The Post-modern Form of Schrodinger’s Equations

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Crucial Subject: Real and Apparent Quantizations

 “Quantum mechanics thus reveals a basic oneness of the universe.”

Erwin Schrodinger

The Background of Post-modern Schrodinger’s Equations

I owe a better part of my understanding of quantization and quantum mechanics to the Schrodinger’s formal description of quantum mechanics. This is why it is an honour for me to post-modernize Schrodinger’s equations, both the time dependent equation and the time-independent equation.

I had in one of my articles discuss my intention to post-modernize quantum mechanics, and in the article linked below I have proceeded to lay the foundation of post-modern quantum mechanics.

In the article above, I explained the kind of energy a free particle truly carries, and it is not the kind of energy a free particle in Schrodinger’s formalism carries. A free atomic particle carries energy in Joules/s2 and not energy in Joules. You must come to this new post-modern understanding of quantum mechanics.

While the Schrodinger’s description still holds, the limit of its validity only lies within the domain of an observer which is the domain of effect, but in the atomic world or in the frame of a free particle, which is the domain of cause, the Schrodinger’s equations transform to the kinds related to energy in Joules/s2 and not to energy in Joules

You should now know that there are two kinds of energy in the universe. The first kind of energy is the energy in Joules which only applies in the macro-world of large bodies. The second kind of energy is the energy in Joules/s which only applies in the micro-world of atoms and particles.

The current form of the Schrodinger’s equations which are related to energy in Joules only applies to an observer outside the atom. This is why I call modern quantum mechanics observer based quantum mechanics.

In post-modern physics, we have a new form of quantum mechanics which I call observer-atom based quantum mechanics, and which is founded on the second correspondence principle in the universe. I have talked about the second correspondence principle in the article above.

erwin-schrodinger life

Post-modern quantum mechanics is concerned with how the principles of quantum mechanics apply directly in the atom which is the domain of cause, and how from the domain of cause proceed observations in the observer’s domain of effect. This is important.

So, what I am about to show you is the true causal form of Schrodinger’s equations and not the observative form of Schrodinger’s equations which was what Schrodinger originally proposed.

The procedure for deriving the causal form of Schrodinger’s equation as it applies to the domain of the particle is the same as the popular method of modern quantum mechanics. The important uniqueness of this new procedure is that the Schrodinger’s equations are based on force and energy in Joules/sand not on momentum and energy in Joules.

The Post-modern Time-Dependent Schrodinger’s Equation

Now, for a particle in an infinite potential well, the wave function of a particle of fixed energy Ea in Joules/s2 could most naturally be written as a linear combination of wave functions of the form,

\Psi_{p} (x, \; \tau)=A e^{i(kx-\omega\tau)}\;\;.\;\;\;.\;\;\;.\;\;\;.\;(1)

Where the tau time τ is used to show that particles have the di-temporal experience of time and not the uni-temporal experience of time that applies outside the atomic world for large bodies. This is very important, and I have purposely refused to denote time by t or t2 in order to show this.

N.B: the subscript “p” in the above expression indicates that it is the Schrodinger’s equation as it applies to an atomic particle. This procedure has become a normal tradition to me as I show you the post-modern rules of the observer-atom based quantum mechanics.

The above represents a standing wave which resulted from the combination of a wave travelling in the positive x-direction and another corresponding wave travelling in the opposite direction. This condition is necessary in order to satisfy the boundary conditions.

Beyond that, I have also informed you that the universe itself is a standing gravi-electromagnetic field. The Theory of the Universe is a theory that generalizes this great truth beyond the atomic world, to even the macro world of large bodies.

You must now know that the above wave function represents a free particle carrying force F=\hbar_a\.k and energy in Joules/s2 E_a=\hbar_a\;\omega_a. In the atomic world, particles don’t carry momentum and energy in Joules. Realizing this, we can then proceed to write that,

\frac{\partial^2\;\Psi_p}{\partial^2\;x}=-k^2\: \Psi_p\;\;.\;\;\;.\;\;\;.\;\;\;.\;(2)

Remember that in the atomic world \hbar_a  is the Joules’s constant, and it is the atomic analog of the Planck’s constant \hbar_c  which now applies in the frame of an observer. These two constants are essential aspects of the post-modern observer-atom based quantum mechanics.

