The 4 bugs, 1.7


In an accurate language, that design has already given the experiment-setup a property that “X is in a mixed state”, meaning that the probabilities assigned to different possible values of X are actually classical probabilities.


In the most basic double-slit experiment, assume that the probability of the going to either slit is \displaystyle{\frac{1}{2}}.

In the standard quantum mechanics language:

When no detector installed or no detector is activated, 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 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}

Instead, mathematically, the particle’s position variable is in the state \displaystyle{| \psi \rangle}, which is a pure state, which is one single state, not a statistical mixture.

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

Physically, it means that although, before measurement, the position variable of the particle is in the state, after measurement, the state will have a probability of \displaystyle{0.5} to become \displaystyle{| \psi_L \rangle}; and a probability of \displaystyle{0.5} to become \displaystyle{| \psi_R \rangle}.

In short, the wave function \displaystyle{| \psi \rangle } will collapse to either \displaystyle{| \psi_L \rangle} or \displaystyle{| \psi_R \rangle}.

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:

— Me@2022-02-22 07:01:40 PM



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