ABSTRACT

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 $\psi =c_1{\psi }_1+c_2{\psi }_2$ is correct only when the supports of the wave functions ${\psi }_1$ and ${\psi }_2$ essentially overlap. When the wave functions ${\psi }_1$ and ${\psi }_2$ 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in this paper) we claim that canonical interpretation of the wave function $\psi =c_1{\psi }_1+c_2{\psi }_2$ is no longer valid for a such cat state.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: $|\text{cat}\rangle =c_1|\text{livecat}\rangle +c_2|\text{deathcat}\rangle$ .

Keywords:

Probability Representation of Quantum States, Schrödinger’s Cat, GRW Collapse Model, Von Neumann Measurement Postulate

1. 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] . To review the essentials, it is sufficient to consider two-state systems. Suppose a nucleus $n\mathrm{,}$ whose Hilbert space is spanned by orthonormal states $|s_i\left(t\right)\rangle ,i=1,2,$ where $|s_1\left(t\right)\rangle =|\text{undecayednucleusatinstant}t\rangle$ and $|s_2\left(t\right)\rangle =|\text{decayednucleusatinstant}t\rangle$ is in the superposition state,

${|{\Psi }_t\rangle }_n=c_1|s_1\left(t\right)\rangle +c_2|s_2\left(t\right)\rangle ,{\left|c_1\right|}^2+{\left|c_2\right|}^2=1.$ (1.1)

A measurement apparatus $A\mathrm{,}$ which may be microscopic or macroscopic, is designed to distinguish between states $|s_i\left(t\right)\rangle$ by transitioning at each instant $t$ into state $|a_i\left(t\right)\rangle$ if it finds that $n$ is in $|s_i\left(t\right)\rangle ,i=1,2.$ Assume that the detector is reliable, implying that the $|a_1\left(t\right)\rangle$ and $|a_2\left(t\right)\rangle$ are orthonormal at each instant $t$ , i.e., $\langle a_1\left(t\right)|a_2\left(t\right)\rangle =0$ and that the measurement interaction does not disturb states $|s_i\rangle$ ―i.e., the measurement is “ideal”. When $A$ measures ${|{\Psi }_t\rangle }_n\mathrm{,}$ the Schrödinger equation’s unitary time evolution then leads to the “measurement state” ${|{\Psi }_t\rangle }_{nA}\mathrm{<mo>\; :\; </mo>}$

${|{\Psi }_t\rangle }_{nA}=c_1|a_1\left(t\right)\rangle +c_2|a_2\left(t\right)\rangle \mathrm{,}{\left|c_1\right|}^2+{\left|c_2\right|}^2=1.$ (1.2)

of the composite system $nA$ following the measurement.

Standard formalism of continuous quantum measurements [2] [3] [4] [5] leads to a definite but unpredictable measurement outcome, either $|a_1\left(t\right)\rangle$ or $|a_2\left(t\right)\rangle$ and that ${|{\Psi }_t\rangle }_n$ suddenly “collapses” at instant $t^{\prime }$ into the corresponding state $|s_i\left(t^{\prime }\right)\rangle \mathrm{.}$ But unfortunately Equation (1.2) does not appear to resemble such a collapsed state at instant $t^{\prime }\mathrm{<mo>\; ?\; </mo>}$ .

The measurement problem is as follows:

(I) How do we reconcile the canonical collapse model that postulates [2] definite but unpredictable outcomes with the “measurement state” ${|{\Psi }_t\rangle }_{nA}.$

(II) How do we reconcile the measurement that postulates definite but unpredictable outcomes with the “measurement state” ${|{\Psi }_t\rangle }_{nA}$ 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.

The canonical interpretations of the experiment

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.

Remark 1.1. Ensemble interpretation is in a good agreement with a canonical interpretation of the wave function ( $\psi$ -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)).

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 ${\psi }_t\left(x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_n\right)$ belonging to the Hilbert space ${\pounds }_2\left({ℝ}^{3n}\right)\mathrm{.}$ (2) At random times, the wave function experiences a sudden jump of the form:

${\psi }_t\left(x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_n\right)\to {\psi }_t\left(x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_n\mathrm{<mo>\; ;\; </mo>}{\tilde{x}}_m\right)=\frac{{\Re }_m\left({\tilde{x}}_m\right){\psi }_t\left(x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_n\right)}{{\Vert {\Re }_m\left({\tilde{x}}_m\right){\psi }_t\left(x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_n\right)\Vert }_2},$ (1.3)

where ${\psi }_t\left(x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_n\right)$ is the state vector of the whole system at time $t\mathrm{,}$ immediately prior to the jump process and ${\Re }_n\left({\tilde{x}}_m\right)$ is a linear operator which is conventionally chosen equal to:

${\Re }_m\left({\tilde{x}}_m\right)={\left(\text{π}{r}_c^2\right)}^{-\mathrm{3\; <mo>\; /\; </mo>4}}exp\left[-\frac{{\left({\hat{x}}_m-{\tilde{x}}_m\right)}^2}{2r_c^2}\right]\mathrm{,}$ (1.4)

where $r_c$ is a new parameter of the model which sets the width of the localization process, and ${\hat{x}}_m$ is the position operator associated to the m-th particle of the system and the random variable ${\hat{x}}_m$ corresponds to the place where the jump occurs. (3) It is assumed that the jumps are distributed in time like a Poissonian process with frequency $\lambda ={\lambda }_{\text{GRW}}$ 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

$\frac{\text{d}}{\text{d}t}\rho \left(t\right)=-\frac{i}{\hslash }\left[\hat{H}\mathrm{,}\rho \left(t\right)\right]-T\left[\rho \left(t\right)\right]\mathrm{.}$ (1.5)

Here $\hat{H}$ is the standard quantum Hamiltonian of the particle, and $T[\cdot ]$ represents the effect of the spontaneous collapses on the particle’s wave function. In the position representation, this operator becomes:

