# Square root of probability, 3.2.2

Eigenstates 3.3.2.2

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Actually, quantum mechanics does NOT allow any violation of these logical laws. A quantum superposition state is NOT any “overlapping” of multiple physical states. A quantum superposition state is ONE single physical state.

Contrary to popular belief, Schrödinger created the thought experiment to illustrate that a quantum superposition state should NOT be regarded as any “overlapping” of multiple physical states.

To explain that, we should use the most basic quantum experiment, the double-slit experiment, instead of the cat experiment, because:

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And more fundamentally:

2.1

The cat-alive state is a well-defined state. The cat-dead state also.

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A quantum state must be a superposition state when in (the definition of) the experiment setup, some definitions of trajectories are missing.

The “definitions” here must be in terms of physical phenomena, because “trajectories” have no objective existence, due to the fact that fundamentally particles have no objective identities; fundamentally, identical particles are indistinguishable.

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`Which trajectory has a particle travelled along`” is a hindsight story.

`The electron at location x_1 at time t_1 and the electron at location x_2 at a later time t_2 are actually the same particle`” is also a hindsight story.

These kinds of post hoc stories do not exist when some definitions of trajectories are missing in the overall definition of the experiment setup. In such a case, we can only use a superposition state to describe the state of the physical system.

Since such a situation has never happened in a classical (or macroscopic) system, it gives a probability distribution that has never existed before. Such a new kind of probability distribution can only be deduced by the mathematical representation of a superposition state. In other words, such a new kind of probability distribution is encoded in and only in the mathematical language of superposition state.

For example, in the double-slit experiment, if the experiment setup definition disallows any kinds of detectors (that can distinguish particle-go-left and particle-go-right), then tautologically, “go-left” and “go-right” are logically indistinguishable; “go-left” itself and “go-right” itself are both physically meaningless. So the system is actually in ONE single state, namely the superposition state.

Since “go-left” and “go-right” both have no physical meanings, “`it is in a superposition state of going-left state and going-right state`” means neither “`the particle goes left and goes right`” nor “`the particle goes left or goes right`“. Instead, it means that

2.11   going-left and going-right are logically indistinguishable so the system is actually logically ONE single physical state;

2.121   the probability distribution is not that of going-left nor that of going-right;

2.122   so this new kind of state requires a never-seen-before probability distribution,

2.123   which can be calculated from the mathematical expression of the superposition state, aka the wave function.

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Indeterminacy in measurement was not an innovation of quantum mechanics, since it had been established early on by experimentalists that errors in measurement may lead to indeterminate outcomes. However, by the later half of the eighteenth century, measurement errors were well understood and it was known that they could either be reduced by better equipment or accounted for by statistical error models. In quantum mechanics, however, indeterminacy is of a much more fundamental nature, having nothing to do with errors or disturbance.

— Wikipedia on `Quantum indeterminacy`

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Quantum indeterminacy is not due to physical limitations. Quantum indeterminacy is a definitional indeterminacy–some necessary definitions are not precise enough or even missing altogether.

It occurs when you do not allow installing any measuring devices (for some variables) in the experiment. It occurs when you define an experiment in such a way that you cannot define, for example, the difference between trajectories based on the difference between possible physical phenomena.

However, since the difference between the cat-alive state and cat-dead state is well-defined, there is no indeterminacy-due-to-undefined-difference (aka quantum indeterminacy). So there is no so-called “`quantum superposition of cat-alive and cat-dead`“. In other words, the cat itself is already a measuring device for the particle state.

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For simplicity, assuming that in the box, there were no measuring devices. So before putting the cat there, the atom was in a superposition of decayed and not-decayed-yet.

That does not mean that the atom is physically in two states at the same time. Instead, it actually means that there is logically no difference between decayed and not-yet-decayed, because there is no existence of a measuring device that can make the definitional physical difference.

superposition

~ lack of the existence of measuring device in the definition of the experimental setup `to define the difference between microscopic events in terms of the difference between observable physical events`

physical definition

~ define the microscopic events in terms of observable physical phenomena such as the change of readings of the measuring device

~ `define` unobservable events in terms of observable events

However, while you are designing the experiment, once you allow the cat to be inside, you have actually provided a physical definition of “`the atom has decayed`” and “`the atom has not decayed yet`“. The cat is actually the measuring device whose existence provides that physical definition. That is the so-called “wave function collapse”.

The “wave function collapse” itself is not a physical process. It is actually a change of probability distribution due to the change of experimental design. `The experiment with measurement device` and `that without measurement device` are actually two distinct experiments with distinct probability distributions.

wave function collapse

~ probability distribution change (replacement) due to replacing “`the experiment without measurement device`” with “`the experiment with measuring device`

2.2

— Me@2022-01-31 08:33:01 AM

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Although the final device in the chain of measurement needs to be macroscopic for a human being to read, the “measuring device” does not have to be macroscopic.

It even can be only one particle, as long as it can store the result of “going left” or “going right”.

In other words, if by adding an object in the experiment during the experiment design process, “go left” and “go right” acquire their physical definitions (physical meanings) by being distinguishable, then that object is a “measuring device”.

measuring device

~ logical case differentiator during the experiment design process

~ physical case differentiator during the experiment

— Me@2022-02-01 11:02:20 AM

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