\Delta s+\nabla s\cdot\nabla s\right)\psi$
$\displaystyle\begin{matrix}{}&{}&{}&{}\\\
\end{matrix}+\left(V+\frac{{{\hbar}^{2}}}{2m}\nabla s\cdot\nabla s\right)\psi$
$\displaystyle={{\widehat{H}}^{\left(ri\right)}}\psi+\frac{{{\hbar}^{2}}}{2m}\psi\nabla
s\cdot\nabla s$

Or in the form

$\sqrt{-1}\hbar\left({{\partial}_{t}}-{{\widehat{w}}^{\left(cl\right)}}\right)\psi={{\hat{H}}^{\left(ri\right)}}\psi+{{E}^{\left(s\right)}}\psi/2$
(12)

where ${{E}^{\left(s\right)}}=\frac{{{\hbar}^{2}}}{m}\nabla s\cdot\nabla s$,
it evidently implies that ${{E}^{\left(s\right)}}$ plays the role of potential
energy. In fact, the (12) supports a statement to the definition 14. In other
words, such definition 14 is a natural extension for the classical results, in
the later discussions, we can surely say about this points.

######  Definition 17.

D-operator based on the Schrödinger equation (1) is
$\widehat{{{D}_{t}}}^{\left(cl\right)}={{\partial}_{t}}-{{\hat{w}}^{\left(cl\right)}}$.

Actually, the D-operator also can be taken in another form

$\widehat{{{D}_{t}}}^{\left(cl\right)}={{\partial}_{t}}+{{\hat{w}}^{\left(cl\right)}}$
(13)

that is, $-{{\hat{w}}^{\left(cl\right)}}\to{{\hat{w}}^{\left(cl\right)}}$ on
above definition. According to the G-operator given by definition 14, thusly,
it yields

######  Definition 18.

G-operator is then defined as

${{\widehat{H}}^{\left(gr\right)}}=\hbar{{\widehat{N}}_{t}}^{\left(cl\right)}={{\widehat{H}}^{\left(cl\right)}}-{{E}^{\left(\operatorname{Im}\right)}}\left({{{\hat{w}}}^{\left(cl\right)}}\right)$

based on definition 17, where the imaginary geomenergy is
${{E}^{\left(\operatorname{Im}\right)}}\left({{{\hat{w}}}^{\left(cl\right)}}\right)=\sqrt{-1}\hbar{{\hat{w}}^{\left(cl\right)}}$.

As a consequence of the definition 14, we figure out that eigenvalue theorem
below.

######  Theorem 6.

The eigenvalue of the G-operator in definition 18 on Schrödinger equation (1)
is

${{E}^{\left(gr\right)}}={{E}^{\left(cl\right)}}-\sqrt{-1}\hbar{{w}^{\left(q\right)}}$

###### Proof.

For a given wave function $\psi$, the eigenvalue equation of the G-operator on
Schrödinger equation (1) is expressed as

$\displaystyle{{\widehat{H}}^{\left(gr\right)}}\psi$
$\displaystyle=\hbar{{\widehat{N}}_{t}}^{\left(cl\right)}\psi={{\widehat{H}}^{\left(cl\right)}}\psi-{{E}^{\left(\operatorname{Im}\right)}}\left({{{\hat{w}}}^{\left(cl\right)}}\right)\psi$
$\displaystyle={{\widehat{H}}^{\left(cl\right)}}\psi-\sqrt{-1}\hbar{{{\hat{w}}}^{\left(cl\right)}}\psi$
$\displaystyle={{E}^{\left(cl\right)}}\psi-\sqrt{-1}\hbar\psi{{w}^{\left(q\right)}}$
$\displaystyle=\left({{E}^{\left(cl\right)}}-\sqrt{-1}\hbar{{w}^{\left(q\right)}}\right)\psi$
$\displaystyle={{E}^{\left(gr\right)}}\psi$

where ${{\widehat{w}}^{\left(cl\right)}}\psi={{w}^{\left(q\right)}}\psi$ has
been used for the proof. Then, we complete the proof. ∎

In fact, if D-operator is taken according to (13), then eigenvalue of the
G-operator on Schrödinger equation (1) is

${{E}^{\left(gr\right)}}={{E}^{\left(cl\right)}}+\sqrt{-1}\hbar{{w}^{\left(q\right)}}$

This is exactly the characters of conjugate complex numbers.