$\langle x|T\left[\rho \left(t\right)\right]|y\rangle =\lambda \left\{1-exp\left[-\frac{{\left(x-y\right)}^2}{4r_c^2}\right]\right\}\langle x|\rho \left(t\right)|y\rangle \mathrm{.}$ (1.6)

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]:

$\text{d}|{\psi }_t\left(x\right)\rangle =\left[-\frac{i}{\hslash }\hat{H}-k{\left(\hat{q}-\langle q_t\rangle \right)}^2\text{d}t\right]|{\psi }_t\left(x\right)\rangle \text{d}t+\sqrt{2k}\left(\hat{q}-\langle q_t\rangle \right)\text{d}{W}_t|{\psi }_t\left(x\right)\rangle \mathrm{.}$ (1.7)

Here $\hat{q}$ is the position operator, $\langle q_t\rangle =\langle {\psi }_t|\hat{q}|{\psi }_t\rangle$ it is its expectation value, 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: $k=\left(m/m_0\right){\lambda }_0,$ where $m_0$ is the nucleon’s mass and ${\lambda }_0$ measures the collapse strength. It is easy to see that Equation (1.5) contains both non-linear and stochastic terms, which are necessary to induce the collapse of the wave function. For example let us consider a free particle ( $\hat{H}=p^2/2m$ ), and a Gaussian state:

${\psi }_t\left(x\right)=\exp \left\{-a_t{\left(x-{\bar{x}}_t\right)}^2+i{\bar{k}}_{t}x\right\}.$ (1.8)

It is easy to see that ${\psi }_t\left(x\right)$ given by Equation (1.6) is solution of Equation (1.5), where

$\frac{\text{d}{a}_t}{\text{d}t}=k-\frac{2i\hslash }{m}{a}_t^2,\frac{\text{d}{\bar{x}}_t}{\text{d}t}=\frac{\hslash }{m}{\bar{k}}_t+\frac{\sqrt{k}}{2\mathrm{Re}\left(a_t\right)}{\stackrel{\dot{}}{W}}_t,\frac{\text{d}{\bar{k}}_t}{\text{d}t}=-\sqrt{k}\frac{\mathrm{Im}\left(a_t\right)}{\mathrm{Re}\left(a_t\right)}{\stackrel{\dot{}}{W}}_t$ (1.9)

The CSL model is defined by the following stochastic differential equation in the Fock space:

$\begin{array}{l}\text{d}|{\psi }_t\left(x\right)\rangle =\left[-\frac{i}{\hslash }\hat{H}-k{\left(\hat{M}\left(x\right)-\langle M_t\left(x\right)\rangle \right)}^2\text{d}t\right]|{\psi }_t\left(x\right)\rangle \text{d}t\\ +\sqrt{2k}\left(\hat{M}\left(x\right)-\langle M_t\left(x\right)\rangle \right)\text{d}{W}_t\left(x\right)|{\psi }_t\left(x\right)\rangle .\end{array}$ (1.10)

2. 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 $\alpha$ -particle has total energy $E$ and is incident on the barrier from the right to left.

The Schrödinger equation in each of regions $I=\left\{x|x<0\right\}\mathrm{,}II=\left\{x|0\le x\le l\right\}$ and $III=\left\{x|x>l\right\}$ takes the following form

$\frac{{\partial }^2\Psi \left(x\right)}{\partial x^2}+\frac{2m}{{\hslash }^2}\left[E-U\left(x\right)\right]\Psi \left(x\right)=0,$ (2.1)

where

$U\left(x\right)\mathrm{<mo>\; =\; </mo>}\{\begin{array}{l}0\text{for}x<0\hfill \\ U_0\text{for}0\le x\le l\hfill \\ 0\text{for}x>l\hfill \end{array}$ (2.2)

The solutions read [8] :

$\begin{array}{l}{\Psi }_{III}\left(x\right)\mathrm{<mo>\; =\; </mo>}{C}_{+}exp\left(ikx\right)+C_{-}exp\left(-ikx\right)\mathrm{,}\hfill \\ {\Psi }_{II}\left(x\right)\mathrm{<mo>\; =\; </mo>}{B}_{+}exp\left(k^{\text{'}}x\right)+B_{-}exp\left(-k^{\text{'}}x\right)\mathrm{,}\hfill \\ \begin{array}{c}{\Psi }_I\left(x\right)=Acos\left(kx\right)\\ =\frac{A}{2}\left[exp\left(ikx\right)+exp\left(-ikx\right)\right]\mathrm{,}\end{array}\hfill \end{array}$ (2.3)

where

$\begin{array}{l}k=\frac{2\text{π}}{\hslash }\sqrt{2mE},\hfill \\ k^{\prime }=\frac{2\text{π}}{\hslash }\sqrt{2m\left(U_0-E\right)}.\hfill \end{array}$ (2.4)

At the boundary $x=0$ we have the following boundary conditions:

$\begin{array}{c}{{\Psi }_I\left(0\right)|}_{x=0}={{\Psi }_{II}\left(0\right)|}_{x=0}\mathrm{,}\\ {\frac{\partial {\Psi }_I\left(x\right)}{\partial x}|}_{x=0}={\frac{\partial {\Psi }_{II}\left(x\right)}{\partial x}|}_{x=0}.\end{array}$ (2.5)

At the boundary $x=l$ we have the following boundary conditions

$\begin{array}{c}{{\Psi }_{II}\left(l\right)|}_{x=l}={{\Psi }_{III}\left(l\right)|}_{x=l},\\ {\frac{\partial {\Psi }_{II}\left(x\right)}{\partial x}|}_{x=l}={\frac{\partial {\Psi }_{III}\left(x\right)}{\partial x}|}_{x=l}.\end{array}$ (2.6)