The above can be written in relation to E_a=\: \frac{F\: ^2}{2\: m}=\: \frac{\hbar_a\:^2\:k^2}{2\: m}  as, 

-\frac{\hbar_a\:^2}{2\;m}\; \frac{\partial^2\;\Psi_p}{\partial^2\;x}=\:\frac{F^2}{2\;m}\: \Psi_p \;\;.\;\;\;.\;\;\;.\;\;\;.\;(3)

And similarly for the relation of the wave function and time,

\frac{\partial\: \Psi_p}{\partial\;\tau}=\:-i\. \omega_a\: \Psi_p \;\;.\;\;\;.\;\;\;.\;\;\;.\;(4)

The above can also be written using E_a=\hbar_a\;\omega_a as,

i\hbar_a\:\frac{\partial\: \Psi_p}{\partial\;\tau}=\hbar_a\: \omega_a\:\psi_p=\:E_a\:\Psi_p \;\;.\;\;\;.\;\;\;.\;\;\;.\;(5)

We now generalize this to the situation in which there is both a kinetic energy in Joules/s2 and a potential energy in Joules/spresent, then E_a=\: \frac{F\: ^2}{2\: m}+ V_a\.(x}) so that we now have a new form of energy equation given as,

E_a\:\Psi_p=\:\frac{F^2}{2\: m}\Psi_p+V_a(x)\:\Psi_p\;\;.\;\;\;.\;\;\;.\;\;\;.\;(6)

In the above, Ψp is now the wave function of a particle moving in the presence of a potential Va (x). Substituting the results from equations (3) and  (5) into the above equation (6), then we will have that,

-\frac{\hbar_a\;^2}{2\;m}\;\; \frac{\partial^2\;\psi_p}{\partial^2\;x}+V_a(x)\:\Psi_p=\:i\hbar_a \;\frac{\partial\;\psi_p}{\partial\:\tau}\;\;.\;\;\;.\;\;\;.\;\;\;.\;(7)

The above is now the post-modern time-dependent Schrodinger’s equation. The great truth emerging from the post-modernization of quantum mechanics is that quantum mechanics according to the frame of the particle is dependent on force and not momentum, and on energy in Joules/s2 and not energy in Joules.

The above equation can be solved, and in fact applied as the modern form of Schrodinger’s equation has been applied. However, there is a critical difference which is that we now know where the probabilistic nature of quantum mechanics come from.

The post-modern description of quantum mechanics and of the Schrodinger’s equation now stands on a new principle of the universe. This new principle is the principle of non-inertia. It is this principle that forms the association between force and energy in Joules/s2 in post-modern quantum mechanics. 

So, we are no longer basing quantum mechanics on the complementarity principle or on the Copenhagen interpretation of quantum mechanics. The new quantum mechanics has discarded all these principles and is founded on the principle of non-inertia.

The principle of non-inertia which states that accelerated rest and accelerated motion are indistinguishable in the atomic world emerged not from post-modern quantum mechanics but from Absolute Relativity. You must understand these things so that you will realize how quantum mechanics has been radically refounded.

The Post-modern Time-Independent Schrodinger Equation

Before I proceed to show the post-modern time-independent Schrodinger equation, I want to briefly clarify what kind of time I am talking about. The time-dependent Schrodinger wave equation is dependent on the di-temporal experience of time in the atomic world. I have talked about this in the article referenced above.

So, for the time independent Schrodinger equation, I am talking about the absence of di-temporal time. We are now dealing with the wave function of the free particle without any relation to its di-temporal experience of time. This is the new basis for analysing Schrodinger’s equations.

