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question: "A spin right $\frac{1}{\sqrt 2}(|0\rangle + |1\rangle)$ is sent through a Hadamard gate, creating the superposition of $|+\rangle$ and $|-\rangle$, given by $\frac{1}{\sqrt 2}(|+\rangle + |-\rangle)$. By making a basis change substitution, show that this is equivalent to producing a $|0\rangle$ state."

How do I represent a basis change substitution mathematically?

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$|+\rangle$ and $|-\rangle$ represented by $|0\rangle$ and $|1\rangle$ basis vectors:

$$|+\rangle = \frac{1}{\sqrt{2}} (|0\rangle + |1\rangle) \qquad |-\rangle = \frac{1}{\sqrt{2}} (|0\rangle - |1\rangle)$$

By substituting these expressions that are written in $|0\rangle$ and $|1\rangle$ basis will prove what is wanted:

$$\frac{1}{\sqrt{2}} (|+\rangle + |-\rangle) = \frac{1}{2}(|0\rangle + |1\rangle + |0\rangle - |1\rangle) = |0\rangle$$

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