From the boundary conditions (2.5)-(2.6) one obtains [8] :

$\begin{array}{l}{B}_{+}=\frac{A}{2}\left(1+i\frac{k}{k^{\prime }}\right),B_{-}=\frac{A}{2}\left(1-i\frac{k}{k^{\prime }}\right),\\ C_{+}=A\left[\text{ch}\left(k^{\prime }l\right)+iD\text{sh}\left(k^{\prime }l\right)\right],C_{-}=i\left(AS\text{sh}\left(k^{\prime }l\right)\exp \left(ikl\right)\right),\\ D=\frac{1}{2}\left(\frac{k}{k^{\prime }}-\frac{k^{\prime }}{k}\right),S=\frac{1}{2}\left(\frac{k}{k^{\prime }}+\frac{k^{\prime }}{k}\right).\end{array}$ (2.7)

From (2.7) one obtain the conservation law

${\left|A\right|}^2={\left|C_{+}\right|}^2-{\left|C_{-}\right|}^2.$

Let us introduce now a function $E_{II}\left(x\mathrm{,}l\right)={\theta }_2\left(x,l\right)E_2\left(x,l\right)$ where

$\begin{array}{l}{E}_2\left(x\mathrm{,}l\right)=\{\begin{array}{l}{\left(\text{π}{r}_c^2\right)}^{-1/4}exp\left(-\frac{x^2}{2r_c^2}\right)\text{for}-\infty \mathrm{<mo>\; <\; </mo>}x\mathrm{<mo>\; <\; </mo>}\frac{l}{2}\hfill \\ {\left(\text{π}{r}_c^2\right)}^{-1/4}exp\left(-\frac{{\left(x-l\right)}^2}{2r_c^2}\right)\text{for}\frac{l}{2}\le x\mathrm{<mo>\; <\; </mo>}\infty \hfill \end{array}\hfill \\ {\theta }_2\left(x\mathrm{,}l\right)=\{\begin{array}{l}1\text{for}x\in \left[\mathrm{0,}l\right]\hfill \\ 0\text{for}x\notin \left[\mathrm{0,}l\right]\hfill \end{array}.\hfill \end{array}$ (2.8)

Assumption 2.1. We assume now that:

(i) at instant $t=0$ the wave function ${\Psi }_I\left(x\right)$ experiences a sudden jump of the form

${\Psi }_I\left(x\right)\to {\Psi }_I^{\mathrm{<mo>\; \#\; </mo>}}\left(x\right)\mathrm{<mo>\; =\; </mo>}\frac{{\Re }_I\left(\hat{x}\right){\Psi }_I\left(x\right)}{{\Vert {\Re }_I\left(\hat{x}\right){\Psi }_I\left(x\right)\Vert }_2}\mathrm{,}$ (2.9)

where ${\Re }_I\left(\hat{x}\right)$ is a linear operator which is chosen equal to:

${\Re }_I\left(\hat{x}\right)={\left(\text{π}{r}_c^2\right)}^{-1/4}{\theta }_1\left(\hat{x}\mathrm{,}l\right)exp\left[-\frac{{\hat{x}}^2}{2r_c^2}\right]\mathrm{<mo>\; ;\; </mo>}$ (2.10)

where

${\theta }_1\left(x\mathrm{,}l\right)\mathrm{<mo>\; =\; </mo>}\{\begin{array}{l}1\text{for}x\in \left[-l\mathrm{,0}\right]\mathrm{,}\hfill \\ 0\text{for}x\notin \left[-l\mathrm{,0}\right]\mathrm{.}\hfill \end{array}$

Remark 2.1. Note that: $\text{supp}\left({\Psi }_I^{\mathrm{<mo>\; \#\; </mo>}}\left(x\right)\right)\subseteq \left[-l\mathrm{,0}\right].$

(ii) at instant $t=0$ the wave function ${\Psi }_{II}\left(x\right)$ experiences a sudden jump of the form

${\Psi }_{II}\left(x\right)\to {\Psi }_{II}^{\mathrm{<mo>\; \#\; </mo>}}\left(x\right)\mathrm{<mo>\; =\; </mo>}\frac{{\Re }_{II}\left(\hat{x}\right){\Psi }_{II}\left(x\right)}{{\Vert {\Re }_{II}\left(\hat{x}\right){\Psi }_{II}\left(x\right)\Vert }_2},$ (2.11)

where ${\Re }_{II}\left(\hat{x}\right)$ is a linear operator which is chosen equal to:

${\Re }_{II}\left(\hat{x}\right)=E_{II}\left(\hat{x}\mathrm{,}l\right)\mathrm{<mo>\; ;\; </mo>}$ (2.12)

Remark 2.2. Note that: $\text{supp}\left({\Psi }_{II}^{\mathrm{<mo>\; \#\; </mo>}}\left(x\right)\right)\subseteq \left[\mathrm{0,}l\right]\mathrm{.}$

(iii) at instant $t=0$ the wave function ${\Psi }_{{}_{III}}\left(x\right)$ experiences a sudden jump of the form

${\Psi }_{III}\left(x\right)\to {\Psi }_{III}^{\mathrm{<mo>\; \#\; </mo>}}\left(x\right)=\frac{{\Re }_{III}\left(\hat{x}\right){\Psi }_{III}\left(x\right)}{{\Vert {\Re }_{III}\left(\hat{x}\right){\Psi }_{III}\left(x\right)\Vert }_2},$ (2.13)

where ${\Re }_{III}\left(\hat{x}\right)$ is a linear operator which is chosen equal to:

${\Re }_{III}\left(\hat{x}\right)\mathrm{<mo>\; =\; </mo>}{\left(\text{π}{r}_c^2\right)}^{-1/4}exp\left[-\frac{{\left(\hat{x}-l\right)}^2}{2r_c^2}\right]$ . (2.14)

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.