Now, since the time dependence entered into the wave function via a complex exponential factor e^{-{\frac {i E_a\.\tau}{\hbar_a}}} . This suggests that we can remove the time dependence in the above Schrodinger’s equation by suggesting a solution of the form 

\Psi_{p} (x, \; \tau)= \psi_p\:_{(x)}\:e^{-{\frac {i E_a\.\tau}{\hbar_a}}}\;\;.\;\;\;.\;\;\;.\;\;\;.\;(8)

If we substitute the above equation (8) into the time-dependent Schrodinger equation (7), we will obtain the expression below:

-\frac{\hbar_a\.^2}{2\;m}\;\;\frac{\partial^2\;\psi_p{(x)}}{\partial^2\;x}+V_a(x)\:\psi_p{(x)}=E_a\psi_p{(x)}\;.\;\;\;.\;\;\;.\;\;\;.\;(9)

After re-arranging the terms above, we will arrive at

-\frac{\hbar_a\.^2}{2\;m}\;\;\frac{\partial^2\;\psi_p{(x)}}{\partial^2\;x}+(E_a-V_a(x)\:)\psi_p{(x)}=0\;\;.\;\;\;.\;\;\;.\;\;\;.\;(10)

The above equation is be the post-modern time-independent Schrodinger’s equation. In the above equation, the energy in Joules/s2 is a free quantity which can take any value within the boundaries determined by the potential energy Va (x).

According to the rules of quantum mechanics, the wave function should be solved at different values of energy in Joules/s2, and only the wave functions that are normalizable and have continuous derivative are accepted as viable solutions to the time-independent Schrodinger equation.

According to quantum mechanics, quantization becomes one of physical consequences of a wave function that satisfy the above two conditions of normalization and continuity. This is the point where we fall back to The Theory of the Universe in order not to err.

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While this new description quantum mechanics reveal that quantization is a phenomenon that is associated with the energy in Joules/s2 and not the energy in Joules as have been thought, there is however, a true and fundamental reason quantization arises in the atomic world.

The true reason quantization exists is buried in absolute relativity and in a basic understanding of the principle of non-inertia. I really hope to discuss it in one of my future articles, but for now, and in this article, we will emphasize this new fact and basic understanding that quantization is associated with the energy in Joules/sand not the energy in Joules.

It has become my tradition to explain certain core mysteries of the universe from the conceptual framework of Absolute Relativity and not from any other conceptual framework. 

Schrodinger’s Equations: The Observer and Particle Frames

It is important to now know and understand the observer-atom based quantum mechanics. The Schrodinger’s equations and especially the time independent equation which you have formerly known, are as they hold relative to an observer.

What I have so far showed you is the post-modern form of Schrodinger’s equation relative to atomic particles. The diagram below clearly shows you how the time-independent Schrodinger’s equation translates between an observer and a bounded particle in a finite potential well.

schrodinger's equations post-modern physics

From the above diagram, you can see that the modern Schrodinger’s equation applies to the observer, while the post-modern Schrodinger’s equation applies to the atomic particle. Now, since the particle’s frame is the domain of cause, it therefore follows that the phenomenon of quantization is not real in the domain of the observer.

This brings us to the concepts of real and apparent quantizations. Real quantization is a phenomenon in the universe associated with energy Ea in Joules/s2, which is shown above to be the free component of the Schrodinger’s equation in the frame of the particle.

Real quantization is a phenomenon in the universe associated with the energy in Joules/s^2 Click To Tweet

Without this form of energy which arises due to the principle of non-inertia the observer outside the atom would not observe quantization. And even the quantization he observes is that associated with energy E in Joules. This quantization is not real, it is what I call apparent quantization.

Apparent quantization is a phenomenon in the universe associated with the energy in Joules.Click To Tweet

Quantization cannot arise by the energy in Joules, which outside the atom is associated with the principle of inertia. I want you to know the origin of quantum mechanics, which is why I have decided in this article to associate the Schrodinger’s formalism to true causes and not to effects.

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Quantum mechanics without post-modern physics and the insights emerging from The Theory of the Universe would be so incomplete. It’s like its other half concerned with causes would be missing.

So, from now henceforth, the observer must know the limits of his reach in investigating the atomic world. Ironically, these limits or qualitative boundaries are necessary to give him the understanding of the atomic world which he seeks.

