# Uncertainty principle

Heisenberg principle

One of the most important principles in quantum mechanics, which asserts that the dispersions of the values of two physical quantities $a$ and $b$ described by non-commuting operators $\widehat{a}$ and $\widehat{b}$ with non-zero commutator $[ \widehat{a} , \widehat{b} ]$ in any state of a physical system cannot be simultaneously very small.

More precisely, let $\phi \in H$, $\| \phi \| = 1$, be a state of a physical system ( $H$ is the Hilbert space of these states and $( \cdot , \cdot )$ is the scalar product in $H$) and let $\Delta _ \phi ^ {a} = [ ( \widehat{a} {} ^ {2} \phi , \phi ) - ( \widehat{a} \phi , \phi ) ^ {2} ] ^ {1/2 }$ be the dispersion of the quantity $a$ in the state $\phi$; $\Delta _ \phi ^ {b}$ is defined similarly. Then always

$$\tag{1 } \Delta _ \phi ^ {a} \Delta _ \phi ^ {b} \geq \ \frac{1}{2} | ( [ \widehat{a} , \widehat{b} ] \phi , \phi ) | .$$

In particular, the coordinates $x$, $y$, $z$ of a quantum particle and the components $p _ {x}$, $p _ {y}$, $p _ {z}$ of its momentum under all standard free quantizations (i.e. choices of the space $H$ and rules for associating self-adjoint operators acting on $H$ with physical quantities) are represented by operators $\widehat{x}$, $\widehat{y}$, $\widehat{z}$ and $\widehat{p} _ {x}$, $\widehat{p} _ {y}$, $\widehat{p} _ {z}$ such that

$$[ \widehat{p} _ {x} , \widehat{x} ] = \ [ \widehat{p} _ {y} , \widehat{y} ] = \ [ \widehat{p} _ {z} , \widehat{z} ] = i \hbar E ,$$

where $E$ is the identity operator on $H$ and $\hbar$ is the Planck constant. Thus, for any $\phi \in H$,

$$\tag{2 } \Delta _ \phi ^ {p _ {x} } \Delta _ \phi ^ {x} \geq \ \frac \hbar {2} ,\ \ \Delta _ \phi ^ {p _ {y} } \Delta _ \phi ^ {y} \geq \ \frac \hbar {2} ,\ \ \Delta _ \phi ^ {p _ {z} } \Delta _ \phi ^ {z} \geq \ \frac \hbar {2} .$$

#### References

 [1] L.D. Landau, E.M. Lifshitz, "Quantum mechanics" , Pergamon (1965) (Translated from Russian)