Definition 2.1. Let $\Psi \left(x\right)$ be a solution of the Schrödinger Equation (2.1). The stationary Schrödinger Equation (2.1) is a weakly well preserved in region $\Gamma \subseteq ℝ$ by collapsed wave function ${\Psi }^{\mathrm{<mo>\; \#\; </mo>}}\left(x\right)$ if there exist an wave function $\Psi \left(x\right)$ such that the estimate

${\displaystyle \underset{\Gamma }{\int }\left\{\frac{{\partial }^2{\Psi }^{\mathrm{<mo>\; \#\; </mo>}}\left(x\right)}{\partial x^2}+\frac{2m}{{\hslash }^2}\left[E-U\left(x\right)\right]{\Psi }^{\mathrm{<mo>\; \#\; </mo>}}\left(x\right)\right\}\text{d}x=O\left({\hslash }^{2+\alpha }\right),}$ (2.15)

where $\alpha \ge \mathrm{1,}$ is satisfied.

Proposition 2.1. The Schrödinger equation in each of regions $I,II,III$ is a weakly well preserved by collapsed wave function ${\Psi }_I^{\mathrm{<mo>\; \#\; </mo>}}\left(x\right)\mathrm{,}{\Psi }_{II}^{\mathrm{<mo>\; \#\; </mo>}}\left(x\right)$ and ${\Psi }_{III}^{\mathrm{<mo>\; \#\; </mo>}}\left(x\right)$ correspondingly.

Proof. See Appendix B.

Definition 2.2. Let us consider the time-dependent Schrödinger equation:

$\begin{array}{l}i\hslash \frac{\partial \Psi \left(x\mathrm{,}t\right)}{\partial t}=\hat{H}\Psi \left(x\mathrm{,}t\right)\\ t\in \left[\mathrm{0,}T\right]\mathrm{,}x\in {ℝ}^{3n}.\end{array}$ (2.16)

The time-dependent Schrödinger Equation (2.16) is a weakly well preserved by corresponding to $\Psi \left(x\mathrm{,}t\right)$ collapsed wave function ${\Psi }^{\mathrm{<mo>\; \#\; </mo>}}\left(x\mathrm{,}t\right)$

$\begin{array}{c}{\Psi }^{\mathrm{<mo>\; \#\; </mo>}}\left(x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_n\mathrm{,}t\right)=\Psi \left(x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_n\mathrm{,}t\mathrm{<mo>\; ;\; </mo>}{\tilde{x}}_{m_1}\mathrm{,}\cdots \mathrm{,}{\tilde{x}}_{m_k}\right)\\ =\frac{{\Re }_{m_1\mathrm{,}\cdots \mathrm{,}{m}_k}\left({\tilde{x}}_{m_1}\mathrm{,}\cdots \mathrm{,}{\tilde{x}}_{m_k}\right)\Psi \left(x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_n\mathrm{,}t\right)}{{\Vert {\Re }_{m_1\mathrm{,}\cdots \mathrm{,}{m}_k}\left({\tilde{x}}_{m_1}\mathrm{,}\cdots \mathrm{,}{\tilde{x}}_{m_k}\right)\Psi \left(x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_n\mathrm{,}t\right)\Vert }_2},\\ {\Re }_{m_1\mathrm{,}\cdots \mathrm{,}{m}_k}\left({\tilde{x}}_{m_1}\mathrm{,}\cdots \mathrm{,}{\tilde{x}}_{m_k}\right)={\displaystyle \underset{i=1}{\overset{k}{\prod }}{\Re }_{m_i}\left({\tilde{x}}_{m_i}\right)}\end{array}$

in region $\Gamma \subseteq {ℝ}^{3d}$ if there exist an wave function $\Psi \left(x\mathrm{,}t\right)$ such that the estimate

$\begin{array}{l}{\displaystyle \underset{\Gamma }{\int }\left\{i\hslash \frac{\partial {\Psi }^{\mathrm{<mo>\; \#\; </mo>}}\left(x\mathrm{,}t\right)}{\partial t}-\hat{H}{\Psi }^{\mathrm{<mo>\; \#\; </mo>}}\left(x\mathrm{,}t\right)\right\}{d}^{3d}x=O\left({\hslash }^{\alpha }\right)\mathrm{,}}\\ t\in \left[\mathrm{0,}T\right]\mathrm{,}x\in {ℝ}^{3d}\end{array}$ (2.17)

where $\alpha \ge \mathrm{1,}$ is satisfied.

Definition 2.3. Let ${\Psi }^{\mathrm{<mo>\; \#\; </mo>}}\left(x\mathrm{,}t\right)={\Psi }^{\mathrm{<mo>\; \#\; </mo>}}\left(x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_d\mathrm{,}t\right)$ be a function

$\Psi \left(x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_d\mathrm{,}t\mathrm{<mo>\; ;\; </mo>}{\tilde{x}}_{{}_1}\mathrm{,}\cdots \mathrm{,}{\tilde{x}}_{{}_d}\right)\mathrm{.}$ Let us consider the Probability Current Law

$\begin{array}{l}\frac{\partial }{\partial t}P\left(\Gamma \mathrm{,}t\right)+{\displaystyle \underset{\partial \Gamma }{\int }J\left(x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_d\mathrm{,}t\right)\cdot n{\text{d}}^{2d}x}=O\left({\hslash }^{\alpha }\right)\mathrm{,}\\ J\left(x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_d\mathrm{,}t\right)=\Psi \left(x\mathrm{,}t\right)\nabla \overline{\Psi \left(x\mathrm{,}t\right)}-\overline{\Psi \left(x\mathrm{,}t\right)}\nabla \Psi \left(x\mathrm{,}t\right)\mathrm{,}\\ t\in \left[\mathrm{0,}T\right]\mathrm{,}x\in {ℝ}^{3d},\end{array}$ (2.18)

corresponding to Schrödinger Equation (2.16). Probability Current Law (2.18) is a weakly well preserved by corresponding to $\Psi \left(x\mathrm{,}t\right)$ collapsed wave function ${\Psi }^{\mathrm{<mo>\; \#\; </mo>}}\left(x\mathrm{,}t\right)$ in region $\Gamma \subseteq {ℝ}^{3d}$ if there exist an wave function $\Psi \left(x\mathrm{,}t\right)$ such that the estimate

