%%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#State%20vectors%20that%20can%20be%20decomposed%20into%20discrete%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#State%20vectors%20that%20can%20be%20decomposed%20into%20discrete%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: $\hat{P}_S = \sum_{i=1}^m |a_i\rangle\langle a_i |\,\,\,\,\mathrm{and}\,\,\,\,\hat{P}_S^{\perp} = \sum_{i=m+1}^n |a_i\rangle\langle a_i |$ ## 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. ### Constructing observables from projection operators Given some [observable](Observable.md) $\hat{O}$ along with its [eigenstates](observable.md#^151c59) and [eigenvalues,](observable.md#^3657bb)  ([... see more](Observable.md#Constructing%20observables)) ### Extracting projection operators from observables Given some [observable](Observable.md) $\hat{O}$ where we only know its [eigenvalues,](observable.md#^3657bb) we may extract a [projector to a particular state](Projection%20operators%20in%20quantum%20mechanics.md#Projections%20onto%20one-dimensional%20subspaces) with the following expression: $\hat{P}_j = \prod_{i \neq j} \frac{(\hat{O} - a_i)}{(a_j - a_i)}$ This is a convenient way of also finding the [eigenstates](observable.md#^151c59) of a system if those eigenstates are otherwise difficult to derive. %%This is from homework 3 problem 1 in Schollwock's quantum 1 course from 2019. additionally it needs to be shown how these projectors are built from the null observable in our particular space.%% ## 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%% # 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 --- # Proofs and Examples ## Extracting spin projection operators from their known eigenvalues %%This is an example inspired from problem 1 homework 3 of Schollwock's qm class. the text here should be quoted from text in the corresponding spin-operator article.%% --- # Recommended Reading #QuantumMechanics/QuantumMeasurement #QuantumMechanics/QuantumMeasurement/QuantumObservables