what is the physical importance of eigen value and eigen state in quantum mechanics?

Answer 1

The reference is a little clearer explaining these concepts. Schroedinger's equations represent the physical properties of a quantum system. Unlike continuous (smooth) equations like Maxwell's equations, the quantum equations have only a finite number of discrete solutions. Each such solution is called an eigenstate. The values of the relevant parameters are called eigenvalues. Unfortunately, knowing the meaning of these terms just takes you a little way toward understanding quantum mechanics.

Answer 2

listed here are some hyperlinks below with fairly undemanding examples. that's quite innovations-blowing that there is not any longer plenty accessible that shows step-with the help of-step the thank you to define eigenvalues and eigenvectors. the final thought is to discover the eigenvalues first. that's undemanding considering the fact which you basically style the function polynomial and discover that's roots. The roots are the eigenvalues. looking the eigenvectors demands some extra desirable paintings, i.e. you nevertheless finally end up having to do matrix manipulations finding on how complicated your matrix is. do you already know the thank you to define the eigenvalues and eigenvectors of a 2x2 matrix? If no longer, I recommend you initiate with this so which you would be able to get the final thought. then you definately can attempt computing the comparable for 3x3 matrices. finally you could start questioning approximately favourite algorithms. Do all this manually (with the help of hand) and easily use utility like Matlab to verify solutions. pay attention of Matlab's rref function in early student variations.

Answer 3

OK, well the short answer is that all stationary states (that is, not changing in time) are eigenstates of the Schroedinger equation, and the eigenvalues are the properties (e.g. energy, momentum) of those states.

Answer 4

The Importance of Operators in Quantum Theory

I think there is a fundamental misunderstanding in the approach of especially Joos and probably most other decoherence advocates. They think, first of all, that operators can be dispensed with and eigenvectors have some meaning apart from them, i.e., they deny the eigenvalue-eigenvector link. I think they have not understood one of the unspoken premises of original quantum theory, or else they have consciously rejected it for a reason I would like to have them explain.

IMHO, we cannot see an eigenvector - rather, we see the eigen-value. That is because we cannot factually observe either a phenomenal object or a noumenal object. What the observer sees is always itself in the object under observation. It must reconstruct the object out of itself, since it has no other capacity than that which is within itself. That is the metaphysical meaning of the operator equation: OE=@E. To dispense with the eigenvalue-eigenvector link is to sever the connection of quantum theory to the real structure of our experience. To have captured it in the first place is the superlative genius of quantum theory.

So, given the premise that operators *are* fundamental, we should follow von Neumann's approach of seeking a new ring of operators for quantum theory. What does that mean? It means that we must inquire how to represent as operators the results of the psycho-physical reduction, which will include:

1) the Subject-Superject System,
2) the Measuring Apparatus, in the light of which 1) will become
3) the Quantum System, and
4) the Observer, embodied as part of the total Observer-Observed system, which will lead to the further consideration of,
5) the Environment of the Observer,
6) the Environment of the Measuring Apparatus, and
7) the Environment of the Quantum System.

One must also consider:

a) the real meaning of the position representation,
b) the real meaning of the momentum representation,
c) the real meaning of the coherent state representation,
d) the inverse representation from the point of view of the embodied observer, rather than the detached observer, and
e) the nondual inverse representation, which establishes the phenomenological meaning of the eigenvalue-eigenvector link.

The mathematics must reflect the real metaphysical character of these subsystems and representations, which can only be gleaned from direct insight into the transcendental reality of our experience. Alfred North Whitehead of course gave us 1), William James gave us 2), and it turns out that Immanuel Kant has given us 4). It is obvious that a complete treatment of these elements will substantially alter and perhaps reverse the conclusions of the decoherence argument, as well as provide a *real* solution to the quantum measurement problem.

I would submit that with this knowledge, which I can now provide, the fulfillment of von Neumann's goal is for the first time within reach, if anyone cares to proceed in the direction that leads to a real, rather than an illusory, advance. Moreover, for the first time the intuitive but unspoken assumptions of the Copenhagen Interpretation can now be fully explained, thus clarifying once and for all the confusion known as the "quantum muddle".