######  Corollary 3.

Based on theorem 6, there exists a pure imaginary geomenergy form
${{E}^{\left(\operatorname{Im}\right)}}\left(\pm{{w}^{\left(q\right)}}\right)=\pm\sqrt{-1}\hbar{{w}^{\left(q\right)}}$
induced by G-dynamics.

##  5 The cases of G-dynamics

###  5.1 For classical Hamiltonian operator

To consider quantum harmonic oscillator (4), as the G-dynamics of it shown by
(6),

${{\widehat{w}}^{\left(cl\right)}}=\frac{\sqrt{-1}}{\hbar}\frac{{{\widehat{p}}^{\left(cl\right)2}}s}{2m}+\frac{\sqrt{-1}}{m\hbar}{{\widehat{p}}^{\left(cl\right)}}s{{\widehat{p}}^{\left(cl\right)}}$

The pure imaginary geomenergy form is

${{E}^{\left(\operatorname{Im}\right)}}\left({{\widehat{w}}^{\left(cl\right)}}\right)=\sqrt{-1}\hbar{{\widehat{w}}^{\left(cl\right)}}=-\frac{{{\widehat{p}}^{\left(cl\right)2}}s}{2m}-\frac{1}{m}{{\widehat{p}}^{\left(cl\right)}}s{{\widehat{p}}^{\left(cl\right)}}$

Let’s see how its characteristic equation shows, it’s given by

$\displaystyle{{\widehat{w}}^{\left(cl\right)}}\psi$
$\displaystyle=\frac{\sqrt{-1}}{\hbar}\psi\frac{{{\widehat{p}}^{\left(cl\right)2}}s}{2m}+\frac{\sqrt{-1}}{m\hbar}{{\widehat{p}}^{\left(cl\right)}}s{{\widehat{p}}^{\left(cl\right)}}\psi$
$\displaystyle=-\frac{\sqrt{-1}\hbar}{2m}\left(\psi\Delta s+2\nabla
s\cdot\nabla\psi\right)$ $\displaystyle=\frac{\hbar}{2\sqrt{-1}m}\left(\Delta
s+2\nabla s\cdot\nabla\right)\psi$

where $\Delta={{\nabla}^{2}}$. Subsequently, it derives the G-dynamics (11),
or in the form
$\frac{2\sqrt{-1}m}{\hbar}{{\widehat{w}}^{\left(cl\right)}}=\Delta s+2\nabla
s\cdot\nabla$, by using (8), we get its another expression in terms of the
G-dynamics shown as

${{D}^{2}}=\frac{2\sqrt{-1}m}{\hbar}{{\widehat{w}}^{\left(cl\right)}}+\Delta+\nabla
s\cdot\nabla s$

###  5.2 For Ri-operator

######  Theorem 7.

The G-dynamics in ${{\hat{H}}^{\left(ri\right)}}$ in coordinates is

$\widehat{w}^{\left(ri\right)}={{\widehat{w}}^{\left(cl\right)}}+w^{\left(s\right)}$

where $w^{\left(s\right)}=\frac{\hbar}{\sqrt{-1}m}\nabla s\cdot\nabla s$ is
denoted.

###### Proof.

For this case, let’s consider
${{\hat{H}}^{\left(ri\right)}}={{\widehat{T}}^{\left(ri\right)}}+V$ given by
Ri-operator 16, then the G-dynamics can be calculated as

$\widehat{w}^{\left(ri\right)}=\frac{1}{\sqrt{-1}\hbar}{{\left[s,{{\hat{H}}^{\left(ri\right)}}\right]}_{QPB}}=\frac{1}{\sqrt{-1}\hbar}{{\left[s,{{\widehat{T}}^{\left(ri\right)}}+V\right]}_{QPB}}$

where $V$ is a real potential function. Thusly, we obtain

$\displaystyle{{\left[s,{{\widehat{T}}^{\left(ri\right)}}\right]}_{QPB}}\psi$
$\displaystyle=-\frac{{{\hbar}^{2}}}{2m}{{\left[s,\Delta+2\nabla
s\cdot\nabla\right]}_{QPB}}\psi$
$\displaystyle=-\frac{{{\hbar}^{2}}}{2m}{{\left[s,\Delta\right]}_{QPB}}\psi-\frac{{{\hbar}^{2}}}{m}{{\left[s,\nabla
s\cdot\nabla\right]}_{QPB}}\psi$

where

$\displaystyle{{\left[s,\Delta\right]}_{QPB}}\psi$
$\displaystyle=s\Delta\psi-\Delta\left(s\psi\right)$ $\displaystyle=-2\nabla
s\cdot\nabla\psi-\psi\Delta s$ $\displaystyle=-\left(2\nabla
s\cdot\nabla+\Delta s\right)\psi$

and

${{\left[s,\nabla s\cdot\nabla\right]}_{QPB}}\psi=-\psi\nabla s\cdot\nabla s$

Therefore, we get

${{\left[s,{{\widehat{T}}^{\left(ri\right)}}\right]}_{QPB}}=\frac{{{\hbar}^{2}}}{2m}\left(2\nabla
s\cdot\nabla+\Delta s+2\nabla s\cdot\nabla s\right)$

