Editing Projection-valued measure (section)

==Application in quantum mechanics==
{{see also|Expectation value (quantum mechanics)}}
In quantum mechanics, given a projection-valued measure of a measurable space <math>X</math> to the space of continuous endomorphisms upon a Hilbert space <math>H</math>,

* the [[Projective Hilbert space|projective space]] <math>\mathbf{P}(H)</math> of the Hilbert space <math>H</math> is interpreted as the set of possible ([[Probability_amplitude#Normalization|normalizable]]) states <math>\varphi</math> of a quantum system,{{sfn | Ashtekar | Schilling | 1999 | pp=23–65}}
* the measurable space <math>X</math> is the value space for some quantum property of the system (an "observable"), 
* the projection-valued measure <math>\pi</math> expresses the probability that the [[observable]] takes on various values.

A common choice for <math>X</math> is the real line, but it may also be 

* <math>\mathbb{R}^3</math> (for position or momentum in three dimensions ), 
* a discrete set (for angular momentum, energy of a bound state, etc.), 
* the 2-point set "true" and "false" for the truth-value of an arbitrary proposition about <math>\varphi</math>.

Let <math>E</math> be a measurable subset of <math>X</math> and <math>\varphi</math> a normalized [[Quantum_state|vector quantum state]] in <math>H</math>, so that its Hilbert norm is unitary, <math>\|\varphi\|=1</math>. The probability that the observable takes its value in <math>E</math>, given the system in state <math>\varphi</math>, is

:<math>
 P_\pi(\varphi)(E) = \langle \varphi\mid\pi(E)(\varphi)\rangle = \langle \varphi\mid\pi(E)\mid\varphi\rangle.</math>

We can parse this in two ways. First, for each fixed <math>E</math>, the projection <math>\pi(E)</math> is a [[self-adjoint operator]] on <math>H</math> whose 1-eigenspace are the states <math>\varphi</math> for which the value of the observable always lies in <math>E</math>, and whose 0-eigenspace are the states <math>\varphi</math> for which the value of the observable never lies in <math>E</math>.  

Second, for each fixed normalized vector state <math>\varphi</math>, the association 

:<math>
P_\pi(\varphi) :
E \mapsto  \langle\varphi\mid\pi(E)\varphi\rangle
</math>
is a probability measure on <math>X</math> making the values of the observable into a random variable.

{{Anchor|Projective measurement}}A measurement that can be performed by a projection-valued measure <math>\pi</math> is called a '''projective measurement'''. 

If <math>X</math> is the real number line, there exists, associated to <math>\pi</math>, a self-adjoint operator <math>A</math> defined on <math>H</math> by

:<math>A(\varphi) = \int_{\mathbb{R}} \lambda \,d\pi(\lambda)(\varphi),</math>

which reduces to

:<math>A(\varphi) = \sum_i \lambda_i \pi({\lambda_i})(\varphi)</math>

if the support of <math>\pi</math> is a discrete subset of <math>X</math>. 

The above operator <math>A</math> is called the observable associated with the spectral measure.