The operator for 1) is related to the fact that the concrescence of the actual entity has unfolded backwards, across the oroboric abyss, from the present subject, which has emerged from the phenomenal flow of time. This backwards unfoldment of the phenomenal flow of time into the meta-physical present subject and then into the physical and mental actual entity of nature must be related to the forward transformation of the future by the present, which the operator for the Subject-Superject System expresses.

2) is the bit of experience, neutral between content and consciousness. Whitehead made very clear that his process theory was deeper than the subjects treated by James, such as consciousness, thought, and sense-perception (in the mode of presentational immediacy). These subjects are, however, the key to the second hypostasis of von Neumann, for what are they but the ideal Measuring Apparatus? In the atomic form, as the bits of experience, neutral between content and consciousness, they are the Measuring Apparatus itself, as an element of the total quantum reality, while as the stream of consciousness they are the operator pertaining to the Measuring Apparatus.

The operator for 4) refers to the Observer observing itself observing. The notion of the *body-world schema* provides the transcendental clue here, since the body is the organism's perception of itself as a phenomenon. The world is the organism's perception of its environment as a phenomenon that is externally real. So this type of operator leads naturally into the consideration of the environment of the Observer, which is 5). The body-world schema can be extended through the notions of noema and dianoia-noesis, as well as noumenon. These lead to the operators for the environment of the Measuring Apparatus, which is 6), and the environment of the Quantum System, which is 7), respectively.

The position representation involves the projection of the classical subject and its enveloping object onto the quantum noumenal reality, as the bra and ket vector, respectively. Thought as the sixth skhanda, or tendency of the Soul, in the astral sub-world of the meta-physical world, is also projected as the seed for the transformation of the systems represented by the wave function into the potential (in the emotional world) for an actual event (in the mental world).

The form of the principle of the psycho-physical parallelism actually employed by von Neumann in his book reduces mental, emotional, and physical worlds, as III, II, and I, into mental, causal, and astral sub-worlds of the physical world. By moving the cut, von Neumann empties III of materiality, until only the "ego sum" remains, which von Neumann somewhat mistakenly calls the abstract "ego". The latter is rather the psychological Freudian ego within the classical order. This "ego sum", however, must be clearly distinguished from the "ego sum" of Descartes and Husserl, which is gained by the phenomenological reduction rather than the psycho-physical reduction.

In any case, the point is that once the "ego sum" is deduced, the systems represented by the wave function are then seen in the context of the "ego sum", which is indeed the essential meaning of the principle of the psycho-physical parallelism as von Neumann defines it. Therefore, the transformed systems as the potential for an actual event have the character of an "I", and the the actual event has the character of an "IT". The actual event, or IT, then further spawns a THOU and a THOUGHT. I, IT, and THOU, are like a I, II, and III, while THOUGHT is a biofeedback circuit, which brings the quantum noumenal reality into accord with the actual event in the mental world and hence effects the state vector reduction implied by the actual event.

To understand the Hilbert space character of the position representation, one must realize that the classical object already contains the quantum notion of Potentiality as something between the classical notions of Extension and Thought, or Matter and Spirit. It also contains the notions of Possibility, Actuality, and Reality. Possibility represents a spiritual transformation of the classical subject and as such a subjective component of an objective reality, i.e., a basis state.

The essential character of the classical order is that it describes the behavior of objects. It does not at all inquire into or establish the existential status of those objects. So, the Operator for the position representation, which is just a projection of classical reality onto quantum noumenal reality, will represent Behavior, i.e., it is the Behavior Operator.