Thusly,

$\displaystyle\widehat{w}^{\left(ri\right)}$
$\displaystyle=\frac{1}{\sqrt{-1}\hbar}{{\left[s,{{\widehat{T}}^{\left(ri\right)}}\right]}_{QPB}}=\frac{\hbar}{2\sqrt{-1}m}\left(2\nabla
s\cdot\nabla+\Delta s+2\nabla s\cdot\nabla s\right)$ (14)
$\displaystyle=\frac{\hbar}{2\sqrt{-1}m}\left(2\nabla s\cdot\nabla+\Delta
s\right)+\frac{\hbar}{\sqrt{-1}m}\nabla s\cdot\nabla s$
$\displaystyle={{\widehat{w}}^{\left(cl\right)}}+w^{\left(s\right)}$

where $w^{\left(s\right)}=\frac{\hbar}{\sqrt{-1}m}\nabla s\cdot\nabla s$ is
denoted. ∎

Obviously,
${{\hat{w}}^{\left(ri\right)}}-{{\hat{w}}^{\left(cl\right)}}={{w}^{\left(s\right)}}$
holds. The theorem 7 can be rewritten in a form

${{E}^{\left(\operatorname{Im}\right)}}\left({\hat{w}^{\left(ri\right)}}\right)=\sqrt{-1}\hbar\widehat{w}^{\left(ri\right)}=\sqrt{-1}\hbar{{\widehat{w}}^{\left(cl\right)}}+{{E}^{\left(s\right)}}$

where ${{E}^{\left(s\right)}}=\frac{{{\hbar}^{2}}}{m}\nabla s\cdot\nabla
s=\sqrt{-1}\hbar{{w}^{\left(s\right)}}$, or expressed by the imaginary
geomenergy

${{E}^{\left(\operatorname{Im}\right)}}\left({\hat{w}^{\left(ri\right)}}\right)={{E}^{\left(\operatorname{Im}\right)}}\left({{\widehat{w}}^{\left(cl\right)}}\right)+{{E}^{\left(s\right)}}$

where
${{E}^{\left(\operatorname{Im}\right)}}\left({{\widehat{w}}^{\left(cl\right)}}\right)=\sqrt{-1}\hbar{{\widehat{w}}^{\left(cl\right)}}$
is the imaginary geomenergy in terms of ${{\widehat{w}}^{\left(cl\right)}}$.
Correspondingly, the structural $c$-energy can be given in a form

$\displaystyle{{T}^{\left(c\right)}}$
$\displaystyle=-\frac{{{\hbar}^{2}}}{2m}\left(\Delta s+{{\left|\nabla
s\right|}^{2}}\right)=-\frac{{{\hbar}^{2}}}{2m}\Delta
s-\frac{{{\hbar}^{2}}}{2m}{{\left|\nabla s\right|}^{2}}$
$\displaystyle=-\frac{{{\hbar}^{2}}}{2m}\Delta s-{{E}^{\left(s\right)}}/2$

where it results in
${{T}^{\left(c\right)}}+{{E}^{\left(s\right)}}/2=-\frac{{{\hbar}^{2}}}{2m}\Delta
s$. There is more we need to say clearly,

${{\widehat{w}}^{\left(ri\right)}}={{\widehat{w}}^{\left(cl\right)}}+{{w}^{\left(s\right)}}={{\widehat{w}}^{\left(cl\right)}}-\sqrt{-1}a\nabla
s\cdot\nabla s$

where $a=\hbar/m$. Then the conjugate transpose of the
${{\widehat{w}}^{\left(ri\right)}}$ is given by

${{\widehat{w}}^{\left(ri\right)\dagger}}={{\widehat{w}}^{\left(cl\right)\dagger}}+\sqrt{-1}a\nabla
s\cdot\nabla s$

