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Observables and measurements

1 Observables
2 Measurements

Observables

Under the notion of observable in quantum mechanics one can understand any property of a given system in some state, which can be measured in some experiment.
Mathematically observables are postulated to be Hermitian operators mapping Hilbert space $\mathcal{H}$ onto itself and have following properties:

Eigenvalues of observables are real and in fact are possible outcomes of measurements of a given observable.
Corresponding eigenvectors or eigenstates span the Hilbert space, which means, that each observable generates an orthonormal basis, which elements will make up the state after measurement.

Here are several examples of observables:

Observables with continuous spectrum $(dim(\mathcal{H})=\infty)$ : momentum $\hat{p}=-i\hbar\frac{\partial}{\partial x}$ and coordinate $\hat{x}=x$ operators.
Observables with descrete spectrum $(dim(\mathcal{H})=2)$ : Pauli matrices : $\sigma_x = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix},$ $\sigma_y = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix},$ $\sigma_z = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}$

Measurements

The abstract definition of a measurement in quantum mechanics is cloesely related with observables.

Observables with descrete non-degenerate spectrum

Let $O=O^{\dagger}$ be an observable with with discrete non-degenerate spectrum $\lambda_1,\lambda_2,\dots,\lambda_n$ and has descrete eigenstates $|\psi_i\rang$ $i=1,\dots,n$ . Now assume the system is prepared in a state $|\phi\rang$ , which can be represented in eigenbasis of the observable
$|\phi\rang=\sum_{i=1}^nc_i|\psi_i\rang$ , where $c_i\in\mathbb{C}$
Each measurement of the observable $O$ will give some outcome $λ i$ with probability $P(\lambda_i)=|c_i|^2=|\lang\psi_i|\phi\rang|^2$ . After one measured, for example $λ i$ the system will be in the state $|\phi^{\prime}\rang=|\psi_i\rang$ , i.e. projected on one of the eigenstates of the observable $O$ .

Observables with continuous non-degenerate spectrum

This case is rather similar to the previous one. Let $O=O^{\dagger}$ be an observable with a non-degenerate continuous spectrum from $(a,b)\subset \mathbb{R}$ . Each eigenvalue $x$ is associated with a unique eigenstate $|\psi(x)\rang$ . The expansion of the state of the system is
$|\phi\rang=\int_a^b c(x)|\psi(x)\rang dx$ ,
where $c (x)$ is a complex values function, such that $| c (x) | 2$ is a probability density function. Probability of having some outcome $y\in(x^{\prime},x^{\prime\prime})$ is given by
$P(x^{\prime}<y<x^{\prime\prime})=\int_{x^{\prime}}^{x^{\prime\prime}} |c(y)|^2 dy$
after measuring some outcome $y$ the system will be in the state $|\psi(y)\rang$

Observables with degenerate spectra

The analysis in the case of degenerate spectra, where there could be several eigenstates corresponding to a given eigenvalue is mathematically a little bit more involving, but essentially the same. It turns out that it is more convinient to talk about eigenspaces of an observable and decompose the hole Hilbert space in a direct sum of these spaces. Then measuring some outcome will correspond again to projection, but in this case it will be projection on the particular eigenspace and not only on the one of the eigenstates. Probability of the latter measurement can be interpreted as a length of the projection on the eigenspace squared.

Mixed states

Let $ρ$ be a mixed state which the system is prepared in. Let $O$ be an observable with eigenvalues $\lambda_1,\lambda_2,\lambda_3,\dots$ and eigenspaces $v_1,v_2,\dots$ . Moreover let $P_1,P_2,\dots$ be projectors on the corresponding eigenspaces. Then each outcome $λ i$ will be measured with probability
$P (λ i) = Tr(P i ρ)$
After measurement the system will be in the state
$\rho^{\prime}=P_i\rho P_i$

Statistics of outcomes

Often it is convinient to talk about mean values and variances of measurements. Usually experiment consists of several measurements and experimentalist deals with statistical quantities rather than with an outcome of a single measurement.
Mean value of a measurement of an observable $O$ in some state (pure ore mixed) is given by $E(O)=\lang O\rang$ and