The momentum representation has to do with the type of process investigated by Whitehead. It starts with the phenomenal flow of time, in the fifth or phenomenal world. The present then ex-ists or stands apart from its phenomenal nexus of past and future in the attempt to be truly present in the existential sense. As such, however, the present subject in the seventh or meta-physical world is subject to both impermanence and contingency. It therefore attempts to define itself as an essential element or actual entity of nature by catapulting itself across the oroboric abyss separating the meta-physical from the physical world. It further expands itself into a mental pole in addition to its physical pole. The dipolar actual entity is later reduced by the principle of the psycho-physical parallelism to an atomic subject in the physical world. This is the bra vector for the momentum representation. The mental pole of the actual entity has the character of energy, and the momentum of the present moment ingresses into the atomic subject in the same way that an eternal object would ingress. As Bohm has declared, NOW is the synthesis of the present and the eternal.

The object, or ket vector, for the momentum representation is a synthesis of the objects which are spawned by the prehensions of all past concrescences by the present actual entity, i.e., the objects are intersubjective connections. These objects are, individually, the components of the ket vector, and the object which synthesizes them in the course of the superjective part of the process is the ket vector. The operator for the momentum representation is therefore a universal operator that has as its eigenvectors all concrescences in the universe.

Once the superjective process, which synthesizes the atomic subject with all of its objects, is completed and the superject is also reduced to the physical world by the principle of the psycho-physical parallelism, then the subject, object, and superject become a settled Subject-Superject System or Concrescence in the second or emotional or psycho-physical world. It is at this point that the Measuring Apparatus and the Observer ingress into the Concrescence as derivative stages of the process. This ingression is another form of reduction by the principle of the psycho-physical parallelism. All three subsystems take on an Observer-Observed character, which is already present as the Subject of the original System.

The Observer-Observed pair that pertains to the Measuring Apparatus then assumes the role of an inverse operator that represents the point of view of the embodied Observer. Even as the detached Observer in the seventh or meta-physical world sees the noumenon in the first or physical world through the phenomenon in the fifth or phenomenal world, the embodied Observer in the second or emotional world sees the phenomenon in the fifth world through the noumenon in the first world. It represents the noumenal perception of the atomic Observer, who has become part of the total quantum reality.

The assumption here is that by regarding the Observer, which is initially an extra-physical element of the observational procedure, as physical, or in this case psycho-physical, according to the principle of the psycho-physical parallelism, we actually generate, in quantum mechanics, such a psycho-physical observer. "Regarding", in quantum theory, is not an idle or passive matter.

It is only the detached Observer who sees everything from the perspective of the Spacetime Continuum, and so the requirement that measurable observables be local does not apply to the embodied Observer. One could say that Configuration Space was reality to the embodied Observer, and to be local in Configuration Space is the only requirement for the measurable observable of an embodied Observer. The embodied Observer sees reality in terms of information rather than out-formation.

The classical object that gets projected onto quantum noumenal reality to generate the position representation is within the classical order, while the quantum objects that get synthesized into the unified quantum object in the generation of the momentum representation are in the quantum explicate order. A classical object within the classical order, which is the physical sub-world of the meta-physical world within the ontological system of worlds that is the quantum explicate order, is therefore small by comparison to the inherently quantal objects that span the entire lower worlds of the quantum explicate order. The unified quantum object actually spans the entire fourth or etheric world, while the objects that are components of that object span the three lowest worlds (mental, emotional, and physical).

So, the only way that the classical object can be realized as identical to the unified quantum object, which it must be since they are the same object in different representations, is to enfold the unified quantum object into the quantum implicate order, where it juxtaposes to the classical object as a component of mind to a component of the material realm. That is because the quantum implicate order is the Mind of the Observer, and the Observer comprises the classical order.

To understand the coherent state, which is an eigenstate of x+ip, one must find the synthesis of the small classical object and the large unified quantum object within the quantum explicate order. That synthesis is none other than the phenomenal object in the fifth or phenomenal world. So, the coherent states represent the phenomenon proper, whereas Bohr uses the term rather loosely to apply to what in fact is the quantum noumenon, not the quantum phenomenon.