$\begin{array}{l}\frac{\partial }{\partial t}P\left(\Gamma \mathrm{,}t\right)+{\displaystyle \underset{\partial \Gamma }{\int }{J}^{\mathrm{<mo>\; \#\; </mo>}}\left(x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_d\mathrm{,}t\right)\cdot n{\text{d}}^{2d}x}=O\left({\hslash }^{\alpha }\right)\mathrm{,}\\ J^{\mathrm{<mo>\; \#\; </mo>}}\left(x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_d\mathrm{,}t\right)={\Psi }^{\mathrm{<mo>\; \#\; </mo>}}\left(x\mathrm{,}t\right)\nabla \overline{{\Psi }^{\mathrm{<mo>\; \#\; </mo>}}\left(x\mathrm{,}t\right)}-\overline{{\Psi }^{\mathrm{<mo>\; \#\; </mo>}}\left(x\mathrm{,}t\right)}\nabla {\Psi }^{\mathrm{<mo>\; \#\; </mo>}}\left(x\mathrm{,}t\right)\\ =O\left({\hslash }^{\alpha }\right)\mathrm{,}\\ t\in \left[\mathrm{0,}T\right]\mathrm{,}x\in {ℝ}^{3d},\end{array}$ (2.19)

where $\alpha \ge \mathrm{1,}$ is satisfied.

Proposition 2.2. Assume that there exists an wave function $\Psi \left(x\mathrm{,}t\right)$ such that the estimate (2.17) is satisfied. Then Probability Current Law (2.18) is a weakly well preserved by corresponding to $\Psi \left(x\mathrm{,}t\right)$ collapsed wave function ${\Psi }^{\mathrm{<mo>\; \#\; </mo>}}\left(x\mathrm{,}t\right)$ in region $\Gamma \subseteq {ℝ}^{3d}\mathrm{,}$ i.e. the estimate (2.19) is satisfied on the wave function ${\Psi }^{\mathrm{<mo>\; \#\; </mo>}}\left(x\mathrm{,}t\right)$ .

3. 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 $\psi =c_1{\psi }_1+c_2{\psi }_2$ is correct only when the supports the wave functions ${\psi }_1$ and ${\psi }_2$ essentially overlap. When the wave functions ${\psi }_1$ and ${\psi }_2$ 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 $\psi =c_1{\psi }_1+c_2{\psi }_2$ is no longer valid for a such cat state (for details see Appendix C).

3.1. Consideration of the Schrödinger’s Cat Paradox Using Canonical Von Neumann Postulate

Let $|s_1\left(t\right)\rangle$ and $|s_2\left(t\right)\rangle$ be

$\begin{array}{l}|s_1\left(t\right)\rangle =|\text{undecayednucleusatinstant}t\rangle ,\\ |s_2\left(t\right)\rangle =|\text{decayednucleusatinstant}t\rangle .\end{array}$ (3.1)

In a good approximation we assume now that

$|s_1\left(0\right)\rangle ={\displaystyle {\int }_{-\infty }^{+\infty }{\Psi }_{II}^{\mathrm{<mo>\; \#\; </mo>}}\left(x\right)|x\rangle \text{d}x}$ (3.2)

and

$|s_2\left(0\right)\rangle ={\displaystyle {\int }_{-\infty }^{+\infty }{\Psi }_I^{\mathrm{<mo>\; \#\; </mo>}}\left(x\right)|x\rangle \text{d}x}\mathrm{.}$ (3.3)

Remark 3.1. Note that:

(i) $|s_2\left(0\right)\rangle =|\text{decayednucleusatinstant}0\rangle =|\text{free}\alpha -\text{particleatinstant}0\rangle \mathrm{.}$

(ii) Feynman propagator of a free $\alpha$ -particle is [9] :

$K_2\left(x\mathrm{,}t\mathrm{,}{x}_0\right)={\left(\frac{m}{2\text{π}i\hslash t}\right)}^{1/2}exp\left\{\frac{i}{\hslash }\left[\frac{m{\left(x-x_0\right)}^2}{2t}\right]\right\}$ . (3.4)