Furthermore, in this scientific article, space and time take experiential meanings even in the mathematical elucidations. The use of space x for both the observer and the particle frame is to reflect the fact that in the absolute relativistic transformations, both the observer and the particle can only experience one form of space, and never both.

However, while the observer can only experience one form of time in the absolute relativistic transformations, the atomic particle experiences the two forms of time in the absolute relativistic transformations.

This great truth makes the time-dependent Schrodinger’s equation for the observer to be of the form below, and which reflects the uni-temporal experience of time outside the atom:

-\frac{\hbar_c\;^2}{2\;m}\;\; \frac{\partial^2\;\psi_o}{\partial^2\;x}+V(x)\:\Psi_o=\:i\hbar_c \;\frac{\partial\;\psi_o}{\partial\:t}\;\;.\;\;\;.\;\;\;.\;\;\;.\;(11)

N.B: the subscript “o” in the above expression indicates that it is the Schrodinger’s equation as it applies to an observer.

Obviously, the above is Schrodinger’s equation, but now it is re-introduced from a deeply qualitative base. So, the time-dependent Schrodinger’s equation which further clarifies the di-temporal experience of time for the particle then takes the form

-\frac{\hbar_a\;^2}{2\;m}\;\;\frac{\partial^2\;\psi_p}{\partial^2\;x}+V_a(x)\:\Psi_p=\:i\hbar_a\;\frac{\partial\;\psi_p}{\partial\:\tau}=\:i\hbar_a\;\frac{\partial\;\psi_p}{\partial\:t^{2}}\;.\;\;(12)

In the equation (12), I have proceeded to replace tau time τ with time t2 to further show you how force arises in the atomic world. I have only avoided the use of time t2 and will probably continue to do so, so that you will be mindful of the fact that our notion of the square of time tarises from the product of the two forms of time which tau time τ represents.

I want you to look carefully at the above two equations like a scientist and see how the time-dependent Schrodinger’s equation in the observer’s frame is dependent on momentum and energy in Joules, while the time-dependent Schrodinger’s equation in the particle’s frame is dependent on force and energy in Joules/s2.

The differences in how the rules of quantum mechanics apply for the observer and for the particle arise because of the fundamental difference in their experience of time. I have talked about this in this article and also in The Theory of the Universe.

Crucial Discussion

If you are familiar with this blog and my methods, you would have known that all my scientific ideas, no matter how physical and relatable I make them, they still arise from a deep metaphysical base.

What you have learnt or seen in this article about the post-modern form of Schrodinger’s equations comes from the true understanding of the metaphysical nature of the universe. I want you to also have this knowledge. It is my passion for starting this blog.

I have this very honest opinion that I want to share with you. If you are familiar with the current state of science, you must have heard top scientists in the world lament that physics is in a crisis, and that there seems to be no way out.

science and universe

BBC.com

I want to let you know that there is a way out, and this way is in metaphysical science. We have come to the time in scientific history when we can no longer neglect or undermine the subtle aspects of reality, and I mean aspects beyond the reach of our tools and physical observations.

We now must know the laws behind the laws, and the principles behind the principles. The crisis which is now over was due to the fact that we had approached the end of relative science.

The time for a new kind of science has come, the kind of science that exposes fully the subtle aspects of the universe. This new kind of science is at its roots metaphysical, but we can relate its findings to the things we physically observe in the universe.

This new kind of science has produced this scientific article which now shows us the true origin of Schrodinger’s equations. This article reveals to us that in the frame of the atom and atomic particles, the Schrodinger’s equations are based on force and energy in Joules/s2.

This is the new description of post-modern quantum mechanics which recognizes, unlike modern quantum mechanics, the role of the observer and that of the particle.

Schrodinger’s equations are iconic descriptions of the atomic world, and in this new era of science, its potential to describe “a basic oneness” of the universe is now been fully discovered.

Until next time.

Take a glance at Schrodinger #winks,

– M. V. Echa



Importantly related article besides the one shown to you in the beginning of this article:

M. V. Echa

M. V. Echa

My message is the universe, my truth is the universe, and this blog contains all you need to know about the universe, from the true nature of reality to the long-sought unity of the cosmos — which is the big picture!