Now, although the coherent states are classical-like insofar as they exhibit the least possible uncertainty, they are not an orthogonal set of basis states. To get something like the coherent state, which is the eigenstate of a synthesis of the conjugate representations, x and p, one must consider the nondual inverse operator. That is like the inverse operator, except that the inverse operator acts within the ontological system of worlds, whereas the nondual inverse operator operates in the synthesis of the ontological system of worlds and the phenomenological system of worlds. This synthesis begins with the physical world of the ontological system of worlds. It then moves across the oroboric abyss between the physical and the meta-physical worlds, but it interprets the abyss and what lies on the other side of the abyss in phenomenological, not ontological, terms. The result is worlds representing I, II, and III, and then 4 leading into another 4, where the second 4 is the consciousness of the actual observer (III), and the first 4 is the connection between actual observer and the consciousness of the actual observer. In this new synthesized system of worlds, I is physical, II is astral, III is causal, the first 4 (the large 4) is mental and etheric, while the second four (the small 4) is the soul world.

Beyond the soul world are the worlds reached by soul travel, and these comprise a 5 and a 6 (five earth realms and six heaven realms). Beyond these are the seven worlds or *lokas* of yogiraj-dhama, and these comprise a 7. Beyond these are Mahakala-dhama and Sadasiva-dhama and then Hari-loka and then Goloka Vrndavana, which is the "central spiritual order" of the "one", often mentioned by Werner Heisenberg (in "Physics and Beyond" for instance). This central spiritual order overshadows the physical world (oroborically) and its action upon the physical world is the action of the nondual inverse operator on the eigenvectors of quantum noumenal reality. Whitehead's Concrescence is in fact a component of the central spiritual order, so the Observer-Observed pairs mentioned earlier are here in the central spiritual order and constitute the nondual inverse operator.

The first 4, in the diagrammatical aspect of the number "4", rises from the eigen-value of such an operator and the second 4 feeds into the eigenvector of such an operator, thus establishing the eigenvalue-eigenvector link. The first 4 symbolizes primal experience, which is correlated to the quantum Possibility for a classical subject. Since the entire Concrescence exists in the context of the momentum representation, the nondual inverse operator of the embodied Observer also constitutes the momentum or conjugate representation, and yet it includes the essence of the position representation as the primal or pure phenomenologically reduced experience. So, this type of representation combines conjugate representations, as does the coherent state representation, but we believe that it will lead to orthogonal basis states, whereas the coherent state representation does not.

It might be mentioned in passing that the four forces are none other than the four corners united diagrammatically by the first 4, and they are fundamentally the objectivized phases of the primal or pure phenomenologically reduced experience: boundless energy (the strong force), bounded energy (the weak force), unconditioned luminosity (the electromagnetic force), and infinite space and consciousness (the gravitational force). These also correspond, within the classical order, as the Possibility for the classical subject, to the four primary excitory neurotransmitters: dopamine, acetylcholine, serotonin, and noradrenaline.


The Total State and the Field

Now, we still have not mentioned the the total system, which will be defined as S + M + O + (MWI) E_M + E_S, where (MWI) E_M means Mental World Image of the Environment of the Measuring Apparatus. This total system is indeed a higher order object, which includes all of the subjective elements of the subsystems. It defines Dasein, and the principles that govern it are Gestalt and, in an overarching sense, Geometry-Of-Divinity (GOD). Gestalt means Structured Form, and it is the Measure Space, defined in terms of certain scalar valued functions, in the passive sense, and the Algebra of the operators of the total system and its subsystems, in the active sense.

The basic idea is that the operator of the total system represents the Organism and its eigenvector represents the Environment of the Observer. It is only the Environment of the Observer that is not absorbed into the total system, and this is exactly what provides the context for the observation of the total system. Here we are not concerned with the Organism's perception of itself or its Environment, but with its real relation to the Environment of the Observer.

Now, the Field in Radically Unified Quantum Field Theory (RUQFT) is also a real relationship of the Organism to the Environment of the Observer. The Organism has stood apart from the Environment in order to ex-ist, and the Field is the bond that sustains the Organism in its separation from the Environment. The Organism introjects the Field and it hence becomes a Bohm-like guidance field, but of the Observer rather than the Observed. It informs the Organism how to construct the noumenal reality within its inner psyche, which is the only place that noumenal reality truly exists from this metaphysical perspective. The real interpenetration of metaphysical systems of worlds allows for this identification of the inner psyche of the Organism/Observer with the quantum noumenal reality.

