In physics we define observables as follows: .md#^72cb87) The way [observables](Observable%20(classical%20mechanics).md) are encoded in quantum mechanics adds additional structure to this definition. Each [[state vector]] that forms part of an [orthonormal basis](Orthonormal%20bases.md) is associated with a class of [Hermitian operators](Hermitian%20operators.md) called _observables_ that belong in its respective [Hilbert Space](Hilbert%20Spaces%20in%20Quantum%20Mechanics). And this state vector is an orthonormalized [eigenvector](Eigenvalues%20and%20eigenvectors.md) of its associated observable. Such normalized eigenvectors are called [eigenstates](State%20vector.md#Decomposing%20state%20vectors%20into%20sets%20of%20orthonormal%20eigenstates). Since the eigenvectors of a Hermitian operator form an [[Orthonormal bases]] we must then use sets of orthonormal vectors in order to [construct](Observable.md#Constructing%20observables) observables. An observable may be bounded (representable as a matrix) or [[unbounded]], where bounded observables correspond with discrete quantities and [[unbounded]] observables correspond with continuous quantities. %%It may be good to put in the basic mathematical expression here that explicitly shows what an observable does. Here it is a bit vague and lengthy.%% # Physical meaning of _Observables_ An experimentally measurable quantity is the corresponding eigenvalue, $a,$ to a given eigenstate, $|a\rangle$. Thus the observable _represents the act of measuring a physical quantity_. Measurable quantities must be given by real numbers and likewise Hermitian operators have [real eigenvalues.](Hermitian%20operators.md#Properties%20of%20Hermitian%20Operators) # Constructing observables Consider an on observable, $\hat{O}$ acting on some [[State vector]], $|\psi\rangle$. Given that the observable must have a set of [eigenstates](State%20vector.md#Decomposing%20state%20vectors%20into%20sets%20of%20orthonormal%20eigenstates), $\{|a\rangle\}$, that form an orthonormal basis, we can always insert those eigenstates in the form of their [completeness relation](State%20vector.md#State%20vector%20completeness%20relation), [$\sum_i |a_i\rangle\langle a_i|=\mathbb{1}$](State%20vector.md#^18c186), giving, $\hat{O}|\psi\rangle=\hat{O}\bigg(\sum_i |a_i\rangle\langle a_i|\bigg)|\psi\rangle$ Eigenstates are eigenvectors of [observables](Observable.md) and thus these eigenstates have corresponding [eigenvalues](Eigenvalues%20and%20eigenvectors.md) $a_i.$ Thus $\hat{O} |a_i\rangle=a_i|a_i\rangle$ Since $\hat{O} |a_i\rangle=a_i|a_i\rangle$, the observable may then be constructed from the sum of the [projection operators](Projection%20operators%20in%20quantum%20mechanics.md), $\hat{P}_i,$ formed by each possible _eigenstate_, $|a_i\rangle$, in its basis, giving $\hat{O}=\sum_i a_i|a_i\rangle\langle a_i | =\sum_i a_i\hat{P}_i$ ^28da19 ## Constructing observables as matrices If the Observable acts on a finite, $n$, dimensional [Hilbert Space](Hilbert%20Spaces%20in%20Quantum%20Mechanics.md#Finite%20Dimensional%20Hilbert%20Spaces) it may be written as an $n\times n$ [Hermitian matrix](Hermitian%20operators.md#Properties%20of%20Hermitian%20matrices). To derive the matrix elements, we decompose $\hat{O}$ with two [completeness relations](State%20vector.md#State%20vector%20completeness%20relation) corresponding to its set of eigenstates $\{|a_1\rangle...|a_j\rangle...|a_n\rangle\}$, which correspond to column vectors, and their dual-space horizontal vectors, $\{\langle a_1|...\langle a_i\rangle...\langle a_n|\}$, which are mapped to each other by their [Adjoint](Adjoint.md). Thus, $\hat{O} = \bigg(\sum_i^n |a_i\rangle\langle a_i |\bigg)\hat{O}\bigg(\sum_j^n |a_j\rangle\langle a_j |\bigg)=\sum_{i,j}^n\langle a_i|\hat{O}|a_j\rangle |a_i\rangle\langle a_j |$ where the matrix elements are $O_{ij}=\langle a_i|\hat{O}|a_j\rangle$, giving the final form, $\hat{O}=\sum_{i,j}^n O_{ij}|a_i\rangle\langle a_j |.$ Finally the matrix form, which we can use when the operator acts on a [finite dimensional Hilbert Space](Hilbert%20Spaces%20in%20Quantum%20Mechanics.md#Finite%20Dimensional%20Hilbert%20Spaces), comes about as a result of the [Vector outer products](Vector%20outer%20products.md), $|a_i\rangle\langle a_j |$, yielding $\hat{O} = \begin{pmatrix} O_{11} &...