Accordingly, it yields
${{\widehat{w}}^{\left(ri\right)}}+{{\widehat{w}}^{\left(ri\right)\dagger}}={{\widehat{w}}^{\left(cl\right)}}+{{\widehat{w}}^{\left(cl\right)\dagger}}=2{{\widehat{w}}^{\left(cl\right)}}$.
Meanwhile, let ${{\widehat{w}}^{\left(g\right)}}$ be denoted as NG operator to
fit

${{\widehat{w}}^{\left(g\right)}}={{\widehat{w}}^{\left(ri\right)}}-{{\widehat{w}}^{\left(ri\right)\dagger}}=-2\sqrt{-1}a\nabla
s\cdot\nabla s\neq 0$

Obviously, this NG operator breaks the Hermiticity as usually understood

###  5.3 G-operator for Ri-operator

To analyze G-dynamics for both situations together, we can obtain some useful
results.

$\displaystyle\widehat{w}^{\left(ri\right)}\psi$
$\displaystyle={{\widehat{w}}^{\left(cl\right)}}\psi+w^{\left(s\right)}\psi$
$\displaystyle=\frac{\hbar}{2\sqrt{-1}m}\left(\Delta s+2\nabla
s\cdot\nabla\right)\psi+\frac{\hbar}{\sqrt{-1}m}\psi\nabla s\cdot\nabla s$

Accordingly, it leads to the imaginary geomenergy

$\sqrt{-1}\hbar\widehat{w}^{\left(ri\right)}\psi=\frac{{{\hbar}^{2}}}{2m}\left(2\nabla
s\cdot\nabla+\Delta s+2\nabla s\cdot\nabla s\right)\psi$

and with the Schrödinger equation (1), it generates

$\displaystyle\sqrt{-1}\hbar{{\partial}_{t}}\psi-\sqrt{-1}\hbar\widehat{w}^{\left(ri\right)}\psi$
$\displaystyle=\sqrt{-1}\hbar\left({{\partial}_{t}}-\widehat{w}^{\left(ri\right)}\right)\psi$
$\displaystyle=-\frac{{{\hbar}^{2}}}{2m}\Delta\psi+V\psi-\frac{{{\hbar}^{2}}}{2m}\left(2\nabla
s\cdot\nabla+\Delta s+2\nabla s\cdot\nabla s\right)\psi$
$\displaystyle=-\frac{{{\hbar}^{2}}}{2m}\left(\Delta+2\nabla
s\cdot\nabla+\Delta s+\nabla s\cdot\nabla
s\right)\psi+\left(V-{{E}^{\left(s\right)}}/2\right)\psi$
$\displaystyle={{\hat{H}}^{\left(ri\right)}}\psi-{{E}^{\left(s\right)}}\psi/2$

As a result of equation (12) together, we get a similar equations given by

$\displaystyle\sqrt{-1}\hbar\left({{\partial}_{t}}-{{\widehat{w}}^{\left(cl\right)}}\right)\psi={{{\hat{H}}}^{\left(ri\right)}}\psi+{{E}^{\left(s\right)}}\psi/2$
(15)
$\displaystyle\sqrt{-1}\hbar\left({{\partial}_{t}}-{{\widehat{w}}^{\left(ri\right)}}\right)\psi={{{\hat{H}}}^{\left(ri\right)}}\psi-{{E}^{\left(s\right)}}\psi/2$

And then it rightly reflects the theorem 7.

######  Corollary 4.

Ri-operator expressed by the G-operator is given by

${{\hat{H}}^{\left(ri\right)}}={{\hat{H}}^{\left(gr\right)}}+{{E}^{\left(s\right)}}/2$

###### Proof.

Since the (9) and theorem 7 hold, then

$\displaystyle{{{\hat{H}}}^{\left(ri\right)}}$
$\displaystyle={{{\hat{H}}}^{\left(cl\right)}}-\frac{{{\hbar}^{2}}}{m}\nabla
s\cdot\nabla+{{{{T}}}^{\left(c\right)}}$
$\displaystyle={{{\hat{H}}}^{\left(cl\right)}}-\sqrt{-1}\hbar{{\widehat{w}}^{\left(ri\right)}}+{{E}^{\left(s\right)}}/2$

With the definition 14 of G-operator, it has

${{\hat{H}}^{\left(gr\right)}}={{\hat{H}}^{\left(cl\right)}}-\sqrt{-1}\hbar{{\widehat{w}}^{\left(ri\right)}}=\sqrt{-1}\hbar{{\widehat{D}}_{t}}^{\left(ri\right)}$

where D-operator here is
${{\widehat{D}}_{t}}^{\left(ri\right)}={{\partial}_{t}}-{{\widehat{w}}^{\left(ri\right)}}$,
thus, then we complete the proof. ∎

