The von Neumann postulate describes the effect of a so called _von Neumann measurement_ or _projective measurement_ on a quantum system given by a [state vector](State%20vector.md). Consider an initial state vector $|\psi\rangle$, which has included in it the [eigenstate](State%20vector.md#Decomposing%20state%20vectors%20into%20sets%20of%20orthonormal%20eigenstates) $|a\rangle.$ Suppose that the result of a measurement is the quantity $a$ [associated with](Observable.md#Physical%20meaning%20of%20_Observables_) $|a\rangle.$ The [[State vector]] following that measurement is given by, $|\phi\rangle = \frac{\hat{P}_a|\psi\rangle}{\sqrt{\langle\psi|\hat{P}_a|\psi\rangle}}=\frac{\hat{P}_a|\psi\rangle}{\sqrt{P_a}}$ ^db1d13 where $\hat{P}_a$ is a [projections onto a one-dimensional subspace](Projection%20operators%20in%20quantum%20mechanics.md#Projections%20onto%20one-dimensional%20subspaces) formed from the particular [eigenstate](State%20vector#Decomposing%20state%20vectors%20into%20sets%20of%20orthonormal%20eigenstates), $|a\rangle.$ Following from the [Born rule,](Born%20rule.md) the probability of obtaining a measured result $a,$ where the state afterwards is $|\phi\rangle$ following a measurement is expressed as  %%This concept seems to be referred to as a selective measurement in Sakurai pg. 25 however Sakurai doesnt mention the normalization factor.%% # Generalization of the von Neumann postulate The formulation of the [von Neumann postulate](von%20Neumann%20postulate.md) follows from a [generalization](measurement%20operator.md#States%20following%20quantum%20measurement) to other [measurement operator](measurement%20operator)s. We can easily see this by placing $\hat{P}_a$ in place of $\hat{M}_a$ where $\hat{M}_a$ is a general measurement operator corresponding with some eigenstate, $|a\rangle$. Thus, a [state following a measurement is expressed as](measurement%20operator.md#States%20following%20quantum%20measurement) $\frac{\hat{M}_a|\psi\rangle}{\sqrt{\langle\psi|\hat{M}_a^{\dagger}\hat{M}_a|\psi\rangle}}=\frac{\hat{P}_a|\psi\rangle}{\sqrt{\langle\psi|\hat{P}^{\dagger}_a\hat{P}_a|\psi\rangle}}=\frac{\hat{P}_a|\psi\rangle}{\sqrt{\langle\psi|\hat{P}_a|\psi\rangle}}$ where $\hat{P}^{\dagger}_a\hat{P}_a=\hat{P}_a^2=\hat{P}_a$ by [Properties](Projection%20operators%20in%20quantum%20mechanics.md#Properties%20of%20projection%20operators) 1. and 2. of the projection operator. ## POVMs  ([... see more](POVM.md)) --- # Recommended reading The notion of a _von Neumann measurement_ is introduced in terms of a broader notion of [measurement operators](measurement%20operator.md) here: * [Nielson, M. A., Chuang, I. L. _Quantum Computation and Quantum Information_, Cambridge University Press, 2010](Nielsen,%20M.%20A.,%20Chuang,%20I.%20L.%20Quantum%20Computation%20and%20Quantum%20Information,%20Cambridge%20University%20Press,%202010.md) pgs 87-88. Von Neumann measurements are typically the first kind of quantum measurement introduced to students, while generalized measurements are usually only introduced in more specialized courses. As such, the definition of von Neumann measurements may not even be named in many introductory textbooks while its associated [formula](von%20Neumann%20postulate.md#^db1d13) is still usually given, such is in the following sources: * [Schollwöck U., _Grundlegende Formulismus 1_, T2: Quantenmechanik Lecture Notes, Winter 2019/2020 (German)](Quantum%20Mechanics/File%20Repository/QM_-2-Formalismus-1_2.pdf) pg 18 (not publicly available) * [Shankar, R., _Principles of Quantum Mechanics_, Plenum Press, 2nd edition, 1994.](Shankar,%20R.,%20Principles%20of%20Quantum%20Mechanics,%20Plenum%20Press,%202nd%20edition,%201994..md) pg 124 #QuantumMechanics/QuantumMeasurement #QuantumMechanics/FoundationsOfQuantumMechanics