Projection operator (quantum mechanics)

A projection operator $\hat{P}$ is a linear operator that transforms a vector in the direction of another vector, i.e., it projects one vector onto another.

In general,

$\hat{P}=\vert\boldsymbol{\mathit{u}}\rangle\langle\boldsymbol{\mathit{v}}\vert$

$\hat{P}\vert\boldsymbol{\mathit{w}}\rangle=(\vert\boldsymbol{\mathit{u}}\rangle\langle\boldsymbol{\mathit{v}}\vert)\vert\boldsymbol{\mathit{w}}\rangle=\vert\boldsymbol{\mathit{u}}\rangle(\langle\boldsymbol{\mathit{v}}\vert\boldsymbol{\mathit{w}}\rangle) =c\vert\boldsymbol{\mathit{u}}\rangle$

where $c$ is a scalar.

It is useful in quantum mechanics to have a projection operator that maps a vector onto another vector, which is part of a complete set of orthonormal basis vectors $\left \{\boldsymbol{\mathit{i}} \right \}$ in a Hilbert space. We define the operator as:

$\hat{P}_{\boldsymbol{\mathit{i}=\boldsymbol{\mathit{m}}}}=\vert\boldsymbol{\mathit{m}}\rangle\langle\boldsymbol{\mathit{m}}\vert$

This allows us to project a vector $\boldsymbol{\mathit{w}}$ onto the basis vector $\boldsymbol{\mathit{m}}$ :

$\hat{P}_{\boldsymbol{\mathit{i}=\boldsymbol{\mathit{m}}}}\vert\boldsymbol{\mathit{w}}\rangle=\vert\boldsymbol{\mathit{m}}\rangle\langle\boldsymbol{\mathit{m}}\vert\boldsymbol{\mathit{w}}\rangle=c\vert\boldsymbol{\mathit{m}}\rangle$

If $\boldsymbol{\mathit{w}}$ is a wavefunction $\psi$ that is a linear combination of a complete set of orthonormal basis functions, i.e., $\psi=\sum_{i=1}^{N}c_i\phi_i$ , then

$\hat{P}_i\psi=\vert\phi_i\rangle\langle\phi_i\vert\psi\rangle=c_i\phi_i$

When we measure an observable of a system whose state is described by $\psi$ , we get an eigenvalue corresponding to an eigenfunction, which is one of the orthonormal basis functions in the complete set. We say that the wavefunction $\psi$ is projected onto (or collapsed into) the eigenfunction $\phi_i$ .

Projection operator (quantum mechanics)

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