The 4 bugs, 1.8


Note that this standard language is a useful shortcut. However, it is for the convenience of daily-life calculations only. In case you want not only to apply quantum mechanics, but also to understand it (in order to avoid common conceptual paradoxes), you can translate the common language to a more accurate version:

physical definition

~ define 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

— Me@2022-01-31 08:33:01 AM

In the experiment-setup design, when no detector (that can record which slit the particle has gone through) is allowed, the only measurement device remains is the final screen, which records the particle’s final position.

If you ask for the wave function \displaystyle{| \phi \rangle} for the variable representing for the particle’s final position on the screen, then \displaystyle{| \phi \rangle} is in one of the eigenstates, where each eigenstate represents a particular location on the final screen.

The wave function \displaystyle{| \phi \rangle} must be an eigenstate because your experiment design has provided a physical definition for different values of \displaystyle{| \phi \rangle}.

When we see a dot appear at a point on the screen, we say that the particle has reached that location.

However, if you ask which of the 2 slits the particle has gone through, it is impossible to answer, not because of our lack of knowledge (of the details of the experiment-setup physical system), but because of our lack of definition of the case “go-left” and that of the case “go-right” (in terms of observable physical phenomena).

In other words, due to the lack of physical definition, “go-left” and “go-right” are actually logically indistinguishable due to being physically indistinguishable. They should be regarded as identical, thus one single case. We represent that one single case by the wave function

\displaystyle{| \psi \rangle = \sqrt{0.5}~| \psi_L \rangle + \sqrt{0.5}~| \psi_R \rangle}.

a physical variable X is in a superposition state

~ X is a physically-undefined property (of the physical system)

For example, the system does not have the statistical property of “go-left“, nor that of “go-right“. The intended possible values of X, “go-left” and “go-right“, do not exist. Only the value “go-through-double-slit-plate” (without mentioning left and right) exists.

— Me@2022-02-23 07:49:47 AM

~ in the experiment-setup design, no measurement device is allowed to exist to provide a definition of different possible values of X

— Me@2022-02-18 02:04:45 PM

The wave function

\displaystyle{| \psi \rangle = \sqrt{0.5}~| \psi_L \rangle + \sqrt{0.5}~| \psi_R \rangle}

is for calculating the probabilities of passing through the double-slit-plate, without specifying which slit a particle has gone through. This is what \displaystyle{| \psi \rangle} actually means.

Passing through the double-slit-plate” is one single physical state. This physical state is not related to the physical state “go-left“, nor the physical state “go-right“, because those two physical cases do not exist in the first place.

The wave function \displaystyle{| \psi \rangle} represents one single physical case. So \displaystyle{| \psi \rangle} is one state. In other words, \displaystyle{| \psi \rangle} is a pure state, not a mixed state.

In this physical case, the particle is not in any of the following states:

1.  \displaystyle{| \psi_L \rangle}

2.  \displaystyle{| \psi_R \rangle}

3.  \displaystyle{| \psi_L \rangle} or \displaystyle{| \psi_R \rangle}

4.  \displaystyle{| \psi_L \rangle} and \displaystyle{| \psi_R \rangle}

And” and “or” only exist when there are more than one cases. The physical case “go-left” and the physical case “go-right” do not exist in the experiment-setup. So applying “and” or applying “or” are both impossible in this case.

The common misunderstanding comes from representing \displaystyle{| \psi \rangle } as a sum of \displaystyle{| \psi_L \rangle } and \displaystyle{| \psi_R \rangle}. But this is not a physical superposition, but a mathematical superposition only.

This mathematical superposition has 3 meanings (applications):

1. The component eigenstates \displaystyle{| \psi_L \rangle } and \displaystyle{| \psi_R \rangle} are logically indistinguishable (due to the lack of physical definition). They should be regarded as one single physical case.

In other words, the plus sign, \displaystyle{+}, can be directly translated to “is indistinguishable from”.


~ “is indistinguishable from”

\displaystyle{| \psi \rangle = \sqrt{0.5}~| \psi_L \rangle + \sqrt{0.5}~| \psi_R \rangle}

~ \displaystyle{| \psi_L \rangle} and \displaystyle{| \psi_R \rangle} are indistinguishable; so they form one single state \displaystyle{| \psi \rangle}.

In a quantum superposition, all component eigenstates have to be indistinguishable. In other words, for the Schrödinger’s cat thought experiment, the cat superposition that some popular science text writes,

\displaystyle{| \text{cat} \rangle =  \sqrt{0.5}~| \text{cat-alive} \rangle + \sqrt{0.5}~| \text{cat-dead} \rangle},

is actually illegitimate, because \displaystyle{| \text{cat-alive} \rangle} and \displaystyle{| \text{cat-dead} \rangle} are observable physical events; they are distinguishable states (physical cases).

— Me@2022-02-22 07:01:40 PM



2022.02.23 Wednesday (c) All rights reserved by ACHK