Square-root-of-probability wave

The 4 bugs, 1.9

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The common quantum mechanics paradoxes are induced by 4 main misunderstandings.

1.  A wave function is of a particle. Wrong.

2.1  A system's wave function exists in physical spacetime. Wrong.

2.2  A superposition state is a physical superposition of a physical state. Wrong.

3.1  Probability value is totally objective. Wrong.

3.2 (2.3)  In some cases, the wave function of a physical variable of the system is in a superposition state at the beginning of the experiment. And then when measuring the variable during the experiment, that wave function collapses. Wrong.

A wave function (for a particular variable) is an intrinsic property of a physical system.

“Physical system” means the experimental-setup design, which includes not just objects and devices, but also operations.

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. 

2.  Although used for calculating probabilities, a wave function \displaystyle{\phi(x)} itself is not probability.

Instead, we have to calculate its squared modulus \displaystyle{ \left| \phi  \right|^2} in order to get a probability density; and then do an integration

\displaystyle{ \int_a^b \left| \phi(x) \right|^2 dx }

in order to get a probability, assuming in this case, the physical variable is a particle’s location \displaystyle{ x }.

At best, a wave function is the (complex) square root of probability (density) only.

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In our double slit experiment, 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, since the possible answers are physically-undefined.

Probability is in some sense “partial reality” before getting a result. Since a wave function (representing a quantum state) is not probability, we cannot regard the wave function as a “partial reality”. So, although \displaystyle{| \psi \rangle } is expressed as a sum of \displaystyle{| \psi_L \rangle } and \displaystyle{| \psi_R \rangle}, it cannot be regarded as a physical superposition.

1.  It is not the case that the particle is split into 2 halves; one half goes left and another goes right.

2.1  It is not an overlapping of 2 physical states.

2.2  It is not an overlapping of 2 realities, or partial realities.

2.3  It is not an overlapping of 2 different universes.

An overlapping of classical results will also give you a classical result; will not give you interference patterns.

Particles that go through the left slit will be part of the left fringe. Particles that go through the right slit will be part of the right fringes. So even if the reality (or half-reality) of a particle going left and the reality of it going right overlap, the particle will reach where the left or the right fringe will be, not where the interference pattern will be.

In other words, any simple overlapping a no-interference reality with another no-interference reality will give you also no interference patterns.

The fundamentals of this kind of errors are:

1.  Consider the electron version of the double-slit experiment.

Even if each electron in some sense really has passed through both slits, its two halves (or two realities) will never annihilate each other when they meet, because they are not anti-particle pair. No destructive interference will happen.

2.  The probability of any possible reality (component physical state, parallel universe) is always not less than 0 and not bigger than 1, because it is the nature of any probability \displaystyle{p}:

\displaystyle{0 \le p \le 1}.

In other words, the “superposition” of any two probabilities (possible realities, component physical states, parallel universes) will not give you the zero probability that is needed for destructive interference to happen.

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A major cause of this kind of errors is the wave function’s misleading name “probability wave”.

A wave function is not probability. At best, a wave function is the (complex) square root of probability (density) only. So at most, we can call it “complex-square-root-of-probability wave” only.

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

A wave function is not probability” (or “a superposition of wave function is not (simple) overlapping of realities“) is the exact reason for the existence of interference patterns.

Also, the mathematical superposition of eigenstates is exactly for calculating those interference patterns.

\displaystyle{  \begin{aligned}    \psi &= \psi_1 + \psi_2 \\ \\    P &= \left| \psi \right|^2 \\       &= \psi^* \psi \\       &= (\psi_1^* + \psi_2^*)(\psi_1 + \psi_2) \\       &= \left| \psi_1 \right|^2 + \left| \psi_2^* \right|^2 + \psi_1^* \psi_2 + \psi_2^* \psi_1  \\     \end{aligned}  }

This is not totally technically correct; for example, the normalization has not been done. However, the formula is still true schematically.

Exactly since “a wave function is not probability”, we have to “square” it in order to get the probability (density). This “squaring” step creates the cross terms \displaystyle{ \psi_1^* \psi_2 + \psi_2^* \psi_1 }. These cross terms are corresponding to the interference effects.

In other words, the interference effects exist exactly because quantum superposition is a mathematical superposition, not a physical superposition of possible worlds.

If the quantum position was a simple overlapping of possible worlds,

\displaystyle{  \begin{aligned}  P &= P_1 + P_2 = \left| \psi_1 \right|^2 + \left| \psi_2^* \right|^2 \\   \end{aligned}  }.

There would have been no cross terms, and then no interference patterns.

3. 

— Me@2022-02-22 07:01:40 PM

— Me@2022-02-25 04:27:37 PM

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2022.02.25 Friday (c) All rights reserved by ACHK