The document discusses the interpretation of wave functions and Schrodinger's equation in quantum mechanics. It proposes that wave functions can be expressed as complex space vectors that rotate on different axes. This allows wave functions like sinusoidal functions to be expressed using Euler's formula and addressed some issues with differentiation. It suggests wave equations can be satisfied when interactions between systems exhibit exponential behavior over time and position, and proposes the wave function solution Ψ=Ae^-mvxi. Further implications and future outlook are discussed.

1. interpretation of
wave function

2. Schrodinger equation
quantum is said to contain property as a wave and
nature as particles.
The wave function that expresses the property of a
wave is interpreted by Schrodinger equation.

3. Popular interpretation
The popular quantum property as a wave
Schrodinger equation expresses is
summarized in following formula.

4. A question about Schrodinger
equation
There is one question in Schrodinger's wave
equation,looking at x and t as a permutation
of some variable in common(Think that there
is a common operator),an unnatural point
appears when differentiating with x or t.

5. Derivative of Ψ with x and t
correlated to each other
Here is the differentiation method when there is an
icorrelation between x and t.
In the following, we will proceed with the article
considered by deriving ihd Ψ/dx=-p Ψ from the
conventional Schrödinger equation as an article under
special conditions.(Hereafter h is Dirac constant.)
ψ(x, t) = A sin(kx − ωt)
When considering differentiating t with respect to x,
when differentiating t, I generally think of replacing the derivative of t with t'=1･
dt/dx. (Think of not as partial differentiation but as ordinary differentiation)
Differentiating according to t,
dΨ/dx=Acos(kx-ωt)(k+ωdt/dx)
Aωcos(kx-ωt)dt/dx an indefinite term appears.
The same applies to the derivative at t.

6. What do you think about the
interpretation of indefinite terms?
Assuming that the position of the quantum at a
certain time t is x, it cannot be said that many of
them actually affect either x or t, so explore the
possibilities that the classical wave equation
including the indefinite term holds.
The common algebraic property of the wave
equation suggested by the Schrodinger
equation is used as a hint.

7. Condition for establishing
classical wave equation
Considering the conditions for the classical wave
equation, the particle function can be partially
differentiated twice with respect to x and t,
is thought to be the case.
By the way, kx and ωt of Schrodinger equation
are both dimensions of M1L2T-1,
satisfies the condition for the classical wave
equation.
1/s^2・∂2Ψ/∂t2=ΔΨ=∂2Ψ/∂x2+∂2Ψ/∂y2+∂2Ψ/∂z2
(s is the phase velocity
Ψ=Asinmx^2/t

8. Reconsidering the conditions for
the wave equation
Considering that the spring motion can be expressed
by the wave equation with the acceleration in the
opposite direction, and the vibration of the string and
the film can be obtained from the sin function as well, I
think that the acceleration in the opposite direction to
the sin function has a hint.
Considering that there is a solution to the problem in
eliminating the discomfort that the time term and the
position term are said to be in the same row in the sin
function, I consider.

9. Reverse acceleration
Looking at the wave equation only by conclusion, in the
one-dimensional wave equation only for x,
You can see this. About this, if you look only at the first
derivative
These two formulas satisfy the condition.
In addition, the basic sin function and cos function take
the opposite acceleration. Also, the motion represented
by F=-kx (having a position variable strength and a
reverse acceleration) also satisfies the condition.
1/s^2∂2u/∂t2=∂2u/dx2
∂u/∂t=s∂u/∂x、∂u/∂t=-s∂u/∂x

10. sin function and Euler's formula
Here, suppose that the sin function is analyzed
by Euler's formula.
The two-fold derivative of e^ix given by is the
opposite sign of the original function, which is
convenient for showing the acceleration in the
opposite direction of position. Therefore, we
aim to add/subtract the angle to x in the formula
to form sinx.
e^ix=cosx+isinx

11. Introduction of complex space
coordinates
From this point, if we consider the real space coordinates
as they are, there will be a disturbance in the
consistency due to the nature of the imaginary numbers,
so we will consider the introduction of special complex
space coordinates.
Let r be the length from the origin, and consider an axis
that indicates the direction from the origin (base axis),
and the angle from the axis on a plane that passes
through that axis and the origin is Θ1, and the angle from
the vertical plane that passes the same axis. Let Θ2 be
Θ2 and Θ2 must be 0 or an imaginary number. (Θ2^2≦0)

12. Vector in complex space
coordinates
In the previously defined complex space coordinates, the
position is uniquely determined in three dimensions (r, Θ1,
Θ2).
Correspondence with xyz coordinates is also established.
However, in the case of making it correspond to the real
number coordinate space, it is necessary to convert the
position where the imaginary number axis overlaps the real
number surface into a real number instead of an imaginary
number. For example, when rsinΘ1sinΘ2 corresponding to
y is associated with the imaginary number axis,
rsinΘ1cosΘ2 corresponding to the x axis needs to be
defined as a real number. (cosiΘ2 = cosΘ2)
x=rsinΘ1cosΘ2,y=rsinΘ1sinΘ2,z=rcosΘ1

