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\section{Formulation}

We are now considering states that are labeled by where they are located.

For example, state $\ket 3$ means that the particle is located at $x=3$. In other words, we are considering wave functions that are on a real line.

Now, let us first consider the scenario that there is only one particle locating at $x=1$. It is totally fine to change the name of the location to, for example, $x=0$. In this sense, we may say that the two representations of this particle $\ket 1$ and $\ket 0$ are equivalent! Technically, we are in a space that is translationally-invariant. So relabelling does not change the physics.

Let us now consider two particles. Particle $A$ is in the state of $\ket 0$ and particle $B$ is in the superposition $\frac{1}{\sqrt{2}}\{ \ket {-1} + \ket {1} \}$. In other words, we are now considering a composite system $\ket {0}_A \otimes ( \ket {-1} + \ket {1} )_B$ if we omit the normalizing factor.

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