Editing Quantum mechanics (section)

=== Uncertainty principle ===

One consequence of the basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum.<ref name="Cohen-Tannoudji">{{cite book |last1=Cohen-Tannoudji |first1=Claude |last2=Diu |first2=Bernard |last3=Laloë |first3=Franck |title=Quantum Mechanics |author-link1=Claude Cohen-Tannoudji |publisher=John Wiley & Sons |year=2005 |isbn=0-471-16433-X |translator-first1=Susan Reid |translator-last1=Hemley |translator-first2=Nicole |translator-last2=Ostrowsky |translator-first3=Dan |translator-last3=Ostrowsky}}</ref><ref name="L&L">{{cite book |last1=Landau |first1=Lev D. |author-link1=Lev Landau |url=https://archive.org/details/QuantumMechanics_104 |title=Quantum Mechanics: Non-Relativistic Theory |last2=Lifschitz |first2=Evgeny M. |author-link2=Evgeny Lifshitz |publisher=[[Pergamon Press]] |year=1977 |isbn=978-0-08-020940-1 |edition=3rd |volume=3 |oclc=2284121}}</ref> Both position and momentum are observables, meaning that they are represented by [[Hermitian operators]]. The position operator <math>\hat{X}</math> and momentum operator <math>\hat{P}</math> do not commute, but rather satisfy the [[canonical commutation relation]]:
<math display=block>[\hat{X}, \hat{P}] = i\hbar.</math>
Given a quantum state, the Born rule lets us compute expectation values for both <math>X</math> and <math>P</math>, and moreover for powers of them. Defining the uncertainty for an observable by a [[standard deviation]], we have
<math display=block>\sigma_X={\textstyle \sqrt{\left\langle X^2 \right\rangle - \left\langle X \right\rangle^2}},</math>
and likewise for the momentum:
<math display=block>\sigma_P=\sqrt{\left\langle P^2 \right\rangle - \left\langle P \right\rangle^2}.</math>
The uncertainty principle states that
<math display=block>\sigma_X \sigma_P \geq \frac{\hbar}{2}.</math>
Either standard deviation can in principle be made arbitrarily small, but not both simultaneously.<ref name="ballentine1970">Section 3.2 of {{Citation |last=Ballentine |first=Leslie E. |title=The Statistical Interpretation of Quantum Mechanics |journal=Reviews of Modern Physics |volume=42 |issue=4 |pages=358–381 |year=1970 |bibcode=1970RvMP...42..358B |doi=10.1103/RevModPhys.42.358 |s2cid=120024263}}. This fact is experimentally well-known for example in quantum optics; see e.g. chap. 2 and Fig. 2.1 {{Citation |last=Leonhardt |first=Ulf |title=Measuring the Quantum State of Light |year=1997 |url=https://archive.org/details/measuringquantum0000leon |location=Cambridge |publisher=Cambridge University Press |bibcode=1997mqsl.book.....L |isbn=0-521-49730-2}}.</ref> This inequality generalizes to arbitrary pairs of self-adjoint operators <math>A</math> and <math>B</math>. The [[commutator]] of these two operators is
<math display=block>[A,B]=AB-BA,</math>
and this provides the lower bound on the product of standard deviations:
<math display=block>\sigma_A \sigma_B \geq \tfrac12 \left|\bigl\langle[A,B]\bigr\rangle \right|.</math>

Another consequence of the canonical commutation relation is that the position and momentum operators are [[Fourier transform#Uncertainty principle|Fourier transforms]] of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of the dependence in position means that the momentum operator is equivalent (up to an <math>i/\hbar</math> factor) to taking the derivative according to the position, since in Fourier analysis [[Fourier transform#Differentiation|differentiation corresponds to multiplication in the dual space]]. This is why in quantum equations in position space, the momentum <math> p_i</math> is replaced by <math>-i \hbar \frac {\partial}{\partial x}</math>, and in particular in the [[Schrödinger equation#Equation|non-relativistic Schrödinger equation in position space]] the momentum-squared term is replaced with a Laplacian times <math>-\hbar^2</math>.<ref name="Cohen-Tannoudji" />