Conversely, note that the G-operator in corollary 4 is also expressed as

${{\hat{H}}^{\left(gr\right)}}={{\hat{H}}^{\left(ri\right)}}-{{E}^{\left(s\right)}}/2$
(16)

By using Ri-operator in corollary 4, (15) can be simplified as

$\displaystyle\sqrt{-1}\hbar\left({{\partial}_{t}}-{{{\hat{w}}}^{\left(cl\right)}}\right)\psi={{{\hat{H}}}^{\left(gr\right)}}\psi+{{E}^{\left(s\right)}}\psi$
(17)
$\displaystyle\sqrt{-1}\hbar\left({{\partial}_{t}}-{{{\hat{w}}}^{\left(ri\right)}}\right)\psi={{{\hat{H}}}^{\left(gr\right)}}\psi$

where G-operator in terms of the Ri-operator is definitely chosen as

${{\hat{H}}^{\left(gr\right)}}=\sqrt{-1}\hbar\left({{\partial}_{t}}-{{{\hat{w}}}^{\left(ri\right)}}\right)=\sqrt{-1}\hbar{{\hat{D}}_{t}}^{\left(ri\right)}$

and
${{\hat{D}}_{t}}^{\left(ri\right)}={{\partial}_{t}}-{{\hat{w}}^{\left(ri\right)}}$.
As a matter of fact, these two equations are equivalent to each other based on
the theorem 7.

###  5.4 Generalized covariant wave equation

Correspondingly, the definition of generalized covariant wave equation as a
revision of classic Schrödinger equation follows based on geometrinetic energy
operator.

######  Theorem 8.

The generalized covariant wave equation is defined as

$\hbar\widehat{{{N}_{t}}}^{\left(ri\right)}\psi={{{\hat{H}}}^{\left(gr\right)}}\psi$

by using (16), where the D-operator is taken as
$\widehat{{{D}_{t}}}^{\left(ri\right)}={{\partial}_{t}}-{{{\hat{w}}}^{\left(ri\right)}}$.

Apparently, the generalized covariant wave equation can be given by employing
(17),

$\sqrt{-1}\hbar\left({{\partial}_{t}}-{{{\hat{w}}}^{\left(cl\right)}}-{{E}^{\left(s\right)}}/\sqrt{-1}\hbar\right)\psi={{\hat{H}}^{\left(gr\right)}}\psi$

where
${{\hat{w}}^{\left(ri\right)}}={{\hat{w}}^{\left(cl\right)}}+{{E}^{\left(s\right)}}/\sqrt{-1}\hbar$
is distinct to be seen from the theorem 7. More precisely about the
generalized covariant wave equation are listed

$\displaystyle{{{\hat{H}}}^{\left(gr\right)}}=-\frac{{{\hbar}^{2}}}{2m}\left(\Delta+2\nabla
s\cdot\nabla+\Delta s+2\nabla s\cdot\nabla
s\right)={{{\hat{H}}}^{\left(ri\right)}}-{{E}^{\left(s\right)}}/2$
$\displaystyle{{{\hat{w}}}^{\left(ri\right)}}=\frac{\hbar}{2\sqrt{-1}m}\left(2\nabla
s\cdot\nabla+\Delta s+2\nabla s\cdot\nabla
s\right)={{{\hat{w}}}^{\left(cl\right)}}+{{E}^{\left(s\right)}}/\sqrt{-1}\hbar$

Obviously, the generalized covariant wave equation is a natural complete
extension of the classic Schrödinger equation.

