Schrödinger’s Cat Paradox Resolution Using GRW Collapse Model: Von Neumann Measurement Postulate Revisited

In his famous thought experiment, Schrôdinger (1935) imagined a cat that measures the value of a quantum mechanical observable with its life. Since Schrödinger’s time, no any interpretations or modifications of quantum mechanics have been proposed which give clear unambiguous answers to the questions posed by Schrödinger’s cat of how long superpositions last and when (or whether) they collapse? In this paper appropriate modification of quantum mechanics is proposed. We claim that canonical interpretation of the wave function 1 1 2 2 c c ψ ψ ψ = + is correct only when the supports of the wave functions 1 ψ and 2 ψ essentially overlap. When the wave functions 1 ψ and 2 ψ have separated supports (as in the case of the experiment that we are considering in this paper) we claim that canonical interpretation of the wave function 1 1 2 2 c c ψ ψ ψ = + is no longer valid for a such cat state. Possible solution of the Schrödinger’s cat paradox is considered. We pointed out that the collapsed state of the cat always shows definite and predictable outcomes even if cat also consists of a superposition: 1 2 cat live cat death cat c c = + .

ψ have separated supports (as in the case of the experiment that we are considering in this paper) we claim that canonical interpretation of the wave function is no longer valid for a such cat state.Possible solution of the Schrödinger's cat paradox is considered.We pointed out that the collapsed state of the cat always shows definite and predictable outcomes even if cat also consists of a superposition:

Introduction
As Weinberg recently reminded us [1], the measurement problem remains a fundamental conundrum.During measurement, the state vector of the microscopic system collapses in a probabilistic way to one of a number of classical states, in a way that is unexplained, and cannot be described by the time-dependent Schrödinger equation [1].0 a t a t = and that the measurement interaction does not disturb states i s -i.e., the measurement is "ideal".When A measures , t Ψ n the Schrödinger equation's unitary time evolution then leads to the "mea- surement state" : (I) How do we reconcile the canonical collapse model that postulates [2] definite but unpredictable outcomes with the "measurement state" .

t A
Ψ n (II) How do we reconcile the measurement that postulates definite but unpredictable outcomes with the "measurement state" t A Ψ n at each instant t and (III) How does the outcome become irreversibly recorded in light of the Schrödinger equation's unitary and, hence, reversible evolution?
This paper deals with only the special case of the measurement problem, known as Schrödinger's cat paradox (Figure 1).For a good and complete explanation of this paradox one can see Leggett [6] and Hobson [7].
In his famous thought experiment [11], Schrôdinger (1935) imagined a cat that measures the value of a quantum mechanical observable with its life.Adapted to the measurement of position of an alpha particle, the experiment is this.A cat, a flask of poison, and a radioactive source are placed in a sealed box.If an internal monitor detects radioactivity (i.e. a single atom decaying), the flask is shattered, releasing the poison that kills the cat.The Copenhagen interpretation of quantum mechanics implies that after a while, the cat is simultaneously alive and dead.Yet, when one looks in the box, one sees the cat either alive or dead, not both alive and dead.
This poses the question of when exactly quantum superposition ends and reality collapses into one possibility or the other?Since Schrödinger's time, no any interpretations or extensions of quantum mechanics have been proposed which gives clear unambiguous answers to the questions posed by Schrödinger's cat of how long superpositions last and when (or whether) they collapse.

Copenhagen interpretation
The most commonly held interpretation of quantum mechanics is the Copenhagen interpretation [12].In the Copenhagen interpretation, a system stops being a superposition of states and becomes either one or the other when an observation takes place.This thought experiment makes apparent the fact that the nature of measurement, or observation, is not well-defined in this interpretation.
The experiment can be interpreted to mean that while the box is closed, the system simultaneously exists in a superposition of the states "decayed nucleus/dead cat" and "undecayed nucleus/living cat", and that only when the box is opened and an observation performed does the wave function collapse into one of the two states.
However, one of the main scientists associated with the Copenhagen interpretation, Niels Bohr, never had in mind the observer-induced collapse of the wave function, so that Schrödinger's cat did not pose any riddle to him.The cat would be either dead or alive long before the box is opened by a conscious observer [13].Analysis of an actual experiment found that measurement alone (for example by a Geiger counter) is sufficient to collapse a quantum wave function before there is any conscious observation of the measurement [14].The view that the "observation" is taken when a particle from the nucleus hits the detector can be developed into objective collapse theories.The thought experiment requires an "unconscious observation" by the detector in order for magnification to occur.

