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adjoints_and_hermitian_operators [2022/09/27 21:03] – [Projection Operators] adminadjoints_and_hermitian_operators [2022/10/06 00:45] (current) – [Projection Operators] admin
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 where $V^{\perp}$ is the orthogonal complement of $V$ in $\mathcal{H}$. where $V^{\perp}$ is the orthogonal complement of $V$ in $\mathcal{H}$.
  
-In order to completely define the operator, we need to say what it does to vectors that are neither in $V$ or $V^{\perp}$.  However, since $\mathcal{H} = V\oplus V^{\perp}$, we can write any vector $\ket{\psi}$ as $\ket{\psi} = a\ket{\phi} + b\ket{\phi^{\perp}}$, where $\ket{\psi}\in V$ and $\ket{\psi^{\perp}}\in V^{\perp}$.  Then, if we want $\hat{P}_V$ to be a //linear// operator, we must have+In order to completely define the operator, we need to say what it does to vectors that are neither in $V$ or $V^{\perp}$.  However, since $\mathcal{H} = V\oplus V^{\perp}$, we can write any vector $\ket{\psi}$ as $\ket{\psi} = a\ket{\phi} + b\ket{\phi^{\perp}}$, where $\ket{\phi}\in V$ and $\ket{\phi^{\perp}}\in V^{\perp}$.  Then, if we want $\hat{P}_V$ to be a //linear// operator, we must have
 \[\hat{P}_V \ket{\psi} = \hat{P}_V \left ( a\ket{\phi} + b\ket{\phi^{\perp}} \right ) = a\hat{P}_V\ket{\phi} + b\hat{P}_V \ket{\phi^{\perp}} = a\ket{\phi}.\] \[\hat{P}_V \ket{\psi} = \hat{P}_V \left ( a\ket{\phi} + b\ket{\phi^{\perp}} \right ) = a\hat{P}_V\ket{\phi} + b\hat{P}_V \ket{\phi^{\perp}} = a\ket{\phi}.\]
 In other words, the projection operator $\hat{P}_V$ simply returns the component of the vector $\ket{\psi}$ that lies in the subspace $V$. In other words, the projection operator $\hat{P}_V$ simply returns the component of the vector $\ket{\psi}$ that lies in the subspace $V$.