Supported by the definition 12, we consider geoperator in terms of the
Hamiltonian $\hat{H}$, let’s see what we get in this way. Thusly, we have
${{\widehat{H}}^{\left(s\right)}}=\widehat{H}+u_{H}$ that can be called the
geomenergy operator, where $u_{H}=\widehat{H}s$ is coupling interaction
between the observable $\hat{H}$ and the environment $s$. In order to better
understand this case, we precisely consider the classical Hamiltonian operator
(3) ${{\hat{H}}^{\left(cl\right)}}$ for the geomenergy operator that is
expressed as
${{\widehat{H}}^{\left(s\right)}}={{\widehat{H}}^{\left(cl\right)}}+u_{H^{(cl)}}$,
where $u_{H^{(cl)}}={{\widehat{H}}^{\left(cl\right)}}s$, more specifically,
the coupling interaction for the classical Hamiltonian operator (3)
${{\widehat{H}}^{\left(cl\right)}}$ and environment is given by
$u_{H^{(cl)}}={{\widehat{H}}^{\left(cl\right)}}s=-{{E}^{\left(\operatorname{Im}\right)}}\left({{w}^{\left(q\right)}}\right)$,
then,
${{E}^{\left(\operatorname{Im}\right)}}\left({{w}^{\left(q\right)}}\right)=-u_{H^{(cl)}}$,
as a result, the geomenergy operator can be rewritten as
${{\widehat{H}}^{\left(s\right)}}={{\widehat{H}}^{\left(cl\right)}}-{{E}^{\left(\operatorname{Im}\right)}}\left({{w}^{\left(q\right)}}\right)$,
when we refer to the (12), we can obtain self-consistent result
${{\widehat{H}}^{\left(s\right)}}\psi=\sqrt{-1}\hbar\left({{\partial}_{t}}-{{\widehat{w}}^{\left(cl\right)}}\right)\psi$
for a given wave function $\psi$, that is to say,
${{\widehat{H}}^{\left(s\right)}}\psi={{\hat{H}}^{\left(ri\right)}}\psi+{{E}^{\left(s\right)}}\psi/2$
holds, accordingly, it yields
${{\widehat{H}}^{\left(s\right)}}\psi-{{E}^{\left(s\right)}}\psi/2={{\hat{H}}^{\left(ri\right)}}\psi$
which is exactly the generalized covariant wave equation as theorem 8 given.

##  6 GEO for generalized quantum harmonic oscillator

The covariant time evolution of operators
$x(t),{{\hat{p}}}^{{\left(ri\right)}}(t)$ depends on the Hamiltonian of the
system. Considering the one-dimensional generalized quantum harmonic
oscillator based upon geomentum operator 11, it’s given by

$\hat{H}^{\left(ri\right)}={\frac{{{\hat{p}}}^{{\left(ri\right)2}}}{2m}}+{\frac{m\omega^{{2}}x^{{2}}}{2}}$
(18)

where $V\left(x\right)=\frac{1}{2}m{{\omega}^{2}}{{x}^{2}}$, the geomentum
operator in one-dimensional is
${{\widehat{p}}^{\left(ri\right)}}=-\sqrt{-1}\hbar\frac{\rm{D}}{dx}$, and
$\frac{\rm{D}}{dx}=d/dx+ds/dx$. Accordingly, the covariant evolution of the
position and geomentum operator based on the QCHS given by definition 7 are
respectively given by:

${\mathcal{D}\over
dt}x(t)={\sqrt{-1}\over\hbar}[\hat{H}^{\left(ri\right)},x(t)]$ (19)
${\mathcal{D}\over
dt}{{\hat{p}}}^{{\left(ri\right)}}(t)={\sqrt{-1}\over\hbar}[\hat{H}^{\left(ri\right)},{{\hat{p}}}^{{\left(ri\right)}}(t)]$

where $[\cdot,\cdot]$ denotes the GGC of two operators. More precisely, the
concrete computational process of the first covariant evolution terms of the
position is shown as

$\left[{{\widehat{H}}^{\left(ri\right)}},x\right]={{\left[{{\widehat{H}}^{\left(ri\right)}},x\right]}_{QPB}}+G\left(s,{{\widehat{H}}^{\left(ri\right)}},x\right)$

Hence, the direct computation leads to a series of results

$\displaystyle{{\left[{{\widehat{H}}^{\left(ri\right)}},x\right]}_{QPB}}$
$\displaystyle={{\left[\frac{{{\widehat{p}}^{\left(ri\right)2}}}{2m},x\right]}_{QPB}}=\frac{1}{2m}{{\left[{{\widehat{p}}^{\left(ri\right)2}},x\right]}_{QPB}}$
$\displaystyle=\frac{-{{\hbar}^{2}}}{2m}{{\left[\frac{{{d}^{2}}}{d{{x}^{2}}}+2\frac{ds}{dx}\frac{d}{dx},x\right]}_{QPB}}$
$\displaystyle=\frac{-{{\hbar}^{2}}}{m}\frac{\rm{D}}{dx}$

where

${{\widehat{p}}^{\left(ri\right)2}}=-{{\hbar}^{2}}\left(\frac{{{d}^{2}}}{d{{x}^{2}}}+2\frac{ds}{dx}\frac{d}{dx}+\frac{{{d}^{2}}s}{d{{x}^{2}}}+{{\left(\frac{ds}{dx}\right)}^{2}}\right)$