Objective collapse theories
According to objective collapse theories, superpositions are destroyed spontaneously (irrespective of external observation) when some objective physical threshold (of time, mass, temperature, irreversibility, etc.) is reached.Thus, the cat would be expected to have settled into a definite state long before the box is opened.This could loosely be phrased as "the cat observes itself", or "the environment observes the cat".
Objective collapse theories require a modification of standard quantum mechanics to allow superpositions to be destroyed by the process of time evolution.This process, known as "decoherence", is among the fastest processes currently known to physics [15].

Ensemble interpretation
The ensemble interpretation states that superpositions are nothing but subensembles of a larger statistical ensemble.The state vector would not apply to individual cat experiments, but only to the statistics of many similarly prepared cat experiments.Proponents of this interpretation state that this makes the Schrödinger's cat paradox a trivial matter, or a non-issue.This interpretation serves to discard the idea that a single physical system in quantum mechanics has a mathematical description that corresponds to it in any way.

The canonical collapse models
In order to appreciate how canonical collapse models work, and what they are able to achieve, we briefly review the GRW model.Let us consider a system of n particles which, only for the sake of simplicity, we take to be scalar and spinless; the GRW model is defined by the following postulates: (1) The state of the system is represented by a wave function , , , . n £  (2) At random times, the wave function experiences a sudden jump of the form: , , , , , , , , , ; , , , , , , , is the state vector of the whole system at time , t im- mediately prior to the jump process and is a linear operator which is conventionally chosen equal to: where c r is a new parameter of the model which sets the width of the localiza- tion process, and ˆm x is the position operator associated to the m-th particle of the system and the random variable ˆm x corresponds to the place where the jump occurs.
Here Ĥ is the standard quantum Hamiltonian of the particle, and [ ] T ⋅ represents the effect of the spontaneous collapses on the particle's wave function.
In the position representation, this operator becomes: Another modern approach to stochastic reduction is to describe it using a stochastic nonlinear Schrödinger equation, an elegant simplied example of which is the following one particle case known as Quantum Mechanics with Universal Position Localization [QMUPL]: Here q is the position operator, and k is a constant, characteristic of the model, which sets the strength of the collapse mechanics, and it is chosen proportional to the mass m of the particle according to the formula: where 0 m is the nucleon's mass and 0 λ measures the collapse strength.It is easy to see that Equation (1.
The CSL model is defined by the following stochastic differential equation in the Fock space:

Generalized Gamov Theory of the Alpha Decay via Tunneling Using GRW Collapse Model
By 1928, George Gamow had solved the theory of the alpha decay via tunneling [8].The alpha particle is trapped in a potential well by the nucleus.Classically, it is forbidden to escape, but according to the (then) newly discovered principles of quantum mechanics, it has a tiny (but non-zero) probability of "tunneling" through the barrier and appearing on the other side to escape the nucleus.Gamow solved a model potential for the nucleus and derived, from first principles, a relationship between the half-life of the decay, and the energy of the emission.
The α -particle has total energy E and is incident on the barrier from the right to left.
where ( ) The solutions read [8]: where ( ) At the boundary 0 x = we have the following boundary conditions: From the boundary conditions (2.5)-(2.6)one obtains [8]: From (2.7) one obtain the conservation law Assumption 2.1.We assume now that: (i) at instant 0 t = the wave function ( ) where ( ) is a linear operator which is chosen equal to: 1 for , 0 , , = 0 for , 0 .

Schrödinger's Cat Paradox Resolution
In this section we shall consider the problem of the collapse of the cat state vector on the basis of two different hypotheses: (A) The canonical postulate of QM is correct in all cases.
(B) The canonical interpretation of the wave function rect only when the supports the wave functions 1 ψ and 2 ψ essentially over- lap.When the wave functions 1 ψ and 2 ψ have separated supports (as in the case of the experiment that we are considering in section II) we claim that canonical interpretation of the wave function for a such cat state (for details see Appendix C).