Therefore from Equations ((3.3), (2.9) and (3.4)) we obtain

$\begin{array}{c}|s_2\left(t\right)\rangle ={\displaystyle {\int }_{-\infty }^{+\infty }{\Psi }_I^{\mathrm{<mo>\; \#\; </mo>}}\left(x\mathrm{,}t\right)|x\rangle \text{d}x},\\ {\Psi }_I^{\mathrm{<mo>\; \#\; </mo>}}\left(x\mathrm{,}t\right)={\displaystyle \underset{-\infty }{\overset{0}{\int }}{\Psi }_I^{\mathrm{<mo>\; \#\; </mo>}}\left(x_0\right)K_2\left(x\mathrm{,}t\mathrm{,}{x}_0\right)\text{d}{x}_0}\\ ={\left(\text{π}{r}_c^2\right)}^{-1/4}\times {\left(\frac{m}{2\text{π}i\hslash t}\right)}^{1/2}\times {\displaystyle \underset{-\infty }{\overset{0}{\int }}{\theta }_1\left(x_0\mathrm{,}l\right)exp\left(-\frac{x_0^2}{2r_c^2}\right)exp\left(-i\frac{2\text{π}}{\hslash }\sqrt{2mE}{x}_0\right)}\\ \times exp\left\{\frac{i}{\hslash }\left[\frac{m{\left(x-x_0\right)}^2}{2t}\right]\right\}\text{d}{x}_0\\ ={\left(\text{π}{r}_c^2\right)}^{-1/4}\times {\left(\frac{m}{2\text{π}i{\hslash }_{\epsilon }t}\right)}^{1/2}\times {\displaystyle \underset{-l}{\overset{0}{\int }}{\theta }_1\left(x_0\mathrm{,}l\right)exp\left(-\frac{x_0^2}{2r_c^2}\right)}\\ \times exp\left\{\frac{i}{\hslash }\left[\frac{m{\left(x-x_0\right)}^2}{2t}-\text{π}\sqrt{4mE}{x}_0\right]\right\}\text{d}{x}_0\\ ={\left(\text{π}{r}_c^2\right)}^{-1/4}\times {\left(\frac{m}{2\text{π}i\hslash t}\right)}^{1/2}\times {\displaystyle \underset{-l}{\overset{0}{\int }}{\theta }_1\left(x_0\mathrm{,}l\right)exp\left(-\frac{x_0^2}{2r_c^2}\right)\times exp\left\{\frac{i}{\hslash }\left[S\left(t\mathrm{,}x\mathrm{,}{x}_0\right)\right]\right\}\text{d}{x}_0},\end{array}$ (3.5)

where

$S\left(t,x,x_0\right)=\frac{m{\left(x-x_0\right)}^2}{2t}-\text{π}\sqrt{8mE}{x}_0.$ (3.6)

We assume now that

$\hslash \ll 2r_c^2\ll l^2<1.$ (3.7)

Oscillatory integral in RHS of Equation (3.5) is calculated now directly using stationary phase approximation. The phase term $S\left(x\mathrm{,}{x}_0\right)$ given by Equation (3.6) is stationary when

$\frac{\partial S\left(t,x,x_0\right)}{\partial x_0}=-\frac{m\left(x-x_0\right)}{t}-\text{π}\sqrt{8mE}=0.$ (3.8)

Therefore

$\begin{array}{c}-\frac{m\left(x-x_0\right)}{t}-\text{π}\sqrt{8mE}=0,\\ -\left(x-x_0\right)=\text{π}t\sqrt{8E/m}\mathrm{,}\end{array}$ (3.9)

and thus stationary point $x_0\left(t\mathrm{,}x\right)$ are

$x_0\left(t,x\right)=\text{π}t\sqrt{8E/m}+x\mathrm{.}$ (3.10)

Thus from Equation (3.5) and Equation (3.10) using stationary phase approximation we obtain

$\begin{array}{l}|s_2\left(t\right)\rangle ={\displaystyle {\int }_{-\infty }^{+\infty }{\Psi }_I^{\mathrm{<mo>\; \#\; </mo>}}\left(x\mathrm{,}t\right)|x\rangle \text{d}x},\\ {\Psi }_I^{\mathrm{<mo>\; \#\; </mo>}}\left(x\mathrm{,}t\right)={\left(\text{π}{r}_c^2\right)}^{-1/4}\times {\theta }_1\left(x_0\left(t\mathrm{,}x\right)\mathrm{,}l\right)exp\left[-\frac{x_0^2\left(t\mathrm{,}x\right)}{2r_c^2}\right]\\ \times exp\left\{\frac{i}{\hslash }\left[S\left(t\mathrm{,}x\mathrm{,}{x}_0\left(t\mathrm{,}x\right)\right)\right]\right\}+O\left(\hslash \right),\end{array}$ (3.11)

where

$S\left(x,x_0\left(t,x\right)\right)=\frac{m{\left(x-x_0\left(t\mathrm{,}x\right)\right)}^2}{2t}-\text{π}\sqrt{8mE}{x}_0\left(t\mathrm{,}x\right)\mathrm{.}$ (3.12)

From Equation (3.10) we obtain

$\overline{{\Psi }_I^{\mathrm{<mo>\; \#\; </mo>}}\left(x\mathrm{,}t\right)}{\Psi }_I^{\mathrm{<mo>\; \#\; </mo>}}\left(x\mathrm{,}t\right)\simeq {\left(\text{π}{r}_c^2\right)}^{-1/2}\times {\theta }_1\left(x+\text{π}t\sqrt{8E/m}\mathrm{,}l\right)exp\left[-\frac{{\left(x+\text{π}t\sqrt{8E/m}\right)}^2}{r_c^2}\right].$ (3.13)

Remark 3.2. From the inequality (3.7) and Equation (3.13) follows that $\alpha$ -particle at each instant $t\ge 0$ moves quasiclassically from right to left by the law

$x\left(t\right)=-\text{π}t\sqrt{8E/m},$ (3.14)

i.e., estimating the position $x\left(t\mathrm{,}{x}_0\mathrm{,}{t}_0\mathrm{<mo>\; ;\; </mo>}\hslash \right)$ at each instant $t\ge 0$ with final error $r_c$ gives $\left|\langle x\rangle \left(t\right)-x\left(t\right)\right|\le r_c\mathrm{,}i=1,\cdots ,d$ with a probability

$P\left\{\left|\langle x\rangle \left(t\mathrm{,0,0\; <mo>\; ;\; </mo>}\hslash \right)-x\left(t\right)\right|\le r_c\right\}=1.$

Remark 3.3. We assume now that a distance between radioactive source and internal monitor which detects a single atom decaying (see Figure 1) is equal to $L\mathrm{.}$

Proposition 3.1. After $\alpha$ -decay at instant $t=0$ the collaps: $|\text{livecat}\rangle \to |\text{deathcat}\rangle$ arises at instant

$T=\frac{L}{\text{π}\sqrt{8E/m}}$ (3.15)

with a probability $P_T\left(|\text{deathcat}\rangle \right)$ to observe a state $|\text{deathcat}\rangle$ at instant $T$ is $P_T\left(|\text{deathcat}\rangle \right)=1.$

