Quantum mechanics does not predict the result of any given measurement, but it does predict the probability of obtaining a particular value. For a given [Quantum system](Quantum%20systems.md) this probability is determined by the _Born rule^[This rule is so called because it is introduced in [Born, M., Zur Quantenmechanik der Stoßvorgänge, Zeitschrift für Physik 37, 863-867, 1926.](Born,%20M.,%20Zur%20Quantenmechanik%20der%20Stoßvorgänge,%20Zeitschrift%20für%20Physik%2037,%20863-867,%201926..md)]._ ^072bea Given a [state vector](State%20vector), $|\psi\rangle$, the Born rule states that: * the result of a [quantum measurement](Quantum%20measurement%20(index).md) is the [eigenvalue](Observable.md#Constructing%20observables), $a$, of a given [observable](Observable.md) and the measurement always [changes](Born%20rule.md#Von%20Neumann%20postulate) the state vector. * The probability of this result is given by $P_a = |\langle a|\psi\rangle|^2$, which is the square of the [[probability amplitude]]. ^628928 * This is equivalently the [expectation value](expectation%20value.md) of a [projection operator](Projection%20operators%20in%20quantum%20mechanics.md), [$\hat{P}_a = |a\rangle\langle a|$](Projection%20operators%20in%20quantum%20mechanics.md#^a213b0) since $P_a = |\langle a|\psi\rangle|^2=\langle \psi|a\rangle\langle a|\psi\rangle$ and [$\langle \psi|\hat{P}_a|\psi\rangle=\langle \psi|a\rangle\langle a|\psi\rangle.$](expectation%20value.md#^6ab61b) * It follows then that given a [[Wavefunction]], $\psi(x)$, the probability of measuring a state $|x\rangle$ is $P_x=|\psi(x)|^2.$ This result is taken to be a [postulate](Postulates%20of%20Quantum%20Mechanics.md) and introduces [probability](Statistics%20and%20Probability%20(index).md#Probability) into quantum mechanics. In an experimental setting, one can estimate a measurement probability by repeated measurements of identical quantum systems (i.e. systems containing the same ingredients with each prepared in the exact same way). %%Please connect this to observables%% # Role of projection operator We may mathematically model the process of measurement as a [projection](Projection%20operators%20in%20quantum%20mechanics.md) onto a subspace containing one vector and thus the probability appears as the square modulus [of the projection](Projection%20operators%20in%20quantum%20mechanics.md#Role%20in%20Quantum%20Measurement). The application of a so called _projection operator_ or _[measurement operator](measurement%20operator.md)_ is what gives rise to the quantum state following a measurement. %%generalize this a bit - the projection operator is only the simplest kind of measurement operator.%% ## Von Neumann postulate The Born rule tells us nothing about the [state vector](State%20vector.md) after a measurement. This is elaborated on at the most elementary level by the [von Neumann postulate](von%20Neumann%20postulate). # Statistics with quantum measurements Since the [Born rule](Born%20rule.md) establishes that quantum measurements are probabilistic, in order to study [quantum systems](Quantum%20systems.md) we apply concepts form [statistics and probability](Statistics%20and%20Probability%20(index).md) in order to derive [uncertainties](Uncertainty%20(quantum%20mechanics).md) and [variances](variance%20(quantum%20mechanics).md) for quantum measurements. #QuantumMechanics/QuantumMeasurement #QuantumMechanics/FoundationsOfQuantumMechanics