Consideration of the Schrödinger's Cat Paradox Using Canonical Von Neumann Postulate
Let ( ) undecayed nucleus at instant , decayed nucleus at instant .
In a good approximation we assume now that 2 0 decayed nucleus at instant 0 free particle at instant 0 .
(ii) Feynman propagator of a free α -particle is [9]: Therefore from Equations ((3.3), (2.9) and (3.4)) we obtain where We assume now that Oscillatory integral in RHS of Equation (3.5) is calculated now directly using stationary phase approximation.The phase term ( ) given by Equation (3.6) is stationary when and thus stationary point Thus from Equation (3.5) and Equation (3.10) using stationary phase approximation we obtain From Equation (3.10) we obtain Remark 3.2.From the inequality (3.7) and Equation (3.13) follows that α -particle at each instant 0 t ≥ moves quasiclassically from right to left by the law ( ) π 8 , i.e., estimating the position ( ) , , ; We assume now that a distance between radioactive source and internal monitor which detects a single atom decaying (see Figure 1) is equal to .L Proposition 3.1.After α -decay at instant 0 t = the collaps: From Equation (3.17) one obtains where Therefore from Equations ((2.11), (2.12) and (3.20), (3.21)) we obtain

T T > = P
Proof.(i)Note that for 0 t > the marginal density matrix ( ) In this case a Schrödinger's cat in fact performs the single measurement of t Ψ n -particle position with accuracy of x l δ = at instant Note that.When Schrödinger's cat has performed this measurement and the result x L l ≈ ± is observed, then the immediate post measurement state of α -particle (by von Neumann postulate C.4) will end up in the state

Thus standard formalism of continuous quantum measurements [2] [3] [4]
[5] leads to a definite but unpredictable measurement outcomes, either ( )  c x t Ψ II present an α II -particle which lives in region II with a pro- bability 2 1 c (see Figure 2).Wave packet x t Ψ I present an α I -particle which lives in region I with a probability

c
(see Figure 2) and moves from the right to the left.Note that .

∩ = ∅ I II
From Equation (3.28) follows that α I -particle at each instant 0 t ≥ moves quasiclassically from right to left by the law ( )  ) given by (for complete explanation and motivation see [16])

T = P
Thus is the collapsed state of the cat always shows definite and predictable outcomes even if cat also consists of a superposition: Contrary to van Kampen's [10] and some others' opinions, "looking" at the outcome changes nothing, beyond informing the observer of what has already happened.
We remain: there are widespread claims that Schrödinger's cat is not in a definite alive or dead state but is, instead, in a superposition of the two.van Kampen, for example, writes "The whole system is in a superposition of two states: one in which no decay has occurred and one in which it has occurred.Hence, the state of the cat also consists of a superposition: The state remains a superposition until an observer looks at the cat" [10].

Conclusions
A new quantum mechanical formalism based on the probability representation of quantum states is proposed (for complete explanation see [17]).This paper in particular deals with the special case of the measurement problem, known as Schrödinger's cat paradox.We pointed out that Schrödinger's cat demands to reconcile Born's rule.Using new quantum mechanical formalism we find that the collapsed state of the Schrödinger's cat always shows definite and predictable outcomes even if cat also consists of a superposition (see [16] [17] [18]) Using new quantum mechanical formalism the EPRB-paradox is considered successfully.We find that the EPRB-paradox can be resolved by nonprincipal and convenient relaxing of the Einstein's locality principle.

Appendix A
The time-dependent Schrodinger equation governs the time evolution of a quantum mechanical system: The average, or expectation, value i x of an observable i x corresponding to a quantum mechanical operator ˆi x is given by: ( ) .
Remark A.2.Note that under conditions given by Equation (A.
Thus under condition given by Equation (A.3) one obtains .

