%%You need an into here and this whole article needs to be reorganized around that. Is there a general statement that can be made about what projections both onto 1D and multi-D spaces does in qm%% # Properties of projection operators 1) $\hat{P}_a^{\dagger}=\hat{P}_a$ ([Hermiticity](Hermitian%20operators.md)) 2) $\hat{P}_a^2 = \hat{P}_a$ ([idempotency](Idempotent%20operators.md)) Properties 1. can be shown in one step by seeing that $\hat{P}_a^{\dagger}=(|a\rangle\langle a|)^{\dagger}=|a\rangle\langle a|=\hat{P}_a$ and Property 2. can also be shown in one step by writing $\hat{P}_a^2=|a\rangle\langle a|a\rangle\langle a|=|a\rangle\langle a|=\hat{P}_a$. Here $\langle a|a\rangle=1$ by the first [definition](Orthonormal%20bases.md#Properties) of orthonormal bases. The proof of Property 1. and Property 2. follow for projections on larger subspaces, $\hat{P}_S = \sum_{i=1}^m |a_i\rangle\langle a_i |$, from [from the antilinearity of the conjugate transpose](Adjoint.md#Conjugate%20transpose%20Properties). # Projections onto one-dimensional subspaces Given elements of an [orthonormal basis](State%20vector.md#Decomposing%20state%20vectors%20into%20sets%20of%20orthonormal%20eigenstates), $\{|a\rangle\}$, A projection operator that projects onto a _one dimensional subspace_ is any operator that has the form: $\hat{P}_a = |a\rangle\langle a|$ ^a213b0 This operator projects a [state vector](State%20vector.md), $|\psi\rangle$, to a particular [eigenstate](State%20vector.md#Decomposing%20state%20vectors%20into%20sets%20of%20orthonormal%20eigenstates) corresponding to the particular projection operator. The corresponding [completeness relation](Orthonormal%20bases.md#Completeness%20relations%20for%20orthonormal%20bases), $\sum_a |a\rangle\langle a|$, of an eigenbasis is the sum of all projection operators that may be formed from that basis. # Projections onto multi-dimensional subspaces %%This actually needs to be the presentation for the general form of the projection operator. Where might these come up? In agular momntum problems where projection is tied to tensor sums of angular momenta.%% Consider an [[Orthonormal bases]] of a [Hilbert Space](Hilbert%20Spaces%20in%20Quantum%20Mechanics.md) $\mathcal{H}$ where its basis set is $\{|a_1\rangle,..., |a_m\rangle, |a_{m+1}\rangle,...,|a_n\rangle\}$ where the Hilbert subspace, $\mathcal{H}_S$ is spanned by $\{|a_1\rangle,..., |a_m\rangle\},$ and an orthogonal to the space, $\mathcal{H}_S^{\perp}$ is spanned by $\{|a_{m+1}\rangle,...,|a_n\rangle\}$ We may write a state $|\psi\rangle$ in $\mathcal{H}$ as $|\psi\rangle=\sum_{i=1}^m c_m|a_i\rangle+\sum_{i=m+1}^n c_m|a_i\rangle$ as a [superposition](Quantum%20superposition.md) of states in $\mathcal{H}_S$ and $\mathcal{H}_S^{\perp}$ written as $|\psi\rangle = |\psi_S\rangle+|\psi_S^{\perp}\rangle = \hat{P}_S|\psi\rangle + \hat{P}_S^{\perp}|\psi\rangle$ where $\hat{P}_S$ and $\hat{P}_S^{\perp}$ each project on the corresponding subspaces and are written as: $\hat{P}_S = \sum_{i=1}^m |a_i\rangle\langle a_i |\,\,\,\,\mathrm{and}\,\,\,\,\hat{P}_S^{\perp} = \sum_{i=m+1}^n |a_i\rangle\langle a_i |$ ## Projections onto degenerate subspaces For [operators](Linear%20transformations%20in%20quantum%20mechanics.md#Linear%20operators%20in%20quantum%20mechanics) $\hat{A}$ and $\hat{B}$ where $a$ and $b$ designate [eigenvalues](Eigenvalues%20and%20eigenvectors.md) of $\hat{A}$ and $\hat{B}$ we may write a _projection onto a degenerate subspace of $\hat{A}$_ as $\hat{P}_{a}=\sum_b |a,b\rangle\langle a,b|$ %%This is from mcgreevy pg 14%% # Role in Quantum Measurement Given a [[State vector]] decomposed into its [Eigenstates](State%20vector#Eigenstates), $|\psi\rangle=\sum_{a} c_a|a\rangle = \sum_{a} \langle a |\psi\rangle|a\rangle$ The [probability](Born%20rule) of measuring a particular eigenstate $|a\rangle$ is given in terms of the projection onto that eigenstate as: $P_a = \langle\psi|\hat{P}_a|\psi\rangle= \langle\psi|a\rangle\langle a |\psi\rangle = |c_a|^2$ ^c36894 which matches the mathematical statement of the [Born rule](Born%20rule). The state of the system after the measurement is also written in terms of the projection operator by the [[von Neumann postulate]]. The projection operator may in fact be considered to be in a broader class of operators called [[measurement operator]]s. # Projection Operators In an experiment Since the projection operator narrows the set of eigenvectors in your a subspace, one may conceptualize the projection operator as being a filter. Examples of experimental components that act as projections: * [Polarizer](Polarizer%20(quantum).md)s --- # Recommended Reading #QuantumMechanics/QuantumMeasurement #QuantumMechanics/QuantumMeasurement/QuantumObservables