& O_{1n}\\ \vdots & \ddots & \\ O_{n1} & ... & O_{nn}\end{pmatrix} = \begin{pmatrix} \langle a_1|\hat{O}|a_1\rangle &...& \langle a_1|\hat{O}|a_n\rangle\\ \vdots & \ddots & \\ \langle a_n|\hat{O}|a_1\rangle & ... & \langle a_n|\hat{O}|a_n\rangle\end{pmatrix}.$ ## Continuous observables # state vector transformation An observable transforms a [[state vector]] as follows. Consider the transformation $|\phi\rangle=\hat{O}|\psi\rangle,$ which represents the [act of measuring.](Observable.md#Physical%20meaning%20of%20_Observables_) We have [sets of orthonormal eigenstates](State%20vector.md#Decomposing%20state%20vectors%20into%20sets%20of%20orthonormal%20eigenstates) $\{|a_i\rangle\}$ and $\{|a_j\rangle\}$ that [compose](State%20vector.md#Decomposing%20state%20vectors%20into%20sets%20of%20orthonormal%20eigenstates) $|\phi\rangle$ and $|\psi\rangle$ respectively. We can write the [probability amplitude](probability%20amplitude.md) for a given measured quantity $a_i$ (with associated eigenstate $|a_i\rangle$ of $|\phi\rangle$) in terms of a sum containing the set of eigenstates of $|\psi\rangle$ and their probability amplitudes $\langle a_j |\psi\rangle$. If we plug in $|\psi\rangle$ [as a sum of eigenstates](State%20vector.md#Decomposing%20state%20vectors%20into%20sets%20of%20orthonormal%20eigenstates) $\langle a_i | \phi \rangle = \langle a_i |\hat{O}|\psi \rangle = \sum_{i} \langle a_i | \hat{O} | a_j \rangle \langle a_j |\psi\rangle$ And equivalently as a matrix transformation in a [finite dimensional Hilbert Space](Hilbert%20Spaces%20in%20Quantum%20Mechanics.md#Finite%20Dimensional%20Hilbert%20Spaces), $\begin{pmatrix} \langle a_1|\phi\rangle\\ \vdots \\ \langle a_n|\phi\rangle \end{pmatrix} = \begin{pmatrix} \langle a_1|\hat{O}|a_1\rangle &...& \langle a_1|\hat{O}|a_n\rangle\\ \vdots & \ddots & \\ \langle a_n|\hat{O}|a_1\rangle & ... & \langle a_n|\hat{O}|a_n\rangle\end{pmatrix}\begin{pmatrix} \langle a_1|\psi\rangle\\ \vdots \\ \langle a_n|\psi\rangle \end{pmatrix}.$ The operator acts on a bra state as follows: $\langle \phi| = \langle \psi |\hat{O}^{\dagger}$ and the matrix transformation proceeds in the same manner. How do we know what [eigenstates](State%20vector.md#Decomposing%20state%20vectors%20into%20sets%20of%20orthonormal%20eigenstates) the state vector $|\phi\rangle$ will be composed of following a measurement? These eigenstates are determined by the [[von Neumann postulate]], which tells us that the set of eigenstates that composes $|\phi\rangle$ corresponds with the result of the measurement which has a [probability](Born%20rule.md) ascribed to it. ### Additional Matrix Element Properties 1. $\hat{O}^{\dagger}=(\hat{O}^T)^*$ 2. $\langle a_i|\hat{O}^{\dagger}|a_j\rangle =\langle a_j|\hat{O}|a_i\rangle^*$, equivalently, $(\hat{O}^{\dagger})_{ij}=O_{ij}^*$ 3. The diagonal elements, $O_{ij}\in\mathbb{R}$ for $i=j$, and $O_{ij}\in\mathbb{C}$ for $i \neq j$. Property 1, is obvious from the fact that the adjoint of a matrix is equivalent to the [conjugate transpose](Adjoint.md#conjugate%20transpose) and property 2. can be seen from directly taking the adjoint of the operator and seeing that the indices switch. # Constructing observables from degenerate eigenstates For each eigenvalue there is one eigenstate unless the eigenvalue is _degenerate_. %%There needs to be a defined correspondence between quantum and classical observables.%% --- # Recommended reading This page follows closely from the following lectures notes: * [Schollwöck U., _Grundlegende Formulismus 1_, T2: Quantenmechanik Lecture Notes, Winter 2019/2020 (German)](Quantum%20Mechanics/File%20Repository/QM_-2-Formalismus-1_2.pdf) (Not publicly available) For a description of [observables](Observable.md) as sums of [projections](Projection%20operators%20in%20quantum%20mechanics.md) as given [here](Observable.md#Constructing%20observables) see * [McGreevy, John. A., Physics 212A Lecture Notes, Fall 2015.](McGreevy,%20John.%20A.,%20Physics%20212A%20Lecture%20Notes,%20Fall%202015..md), pgs. 12-15. #QuantumMechanics/QuantumMeasurement/QuantumObservables