Appendix C. Generalized Postulates for Continuous Valued Observables
Suppose we have an n-dimensional physical quantum system with continuous observables.
I. Then we claim the following: C.I. Any given n -dimensional quantum system is identified by a set where: (i) H that is some infinite-dimensional complex Hilbert space, ,, ℑ = Ω P that is complete probability space, (iii) ( )

( )
X ω be random variable then we denote such random variable by Suppose we have an observable Q of a quantum system that is found through an exhaustive series of measurements, to have a set ℑ of values q ∈ ℑ such that ( ) that in practice any observable Q is measured to an accuracy q δ determined by the measuring device.We represent now by q the idealized state of the system in the limit 0, q δ → for which the observable definitely has the value .q II.Then we claim the following: C.II.1.The states { } : q q∈ ℑ form a complete set of δ -function norma- lized basis states for the state space ℑ H of the system.
That the states { } : q q∈ ℑ form a complete set of basis states means that any state [ ] ψ ℑ ℑ ∈ H of the system can be expressed as: ⊆ ℑ and while δ -function normalized means that ( ) The completeness condition can then be written as , d ; P q q q ψ + ℑ of obtaining the result q ∈ ℑ lying in the range ( ) → H H whose eigenvalues are the possible results { } : , q q∈ ℑ of a measurement of , Q ℑ and the associated eigenstates are the states { } : , q q∈ ℑ i.e. ˆ, .Q q q q q ℑ = ∈ ℑ Remark C.6.Note that the spectral decomposition of the operator Qℑ is then  .Suppose we have an observable Q Θ of a system that is found through an ex- haustive series of measurements, to have a continuous range of values , d ; P q q q ψ + Θ of obtaining the result q lying in the range ( ) , d ; | d d .P q q q q q c q q ψ ψ ψ , q q ψ ψ Θ = Θ and the value of observable Q Θ is measured once each on many identically prepared system, the average value of all the measurements will be ( ) The completeness condition can then be written as  ( )  , then the system will end up in the state

ˆ, d
q q q q q q q q q P q q P q q q q IV.We claim the following: ; a a q q ψ ψ Θ = Θ and the value of observable Q Θ is measured once each on many identically prepared system, the average value of all the measurements will be ( ) , ; d a P q q dq q ψ + Θ of obtaining the result q lying in the range ( ) , d ; d d .
a P q q q q a c q a q , q δ the result is  , then the system immediately after measurement will end up in the state q q q q q q q q q q q q q q q P q q P q q q q a q q q q q δ δ δ   ; , , , a a a a q q q Ψ Θ Θ = Ψ Θ Θ ∈Θ ∪Θ and the value of observable , Q Θ Θ is measured once each on many identically prepared system, the average value of all the measurements will be ( ) ; , d .
C.V. 2. The probability of getting a result q with an accuracy q δ such that ( ) ( ) a a q q q q q q δ ψ ψ i a a i i a a q q q q q q q q q q q q i i P q q P q q q q q q q q q q a q q a q q q q q q q i δ δ δ δ δ ψ δ ψ P q q q a c q a q P qq q q ψ ψ ψ when the supports of the wave functions 1 ψ and 2 ψ essentially overlap.When the wave functions 1 ψ and 2

Remark 1 . 1 .
Ensemble interpretation is in a good agreement with a canonical interpretation of the wave function (ψ -function) in canonical QM-measurement theory.However under rigorous consideration of a dynamics of the Schrödinger's cat, this interpretation gives unphysical result (see Proposition 3.2.(ii)).

( 3 )
It is assumed that the jumps are distributed in time like a Poissonian process with frequency GRW λ λ = this is the second new parameter of the model.(4) Between two consecutive jumps, the state vector evolves according to the standard Schrödinger equation.The 1-particle master equation of the GRW model takes the form

Figure 2 .
Figure 2. The particle has total energy E and is incident on the barrier ( ) V x from