13. Correspondence with Euler's
formula
Hereinafter, it is defined that cosi=cos1 and the x-axis
corresponds to the base axis, the y-axis corresponds
to the imaginary axis, and the z-axis corresponds to
the real number axis.
Euler's formula is,
Can be transformed. Therefore, any of the wave
equations represented by Asin(bx-ct) or Asinkx^2/t can
be expressed in the form of e^ix in the complex space
without contradiction. e^ix=cosx+isinx = √2sin(x-¼π-
½π)
e^ix=cosx+isinx
=√2sin(x-¼π-½π)

14. The nature of Euler's formula
By the way, the Euler's formula for the derivative of
x and the relation to the exponent are equivalent
on both sides. In other words, (e^iax)'=aie^ix and
(sin(ax+Θ))'=acos(ax+Θ) also holds, and
(e^ix)^n=e^inx and (sinx) ^n=sin^(n)x=sinnx also
holds. The calculus of this will not collapse.
(Strictly speaking, in this coordinate system, i → 1,
sin Θ → cos Θ, isin Θ → cos Θ are equivalent to
rotation only with different directions.)

15. The sin function and e^x are
essentially the same
The sin function and e^x have similar
calculus forms, and their coefficients and
power law are similar. It is natural to think
that there is a function for a certain quantity,
and if it exceeds the condition, it becomes
e^x, and if it does not exceed the condition,
it becomes a sin function.

16. Mathematical analysis of wave
equation
Since the form of the second derivative is long,
consider only the first derivative. Considering the
wave equations for x and t that are correlated with
each other,
For u(x,t), x=X(τ),t=T(τ), if ∂u/dt=s ∂u/∂x is rewritten
differentially,
du/dt=sdu/dx s= dx/dt (s is a constant ≠ 0 if the wave
equation is satisfied)
At this time, du/dt=s・du/dx. Integrating this with
respect to x, from ∫du/dt・dx=su, if ∫dx・du/dt=su ∫dx
is F(x,t), du/dt=su/F
F is the position of x. Which is a constant or linear in
a system in which the variables are preserved,
changes when x and t are considered when
considering inflow and outflow from the outside or
when considering interaction with different systems.

17. y'=f(x)·y solution
Therefore, y'=f(x)y behaves exponentially with
F(x) as the exponent. In the wave equation, it is
considered that the time variable is satisfied
when the position variable with respect to time
exhibits exponential behavior with ∫s/F(=x1-
x0)dt as an exponent. For position variables,
replace position with time.
It is transformed into y'/y=f(x). Differentiate both sides with
respect to x.
From logy=F(x), y=e^F(x)

18. Condition for establishing wave
equation
In general, ihdΨ/dt=HΨ and ihdΨ/dx=-pΨ are
considered to be quantumally valid.
Interpreting this as a wave equation, the
behavior of the mass x is exponential, with
∫H/hidt being the exponential for time change
and ∫-p/hidx being exponential for position
change. The wave equation can be established
when considering the behavioral behavior, or
when the interactions have the same behavior.

19. ∫H/hidt and ∫-p/hidx are dimensionally
equivalent, and equivalent to the momentum
momentum in Newton's equation of motion.
From the consideration so far, I propose
Ψ=Ae^-mvxi=Ae^Eti as one of the solutions of
the wave function.
Since mvx placed here can be expressed as a
space vector, it is assumed that Ψ can also be
expressed as a space vector.

20. What is meant by e^-mvxi on the complex
space is the rotation of mvx of a vector of
size A along the base axis, since it is
equivalent to sin-mvx. The meaning of
e^mn=(e^m)^n is that the size of the rotation
angle can be freely changed by changing
the size of the rotating vector.

21. Implications of dΨ/dt/Ψ
dΨ/dt/Ψ=f(t) has a dimension of f(t), and Ψ=e^f(t)
represents that Ψ undergoes a large change
through the inflow and outflow of f(t) into the
system. I am. It represents a physical change
close to divergence (∇・) or rotation (∇×) in
Maxwell's equations. The divergence (∇・) of
vector analysis is often expressed as dlogy/dt.
dlogy/dx often expresses the rotation (∇ ×) of
vector analysis.

22. Future outlook
From the law of conservation of energy, the
conservation law of the sum of reciprocal
moments can be derived. It is possible to use this
to guide the equivalence of the principle of
invariance of light velocity and the principle of
uncertainty, and to discuss motions that violate
both principles.
It may be possible to verify the prerequisites for
the theory of relativity.
It seems that the generalization of the wave
equation can be increased.

23. I plan to make more after a while. Thank
you.