and the QGB in terms of ${{\widehat{H}}^{\left(ri\right)}},x$ is

$\displaystyle G\left(s,{{\widehat{H}}^{\left(ri\right)}},x\right)$
$\displaystyle=-x{{\left[s,{{\widehat{H}}^{\left(ri\right)}}\right]}_{QPB}}=x{{\left[{{\widehat{H}}^{\left(ri\right)}},s\right]}_{QPB}}$
$\displaystyle=-\frac{{{\hbar}^{2}}}{m}x\frac{ds}{dx}\frac{d}{dx}-\frac{{{\hbar}^{2}}}{2m}x\left(\frac{{{d}^{2}}}{d{{x}^{2}}}s+2{{\left(\frac{ds}{dx}\right)}^{2}}\right)$

Then the QCPB about ${{\widehat{H}}^{\left(ri\right)}},x$ specifically shows

$\displaystyle\left[{{\widehat{H}}^{\left(ri\right)}},x\right]$
$\displaystyle={{\left[{{\widehat{H}}^{\left(ri\right)}},x\right]}_{QPB}}+G\left(s,{{\widehat{H}}^{\left(ri\right)}},x\right)$
$\displaystyle=\frac{-{{\hbar}^{2}}}{m}\frac{\rm{D}}{dx}-\frac{{{\hbar}^{2}}}{m}x\frac{ds}{dx}\frac{d}{dx}-\frac{{{\hbar}^{2}}}{2m}x\left(\frac{{{d}^{2}}}{d{{x}^{2}}}s+2{{\left(\frac{ds}{dx}\right)}^{2}}\right)$
$\displaystyle=-\frac{{{\hbar}^{2}}}{m}\left(\frac{\rm{D}}{dx}+x\frac{ds}{dx}\frac{d}{dx}+\frac{1}{2}x\left(\frac{{{d}^{2}}}{d{{x}^{2}}}s+2{{\left(\frac{ds}{dx}\right)}^{2}}\right)\right)$

As a result, the covariant evolution of the position is given by

$\displaystyle\frac{\mathcal{D}}{dt}x\left(t\right)$
$\displaystyle=\frac{\sqrt{-1}}{\hbar}\left[{{\widehat{H}}^{\left(ri\right)}},x\right]$
$\displaystyle=-\frac{\sqrt{-1}\hbar}{m}\left(\frac{\rm{D}}{dx}+x\frac{ds}{dx}\frac{d}{dx}+\frac{1}{2}x\left(\frac{{{d}^{2}}}{d{{x}^{2}}}s+2{{\left(\frac{ds}{dx}\right)}^{2}}\right)\right)$

In other words, the covariant evolution of the position based on the QCHS is

$\frac{\mathcal{D}}{dt}x\left(t\right)=\frac{\sqrt{-1}}{\hbar}\left[{{\widehat{H}}^{\left(ri\right)}},x\right]=\frac{{{\widehat{p}}^{\left(ri\right)}}}{m}+x\widehat{w}^{\left(ri\right)}$

where G-dynamics is equal to

$\widehat{w}^{\left(ri\right)}=-\frac{\sqrt{-1}\hbar}{2m}\left(2\frac{ds}{dx}\frac{d}{dx}+\frac{{{d}^{2}}}{d{{x}^{2}}}s+2{{\left(\frac{ds}{dx}\right)}^{2}}\right)$
(20)

Definitely, G-dynamics (20) in terms of $\widehat{w}^{\left(ri\right)}$ is the
equivalent expression of the (14) in one dimensional case. Accordingly, the
imaginary geomenergy follows

${{E}^{\left(\operatorname{Im}\right)}}\left(\widehat{w}^{\left(ri\right)}\right)=\sqrt{-1}\hbar\widehat{w}^{\left(ri\right)}=\frac{{{\hbar}^{2}}}{2m}\left(2\frac{ds}{dx}\frac{d}{dx}+\frac{{{d}^{2}}}{d{{x}^{2}}}s+2{{\left(\frac{ds}{dx}\right)}^{2}}\right)$

Similarly, the covariant evolution of the geomentum operator is

${\mathcal{D}\over
dt}{{\hat{p}}}^{{\left(ri\right)}}(t)={\sqrt{-1}\over\hbar}[\hat{H}^{\left(ri\right)},{{\hat{p}}}^{{\left(ri\right)}}(t)]$