Proof. Note that. In this case Schrödinger’s cat in fact performs the single measurement of $\alpha$ -particle position with accuracy of $\delta x=l$ at instant $t=T$ (given by Equation (3.15)) by internal monitor (see Figure 1). The probability of getting a result $L$ with accuracy of $\delta x=l$ given by

${\displaystyle {\int }_{\left|L-x\right|\le l/2}{\left|\langle x|s_2\left(T\right)\rangle \right|}^2\text{d}x}=1.$ (3.16)

Therefore at instant $T$ the $\alpha$ -particle kills Schrödinger’s cat with a probability $P_T\left(|\text{deathcat}\rangle \right)=1.$

Remark 3.4. Note that. When Schrödinger’s cat has performed this measurement the immediate post measurement state of $\alpha$ -particle (by von Neumann postulate C.4) will end up in the state

$\begin{array}{c}|{\Psi }_T\rangle =\frac{{\displaystyle {\int }_{\left|L-x\right|\le l/2}|x\rangle \langle x|s_2\left(T\right)\rangle \text{d}x}}{\sqrt{{\displaystyle {\int }_{\left|L-x\right|\le l/2}{\left|\langle x|s_2\left(T\right)\rangle \right|}^2\text{d}x}}}\\ ={\displaystyle {\int }_{\left|L-x\right|\le l/2}|x\rangle \langle x|s_2\left(T\right)\rangle \text{d}x}\in S_{\Theta },\\ \Theta =\left\{x|\left|L-x\right|\le l/2\right\}.\end{array}$ (3.17)

From Equation (3.17) one obtains

$\begin{array}{c}{x}^{\prime }|{\Psi }_T\rangle ={\displaystyle {\int }_{\left|L-x\right|\le l/2}\langle x^{\prime }|x\rangle \langle x|s_2\left(T\right)\rangle \text{d}x}\\ ={\displaystyle {\int }_{\left|L-x\right|\le l/2}\delta \left(x^{\prime }-x\right)\langle x|s_2\left(T\right)\rangle \text{d}x}={\Psi }_I^{\mathrm{<mo>\; \#\; </mo>}}\left(x^{\prime }\mathrm{,}t\right).\end{array}$ (3.18)

Therefore the state $|{\Psi }_T\rangle$ again kills Schrödinger’s cat with a probability $P_T\left(|\text{deathcat}\rangle \right)=1.$

Suppose now that a nucleus $n\mathrm{,}$ whose Hilbert space is spanned by orthonormal states $|s_i\left(t\right)\rangle ,i=1,2,$ where $|s_1\left(t\right)\rangle =|\text{undecayednucleusatinstant}t\rangle$ and $|s_2\left(t\right)\rangle =|\text{decayednucleusatinstant}t\rangle$ is in the superposition state

${|{\Psi }_t\rangle }_n=c_1|s_1\left(t\right)\rangle +c_2|s_2\left(t\right)\rangle ,{\left|c_1\right|}^2+{\left|c_2\right|}^2=1$ (3.19)

Remark 3.5. Note that: (i) $\begin{array}{c}|s_1\left(0\right)\rangle =|\text{undecayednucleusatinstant}t=0\rangle \\ =|\alpha -\text{particleisideregion}\left(\mathrm{0,}l\right]\text{atinstant}t=0\rangle \mathrm{.}\end{array}$

(ii) Feynman propagator of $\alpha$ -particle inside region $\left(\mathrm{0,}l\right]$ are [9] :

$K_2\left(x,t,x_0\right)={\left(\frac{m}{2\text{π}i\hslash t}\right)}^{1/2}\exp \left\{\frac{i}{\hslash }\left[S\left(t,x,x_0\right)\right]\right\},$ (3.20)

where

$S\left(t\mathrm{,}x\mathrm{,}{x}_0\right)=\frac{m{\left(x-x_0\right)}^2}{2t}+mt\left(U_0-E\right).$ (3.21)

Therefore from Equations ((2.11), (2.12) and (3.20), (3.21)) we obtain

$\begin{array}{c}|s_1\left(t\right)\rangle ={\displaystyle {\int }_{-\infty }^{+\infty }{\Psi }_{II}^{\mathrm{<mo>\; \#\; </mo>}}\left(x\mathrm{,}t\right)|x\rangle \text{d}x}\\ {\Psi }_{II}^{\mathrm{<mo>\; \#\; </mo>}}\left(x\mathrm{,}t\right)={\displaystyle \underset{0}{\overset{l}{\int }}{\Psi }_{II}^{\mathrm{<mo>\; \#\; </mo>}}\left(x_0\right)K_2\left(x\mathrm{,}t\mathrm{,}{x}_0\right)\text{d}{x}_0}\\ ={\left(\frac{m}{2\text{π}i\hslash t}\right)}^{1/2}{\displaystyle \underset{0}{\overset{l}{\int }}E\left(x_0\mathrm{,}l\right){\Psi }_{II}\left(x_0\right){\theta }_l\left(x_0\right)exp\left\{\frac{i}{\hslash }\left[S\left(t\mathrm{,}x\mathrm{,}{x}_0\right)\right]\right\}\text{d}{x}_0\mathrm{,}}\end{array}$ (3.22)

where

${\theta }_l\left(x\right)=\{\begin{array}{l}1\text{for}x\in \left[\mathrm{0,}l\right]\hfill \\ 0\text{for}x\notin \left[\mathrm{0,}l\right]\hfill \end{array}$

Remark 3.6. We assume for simplification now that

$k^{\prime }\le 1.$ (3.23)

Therefore oscillatory integral in RHS of Equation (3.22) is calculated now directly using stationary phase approximation. The phase term $S\left(x\mathrm{,}{x}_0\right)$ given by Equation (3.21) is stationary when