PRemark 3 . 4 .
to observe a state death cat at instant T is that.In this case Schrödinger's cat in fact performs the single measurement of α -particle position with accuracy of x l δ = at instant t T = (given by Equation (3.15)) by internal monitor (see Figure1).The probability of getting a result L with accuracy of x l δ = given by Therefore at instant T the α -particle kills Schrödinger's cat with a proba- Note that.When Schrödinger's cat has performed this measurement the immediate post measurement state of α -particle (by von Neumann postulate C.4) will end up in the state in RHS of Equation(3.22) is calculated now directly using stationary phase approximation.The phase term ( ) 25) Thus from Equation (3.22) and Equation (3.25) using stationary phase ap-proximation we obtain 26) Therefore from Equation (3.22) and Equation (3.26) we obtain Assume now a Schrödinger's cat has performed the single measurement of t Ψ n -particle position with accuracy of x l δ = at instant col T T = (given by Equation (3.29)) by internal monitor (see Figure1) and the result x L l ≈ ± is not observed by Schrödinger's cat.Then the collaps: Equation (3.29)) by internal monitor (see Figure 1).The probability of getting a result L at instant col T T ≈ with accuracy of x l δ = given by ii) The probability of getting a result L at any instant col T T > with accuracy of x l δ = by Equation (3.31) and Equation (3.13) given by in this case Schrödinger's cat in fact performs a single measurement of t Ψ n -particle position with accuracy of x l δ = at instant col 35)) by internal monitor (see Figure 1).The probability of getting the result L at instant col t T = with accuracy of x l δ = by Remark 3.7 and by postulate C.V.2 and by postulate C.IV.3 (see Appendix C 3.36)    Note that, when Schrödinger's cat has performed this measurement and the result x L l ≈ ± is observed, then the immediate post measurement state of α -particle (by generalized Von Neumann postulate C.V.3) will end up in the state d Equation (A.5) into Equation (A.12) gives

1 .. 1 .Remark C. 4 . 3 .
for short.The probability density of random variable The classical pure states correspond to vectors ∈ Thus the set of all classical pure states corresponds to the unit sphere ∞ ⊂ S H in a Hilbert space H . Definition C.2.The projective Hilbert space ( ) P H of a complex Hilbert space H is the set of equivalence classes [ ] v of vectors v in H , with The equivalence classes for the relation P  are also called rays or projective rays.Remark C.3.The physical significance of the projective Hilbert space ( ) P H is that in canonical quantum theory, the states ψ and λ ψ represent the same physical state of the quantum system, for any 0 λ ≠ .It is conventional to choose a state ψ from the ray ψ     so that it has unit norm | In contrast with canonical quantum theory we have used instead contrary to P  equivalence relation , The non-classical pure states correspond to the vectors ∈ v H of a norm 1 ≠ v .Thus the set of all non-classical pure states corresponds to the set \ ∞ ⊂ H S H in the Hilbert space H .
be parti- tioned into two nonempty subsets which are open in the relative topology induced on the set.Equivalently, it is a set which cannot be partitioned into two nonempty subsets such that each subset has no points in common with the set closure of the other.Definition C.5.The well localized pure states set in  Thus the set of all well localized pure states corres- ponds to the unit sphere ∞ we claim the following: C.III.1.For the system in well localized pure state [ ] ψ Θ such that:

4 .
If the system is in well localized pure state there must be a non-zero probability to get some result on measuring observable .Q Θ C.III.5.(von Neumann measurement postulate) Assume that (i) ψ ∞ Θ ∈ S and (ii) ( ) a ψ Θ described by a wave function ( ) [ ]

7 .
Note that C.IV.3 immediately follows from C.IV.1 and C.III.2.C.IV.4.(Generalized von Neumann measurement postulate) If on performing a measurement of observable Q Θ with an accuracy next manuscript to SCIRP and we will provide best service for you:Accepting pre-submission inquiries through Email, Facebook, LinkedIn, Twitter, etc.A wide selection of journals (inclusive of 9 subjects, more than 200 journals) Providing 24-hour high-quality service User-friendly online submission system Fair and swift peer-review system Efficient typesetting and proofreading procedure Display of the result of downloads and visits, as well as the number of cited articles Maximum dissemination of your research work Submit your manuscript at: http://papersubmission.scirp.org/Or contact jamp@scirp.org To review the essentials, it is sufficient to consider two-state systems.Suppose a nucleus , n whose Hilbert space is spanned by or- .14) Remark 2.3.Note that we have chosen operators (2.10), (2.12) and (2.14) such that the boundary conditions (2.5), (2.6) are satisfied.

Resolution of the Schrödinger's Cat Paradox Using Generalized Von Neumann Postulate
1 s t or ( ) 2 s t and thus t Ψ n suddenly "collapses" at unpredictable instant t′ into the corresponding state ( ) , 1, 2.i s t i ′ = 3.2.T = P .V. 3. Assume that the system is initially in the state 11) Remark C.8.Note that C.IV.3 immediately follows from C.III.3.C .7.From statement D.5 it follows: for any state , . (D.2) where t x and t σ are given functions which depend only on variable .t D