The 4 bugs, 1.10

EPR paradox, 11.9

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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 physical states. 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. 

3.  Besides calculating interference patterns in our system (\displaystyle{A}), the coefficients in the superposition are also useful for another system (\displaystyle{B}), which is identical to \displaystyle{A} but with a detector activated.

In our double slit experiment (system \displaystyle{A}), no detector is activated. So the particle’s position variable is in a superposition state

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

where \displaystyle{|\psi_L \rangle} and \displaystyle{| \psi_R \rangle} are eigenstates of going-left and that of going-right respectively. The wave function \displaystyle{| \psi \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 still physically-undefined.

Since system \displaystyle{B} has a detector to provide the physical definitions of “going-left” and “going-right”, the wave function for \displaystyle{B} is not a superposition. Instead, it is schematically

\displaystyle{| \phi \rangle = | \psi_L \rangle~\text{or}~| \psi_R \rangle},

where the corresponding probabilities are given by the squares of each superposition coefficient in \displaystyle{| \psi \rangle}. In other words, \displaystyle{| \phi \rangle} has 0.5 probability being \displaystyle{| \psi_L \rangle} and 0.5 probability being \displaystyle{| \psi_R \rangle}. Instead of being a superposition state, \displaystyle{| \phi \rangle} is a statistical mixture, which is called a “mixed state”.

1.  pure state

1.1  eigenstate

1.2  superposition (of eigenstates)

2.  mixed state

Formally, to represent any kind of states, we need to use the mathematics formalism density matrix.

For the system \displaystyle{A} (with superposition state \displaystyle{\psi}), the density matrix is

\displaystyle{  \begin{aligned}    \rho_A &= | \psi \rangle \langle \psi | \\      &= \left( \frac{1}{\sqrt{2}} | \psi_L \rangle + \frac{1}{\sqrt{2}} | \psi_R \rangle \right) \left( \frac{1}{\sqrt{2}} \langle \psi_L | + \frac{1}{\sqrt{2}} \langle \psi_R | \right) \\     \end{aligned}}

For simplicity, assume that the eigenstates \displaystyle{ \{ |\psi_L\rangle, |\psi_R\rangle \}} form a complete orthonormal set. If we use \displaystyle{\{ | \psi_L \rangle, |\psi_R \rangle \}} as basis,

\displaystyle{  \begin{aligned}    \left[ \rho_A \right]     &=   \begin{bmatrix}   \frac{1}{\sqrt 2} \\   \frac{1}{\sqrt 2}   \end{bmatrix}   \begin{bmatrix} \frac{1}{\sqrt 2} & \frac{1}{\sqrt 2} \end{bmatrix} \\    &=   \begin{bmatrix} \frac{1}{\sqrt 2} & \frac{1}{\sqrt 2} \\   \frac{1}{\sqrt 2} & \frac{1}{\sqrt 2}   \end{bmatrix}       \end{aligned}}

For the system \displaystyle{B} (with state \displaystyle{| \phi \rangle}), the density matrix is

\displaystyle{  \begin{aligned}    \rho_B &= \frac{1}{2} | \psi_L \rangle \langle \psi_L | + \frac{1}{2} | \psi_R \rangle \langle \psi_R | \\     \left[ \rho_B \right] &= \begin{bmatrix} \frac{1}{\sqrt 2} & 0 \\ 0 & \frac{1}{\sqrt 2} \end{bmatrix}   \end{aligned}}

Since the mixed state coefficients of system \displaystyle{B} are provided by the superposition coefficients of system \displaystyle{A}, we have a language shortcut in quantum mechanics:

For a system \displaystyle{A} in a superposition state

\displaystyle{| \psi \rangle = a~| \psi_L \rangle + b~| \psi_R \rangle},

if we install and activate a detector to measure which slit the particle goes through, there are two possible results. One possible result is “left”, with probability \displaystyle{|a|^2}; another is “right”, with probability \displaystyle{|b|^2}.

In other words, the wave function \displaystyle{| \psi \rangle} has a chance of \displaystyle{|a|^2} to collapse to \displaystyle{| \psi_L \rangle} and a chance of \displaystyle{|b|^2} to collapse to \displaystyle{| \psi_R \rangle}.

Note that this kind of language shortcut should be used as a shortcut (for the calculations in daily-life quantum mechanics applications) only. Do not take those words, especially the word “collapse”, literally. If you regard the shortcut presentation as more than shortcut, your understanding of quantum mechanics fundamental concepts will be fundamentally wrong.

If not for daily-life quantum mechanics, but for lifelong quantum mechanics understanding, you have to learn the longcut version.

— Me@2022-02-22 07:01:40 PM

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2022.02.27 Sunday (c) All rights reserved by ACHK