Symmetric equations of the light quantum systems
This is explained
in the article: <Using metric views to establish special relativity>,
the oscillating energy of the light quantum system of a particle is the result
of Lorentz transformation. For the light quantum system of a free particle, the
particle must have vibration energy
when observed in the coordinate
system of relative motion of the particle.
Conversely, when vibration energy is observed, there must be a relative motion between the particle and the coordinate system.
The
coordinate system 
is
a rest system. Suppose the coordinate system
is the relative rest coordinate
system of particle
.
is the center of
particle
. Coordinate
system is moving
with respect to coordination system at velocity 
. From the system, particleis in vibration state, and the
vibration
frequency is
，
.
Take
small volume elements
in the region
around the center of the particle. 
, and momentum
，
is the velocity of the particle,
constant in the volume element. The subscript
“ f ” indicates the reading measured by the flat metric.
The subscript “c” indicates the reading measured by the curved metric.
In the region around the center of the particle, the distribution of matter is not uniform, the velocity of the light quantum of the particle is
, and the vibration
energy is
.
is the reading of density
of the light quantum measured by the flat metric, which is a constant. 
is the reading of
velocity of the particle
measured by the curved metric.
In higher mathematics
there is a function
,
，
,
.
From the point of view
of classical mechanics, the mass of a particle is concentrated at the center of
the particle: 
.
The mass of
a particle from the perspective of the de Broglie
hypothesis in quantum mechanics is 
. This is 
.
It can be seen from the formula 
that 
and 
can replace each other under the integral
symbol. And what we get is the classical value of the whole particle at
point 0 as measured by the curve metric.
According to the formula
we
can get: under the integral sign
, and
can
be substituted for each other.
And
what we get is the classical value of the density of light quantum of whole
particle at point 0 as measured by the curve metric.
According to the formula so
once again we can get:
,
and
all are
the classical values of the whole particle at point 0 as
measured by the curve metric.
Similarly,
for any function
,
we can have:
.
is the classical value of the
whole particle at point 0 as measured by the curve metric. Let’s call this
principle as
principle,
and we’ll use it a
lot in the future.
Now let’s talk about the vibrational frequencies of a particle. Already mentioned above: In the region around the center of the particle, the vibrational
energy
of light quantum is
. For the whole
particle:
In the formula
is the radius of curvature of the
curvature line on the distributing surface of light quantum. The
above
principle
is used in the
derivation. Among them
,
all are the classical values of the
whole particle at point 0 according to the
principle.
is an
equiprobability vector.
Take the radius vector of the coordinate system as
the
axis of
the local coordinate system, we can get
.
“”is a constant said above. Take
, then get 
.
Now back to quantum mechanics, where the state function of a free
particle is 
. But
now there is
.
Need to introduce “auxiliary momentum”
. Let
,
.
is
the mass of the particle.
After introducing “auxiliary momentum” , Quantum mechanics has 
。
.
Among,
, 
.
Similarly：
.
Among
them,
,
.
Take 
, 
. So From the
equation above we can get：
.
The state function of a free
particle is: 
，
,
。
,
, 
。
The above are
，there has been
, and 
above,
，
From
the equation： above，
，
There
has been, so get: 
. Hence the wave
equation for the particle we have gotten is：
, among it 
.
Free particles are
spherically symmetric. It can be expressed in spherical coordinates.
In terms of spherical coordinates, can get: 
,
. Choose the appropriate units that can be
considered 
. Take
，then the wave equation is：
。
Where 
is the third component of
the angular momentum quantum number which corresponds to the charge of
particle.
Equation (1）is the wave equation of free particle. Equation（2）is the wave equation of free particle in spherical coordinate system.
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