The calculation process is as follows

$\displaystyle{{\left[{{\widehat{H}}^{\left(ri\right)}},{{\widehat{p}}^{\left(ri\right)}}\right]}_{QPB}}$
$\displaystyle={{\left[V,{{\widehat{p}}^{\left(ri\right)}}\right]}_{QPB}}=\frac{1}{2}m{{\omega}^{2}}{{\left[{{x}^{2}},{{\widehat{p}}^{\left(ri\right)}}\right]}_{QPB}}$
$\displaystyle=\frac{1}{2}m{{\omega}^{2}}{{\left[{{x}^{2}},{{\widehat{p}}^{\left(ri\right)}}\right]}_{QPB}}=\sqrt{-1}\hbar
m{{\omega}^{2}}x$

The QGB in terms of
${{\widehat{H}}^{\left(ri\right)}},{{\widehat{p}}^{\left(ri\right)}}$ is

$G\left(s,{{\widehat{H}}^{\left(ri\right)}},{{\widehat{p}}^{\left(ri\right)}}\right)={{\widehat{H}}^{\left(ri\right)}}{{\left[s,{{\widehat{p}}^{\left(ri\right)}}\right]}_{QPB}}-{{\widehat{p}}^{\left(ri\right)}}{{\left[s,{{\widehat{H}}^{\left(ri\right)}}\right]}_{QPB}}$

where
${{\left[s,{{\widehat{p}}^{\left(ri\right)}}\right]}_{QPB}}=\sqrt{-1}\hbar\frac{ds}{dx}$
has been used, and the QGB in details becomes

$G\left(s,{{\widehat{H}}^{\left(ri\right)}},{{\widehat{p}}^{\left(ri\right)}}\right)=\sqrt{-1}\hbar{{\widehat{H}}^{\left(ri\right)}}\frac{ds}{dx}-{{\widehat{p}}^{\left(ri\right)}}{{\left[s,{{\widehat{H}}^{\left(ri\right)}}\right]}_{QPB}}$

As a consequence, the QCPB here is given by

$\displaystyle\left[{{\widehat{H}}^{\left(ri\right)}},{{\widehat{p}}^{\left(ri\right)}}\right]$
$\displaystyle={{\left[{{\widehat{H}}^{\left(ri\right)}},{{\widehat{p}}^{\left(ri\right)}}\right]}_{QPB}}+G\left(s,{{\widehat{H}}^{\left(ri\right)}},{{\widehat{p}}^{\left(ri\right)}}\right)$
$\displaystyle=\sqrt{-1}\hbar
m{{\omega}^{2}}x+\sqrt{-1}\hbar{{\widehat{H}}^{\left(ri\right)}}\frac{ds}{dx}-{{\widehat{p}}^{\left(ri\right)}}{{\left[s,{{\widehat{H}}^{\left(ri\right)}}\right]}_{QPB}}$

It turns out that the covariant evolution of the geomentum operator is derived
as follows

$\displaystyle\frac{\mathcal{D}}{dt}{{\widehat{p}}^{\left(ri\right)}}$
$\displaystyle=\frac{\sqrt{-1}}{\hbar}\left[{{\widehat{H}}^{\left(ri\right)}},{{\widehat{p}}^{\left(ri\right)}}\right]$
$\displaystyle=-m{{\omega}^{2}}x-{{\widehat{H}}^{\left(ri\right)}}\frac{ds}{dx}-\frac{\sqrt{-1}}{\hbar}{{\widehat{p}}^{\left(ri\right)}}{{\left[s,{{\widehat{H}}^{\left(ri\right)}}\right]}_{QPB}}$

where the G-dynamics reads

$\widehat{w}^{\left(ri\right)}=-\frac{\sqrt{-1}}{\hbar}{{\left[s,{{\widehat{H}}^{\left(ri\right)}}\right]}_{QPB}}$

which is equivalent to the (20) as a precise expression. The G-dynamics seen
from the (20) can evidently say how it forms and works associated with the
structure function.

##  7 Conclusions

In this paper, we have proposed a series of new concepts based on the analysis
of the G-dynamics included in the covariant dynamics defined by the QCHS in
QCPB. On the solid foundation of the QCPB, we have gained some valuable
consequences centred on the G-dynamics. By using the G-dynamics, we define new
operators such as the D-operator, T-operator, G-operator, Ri-operator etc, we
find their complex connections, in particular, we mainly focus on the their
eigenvalues equation. With the help of the Schrödinger equation, we analyze
the G-dynamics and its related properties. We concretely discuss the different
cases of the G-dynamics, in this way, we present the extension of the
Schrödinger equation, naturally. As an implication, we derive the generalized
quantum harmonic oscillator and the covariant dynamics on it based on the QCHS
theory.

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