$\frac{\partial S\left(t,x,x_0\right)}{\partial x_0}=-\frac{m\left(x-x_0\right)}{t}=0.$ (3.24)

and thus stationary point $x_0\left(t\mathrm{,}x\right)$ are

$\begin{array}{c}-x+x_0=0\\ x_0\left(t,x\right)=x.\end{array}$ (3.25)

Thus from Equation (3.22) and Equation (3.25) using stationary phase approximation we obtain

$\begin{array}{c}{\Psi }_{II}^{\mathrm{<mo>\; \#\; </mo>}}\left(x\mathrm{,}t\right)\\ =E\left(x_0\left(t\mathrm{,}x\right)\mathrm{,}l\right){\Psi }_{II}\left(x_0\left(t\mathrm{,}x\right)\right){\theta }_l\left(x_0\left(t\mathrm{,}x\right)\right)exp\left\{\frac{i}{\hslash }\left[S\left(t\mathrm{,}x\mathrm{,}{x}_0\left(t\mathrm{,}x\right)\right)\right]\right\}+O\left(\hslash \right)\\ =E\left(x\mathrm{,}l\right){\Psi }_{II}\left(x\right){\theta }_l\left(x\right)exp\left\{\frac{i}{\hslash }\left[mt\left(U_0-E\right)\right]\right\}+O\left(\hslash \right)\\ =E\left(x\mathrm{,}l\right){\theta }_l\left(x\right)O\left(1\right)exp\left\{\frac{i}{\hslash }\left[mt\left(U_0-E\right)\right]\right\}+O\left(\hslash \right).\end{array}$ (3.26)

Therefore from Equation (3.22) and Equation (3.26) we obtain

${\left|{\Psi }_{II}^{\mathrm{<mo>\; \#\; </mo>}}\left(x\mathrm{,}t\right)\right|}^2=E^2\left(x,l\right){\theta }_l\left(x\right)O\left(1\right)+O\left(\hslash \right)$ . (3.27)

Remark 3.7. Note that for each instant $t>0:\text{supp}\left({\Psi }_{II}^{\#}\left(x,t\right)\right)\cap \text{supp}\left({\Psi }_I^{\#}\left(x,t\right)\right)=\varnothing \mathrm{.}$

Remark 3.8. Note that, from Equations ((3.11), (3.13), (3.19), (3.22)-(3.27) and (A.13)) by Remark 3.7 we obtain

$\begin{array}{c}{}_n\langle {\Psi }_t|\hat{x}{|{\Psi }_t\rangle }_n={\left|c_1\right|}^2\langle s_1\left(t\right)|\hat{x}|s_1\left(t\right)\rangle +{\left|c_2\right|}^2\langle s_2\left(t\right)|\hat{x}|s_2\left(t\right)\rangle \\ +c_1{c}_2^{\ast }\langle s_2\left(t\right)|\hat{x}|s_1\left(t\right)\rangle +c_1^{\ast }{c}_2\langle s_2\left(t\right)|\hat{x}{|s_1\left(t\right)\rangle }^{\ast }\\ ={\left|c_1\right|}^2\langle s_1\left(t\right)|\hat{x}|s_1\left(t\right)\rangle +{\left|c_2\right|}^2\langle s_2\left(t\right)|\hat{x}|s_2\left(t\right)\rangle \\ ={\left|c_1\right|}^{2}l+{\left|c_2\right|}^{2}T\text{π}\sqrt{8E/m}\mathrm{.}\end{array}$ (3.28)

Proposition 3.2. (i) Suppose that a nucleus $n$ is in the superposition state ${|{\Psi }_t\rangle }_n$ ( ${|{\Psi }_t\rangle }_n$ -particle) given by Equation (3.19). Then the collaps: $|\text{livecat}\rangle \to |\text{deathcat}\rangle$ arises at instant

$T_{\text{col}}\approx \frac{L\pm l}{{\left|c_2\right|}^2\sqrt{8{\text{π}}^{2}E/m}}$ (3.29)

with a probability $P_{T_{\text{col}}}\left(|\text{deathcat}\rangle \right)$ to observe a state $|\text{deathcat}\rangle$ at instant $T_{\text{col}}$ is $P_{T_{\text{col}}}\left(|\text{deathcat}\rangle \right)={\left|c_2\right|}^2\mathrm{.}$

(ii) Assume now a Schrödinger’s cat has performed the single measurement of ${|{\Psi }_t\rangle }_n$ -particle position with accuracy of $\delta x=l$ at instant $T=T_{\text{col}}$ (given by Equation (3.29)) by internal monitor (see Figure 1) and the result $x\approx L\pm l$ is not observed by Schrödinger’s cat. Then the collaps: $|\text{livecat}\rangle \to |\text{deathcat}\rangle$ never arises at any instant $T>T_{\text{col}}$ and a probability $P_{T\mathrm{<mo>\; >\; </mo>}{T}_{\text{col}}}\left(|\text{deathcat}\rangle \right)$ to observe a state $|\text{deathcat}\rangle$ at instant $T>T_{\text{col}}$ is $P_{T>T_{\mathrm{col}}}\left(|\text{deathcat}\rangle \right)=0.$

Proof.(i) Note that for $t>0$ the marginal density matrix $\rho \left(t\right)$ is diagonal

$\rho \left(t\right)=\left(\begin{array}{ll}{\left|c_1\right|}^2{\left|{\Psi }_{II}^{\mathrm{<mo>\; \#\; </mo>}}\left(x\mathrm{,}t\right)\right|}^2\text{d}x\hfill & 0\hfill \\ 0\hfill & {\left|c_2\right|}^2{\left|{\Psi }_I^{\mathrm{<mo>\; \#\; </mo>}}\left(x\mathrm{,}t\right)\right|}^2\text{d}x\hfill \end{array}\right).$