We assume that the cat in a box paradox may allow an additional approach to include multiple cats in the same box.
We can now see two seemingly different views, quantum and classical.
The difference is huge and compares quantum probability to modern statistical probability.
What is quite striking is that when the Schrödinger wave equation remains inactive in the face of many physical and mathematical situations, its square PDE steps forward and does the work.
Contrary to what many mathematicians and physicists think, the proposed equation for Ψ^2 is unified field theory and not that of Ψ itself.
Numerical examples are countless but we present the following:
In the 3D + t space of the Schrödinger equation we must assume, without rigorous reasoning, that the separation of variables expresses f(t) as,
The probability density function of an exponential distribution is
𝑓(𝑥;𝜆)=(𝜆[𝑒−𝜆𝑥]) . . . (1) for 𝑥 ≥ 0
,
𝑓(𝑥;𝜆)= 0
for 𝑥< 0.
Here λ > 0 is the parameter of the distribution, often called the rate parameter.
and the,
Cumulative distribution function exponential distribution like,
𝐹(𝑥;𝜆)=(1−[𝑒−𝜆𝑥])
for 𝑥≥ 0 . . . . (2)
,
𝐹(𝑥;𝜆)=0
for 𝑥 < 0.
NOTE THAT expressions above 1,2 already appear [1,2] via B matrix strings in an alternative parameter:
𝑓(t;T)=(𝜆[𝑒−𝑥/T]) . . . (3) for 𝑥 ≥ 0
called the cooling curve.
𝐹(𝑥;T)= T max(1−[𝑒−𝑥/T]) . . .(4),
𝐹(𝑥;T)=0
for 𝑥 < 0.
called heating curve.
α = Log (0.5+0.5 OR). . . . . . . . . .(5)
α =[0,log1/2]
α Max = log1/2 Conductor
α Min = 0 Insulator
1/2+1/2 RO=Exp(α)
RO= 2*Exp(α)-1
RO Max = 2*1/2-1=0
Note that the necessary and sufficient conditions for deriving equations 3,4 are that the transition matrix B for the heat diffusion equation and the transition matrix Q for Ψ^2 in quantum mechanics satisfy:
B ^ N .I is exponential, which has been shown to be true [1].
To be continued.
[1] A rigorous experimental technique for measuring the thermal diffusivity of metals, Researchgate, IJISRT journal,
August 2022
There's no paradox involving a cat in a box and the corresponding experiment has been done with other setups, typically magnetic moments-the problem with the cat is that it's, still, not possible to define with a precision comparable to other binary choices, what it means for the cat to be alive.
The difference between the quantum system and its classical limit isn't with how probabilities are defined (there's no such distinction between ``quantum'' vs. ``classical'' probabilities) but, rather with the space of states of the quantum system, which is infinitely larger than the space of states of its classical limit. For the cat the classical space of states contains two states, ``|alive>'' and ``|dead>''; the qantum space of states is defined as the space of all linear combinations of these states, viz. a|alive>+b|dead>, with the constraint |a|2+|b|2=1, which defines a 3-sphere. So the quantum space of states is the 3-sphere, of unit radius; however this overcounts, since multiplying the complex numbers a and b by a phase doesn't change anything; so the quantum space of states is, in fact, the unit 2-sphere while the classical limit corresponds to any two antipodal points on this sphere.
The Schrödinger equation with the Bohr/Copenhagen interpretation is the central philosophy of modern quantum mechanics describing the wave function Ψ.
However, this is only valid for subatomic particles in infinite free space.
It can well describe the quantified energy levels in the hydrogen atom and the infinite potential wells.
Obviously, this is not enough to solve the energy density distribution function for material media such as the heat diffusion equation and sound intensity in audio rooms.
On the other hand, the square of the wave function Ψ^2=Ψ.Ψ*=U (energy density in 4D unit space) has its own statistical transition matrix (B-matrix for classical physics such as heat diffusion, sound intensity, EMW fields etc. and Q-matrix for 1D, 2D, 3D quantum mechanics situations in vacuum).
Additionally, Q matrix chains explain the formation of the Big Bang in a vacuum over millions or billions of years.
The properties of the chains of the matrix B and those of the chains of the matrix Q are remarkable and justify the hypothesis that they constitute the theory of unified fields.
The properties of matrix B valid for material supports are:
1-B^N tends to zero as N tends to infinity.
2-Solution vector U(x,y,z,t)=b .(B+B^2+B^3 + . . . B^N)
Where b is the Dirichlet boundary conditions.
3-B+B^2+B^3 + . . . B^N = 1/(I-B)
For N sufficiently large.
II-Properties of the Q-Matrix valid for the infinite vacuum are:
1-Q=[(1+Constant ).I -B]
I is the unitary matrix.
2-Norm Q^N=Norm Q. . for all N
3-The Dirichlet boundary conditions for an isolated quantum system in infinite free space are Ψ^2(-∞)=Ψ^2(+∞)=0
To be continued.
The starting point for obtaining the modified Schrödinger PDE describing ψ^2(x,y,z,t) cannot be the Schrödinger PDE itself because it is not complete.
The starting point is the following PDE,
dU/dt)partial=D.Nabla^2 U +S(x,y,z,t) . . . (1)
Where U=ψ^2=ψψ*.
Equation 1, just like Schrödinger's classic PDE, applies to infinite free space and must be supplemented by rules of vacuum dynamics, the most important of which is:
S(x,y,z,t) =Constant * V(x,y,z,t) . . . . (2)
In addition to other commonly accepted rules of vacuum dynamics.
In this case, in simple terms, ∫E∙dt can also be dimensionless.
In such a context, if we consider that time is discretized and woven into the 4D unit space, this would quite indicate a quantized, complete and simultaneous quantum state of the system.
Additionally, the principle of least action is inherent to the solution, which is not always the case for the classic Schrödinger equation.
To be continued.
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