In any case, this Field can be expanded in terms of fundamental operators, which include Destruction and Creation operators but are not limited to them. We indeed need a new algebra, transcending Destruction and Creation operators as coefficients of momentum eigenstates. The reason is that the real elements in RUQFT are not just particles created out of nothing (the vacuum or ground state) - they are also real things projected into reality from the universal idea, as Plato believed. This universal idea is closely related to the universal substance, which is the GOD of Spinoza, already encountered in our discussion of the total state.

So, the point is that the full set of fundamental operators will be: Destruction, Ideation, Reduction, Affection, and Creation (DIRAC). They will not just create or destroy particles in momentum eigenstates. Rather, one must inquire deeply into the meaning of motion as holomovement and redefine quantum kinematics at the fundamental level. What exists is intimately related to the holomovement itself, so one cannot just talk about the creation or destruction of particles in motion. At a fundamental level the particles are the motion, as holomovement, so there is plenty of room to redefine the fundamental constituents of our theory in a way that will be consistent with the inclusion of Ideas and Ideation as essential elements of reality.

This injection of Idealism into Science is exactly what it needs to get it off the ground of physicalism and allow it to become a truly dialectical and metaphysical reflection of our actual experience, in all of its infinite depth. This will also allow the formulation of a "final theory", that is both fundamental and entirely self-consistent. Its explanatory power will be complete.


NEW LAW for Observables/Beables

The total state is useful mainly for showing the connection between measurement theory and the field of RUQFT. To consider observables, which turn out to be also beables, it is useful to consider only the composite system S + M + O. This total Concrescence, including the derivative stages implied by the ingression of M and O, has several functions in diverse contexts, but the one to be considered here has to do with its dynamical development and its subsequent observability.

We will state without proof, at this point, what the new law for probabilities is in the new theory I am proposing here. It is simply:

Trace (RAMA e^iHt D-M KRISHNA e^-iHt O_D),

where RAMA is a destruction operator of the unified field, H is the Hamiltonian of S + M + O, D-M is the Density Matrix for the preparation of S + M + O, KRISHNA is a creation operator for the unified field, and O_D is the Density Operator for the observation (which is essentially a Projection Operator, O|P, with its own non-Schrodinger type Time-Development Operator, O/T-D).

Although the above clearly describes an observable, to see what that observable is and to see that it is in fact a beable, we rewrite the above formula as:

Trace (e^-iHt O_D RAMA e^iHt D-M KRISHNA),

using the well known property of the trace that it is invariant under the permutation of its terms. This property is especially applicable in an oroborically linked metaphysical system of worlds, such as I have been proposing.

It is then clear that what is being observed is the probability for a creation in the quantum world. This is a real beable that is revealed here as the source of our human phenomenal experience, as represented by the first formula. We can add some brackets to the first formula to further reveal its meaning:

Trace ([RAMA {e^iHt D-M KRISHNA} e^-iHt] O_D).

The inner core involves the act of creation in the quantum world, while the surrounding destruction is a nihilation (Sartre's term) that isolates the core of being under observation and, in modern parlance, shields it from environmental decoherence.

This is the central conclusion and demonstrates that the reality of our human phenomenal experience is indeed the reality of the underlying quantum world, untouched by environmental decoherence. Our phenomenal experience is of course structured classically, but the reality, or essence, of it is quantal. If we could then build measuring devices that truly captured our human mode of perception and even amplified it or if we could improve our own human perception through some practice or discipline, then we could easily prove the above formulas to be correct and to be the natural modes of observation present in nature. Ordinarily, of course, we do not perceive so finely as to recognize that we are witnessing creations in the quantum world, but it is said that through yoga or meditation one can indeed gain that type of finely honed perception, even to the extent of recognizing the individual acts of creation.

You can also see http://en.wikipedia.org/wiki/Eigenvalue for more information