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 $ \hat{a} $ and $ \hat{b} $ with non-zero commutator $ [ \hat{a} , \hat{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} = [ ( \hat{a} {} ^ {2} \phi , \phi ) - ( \hat{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

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

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 $ \hat{x} $, $ \hat{y} $, $ \hat{z} $ and $ \hat{p} _ {x} $, $ \hat{p} _ {y} $, $ \hat{p} _ {z} $ such that

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

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)

Comments

W. Heisenberg [a1] presented this uncertainty principle in 1927. In the same year E.H. Kennard [a2] found relation (2), while the general relation (1) was proved by H.P. Robertson [a3] in 1929. Despite the fact that it is by no means clear why the dispersions (standard deviations) should be the correct measures for the uncertainties, almost all authors of textbooks on quantum mechanics used (1) and (2) as the mathematical formulation of Heisenberg's uncertainty principle. It is, however, not difficult to show that even in the most simple illustrations of the uncertainty principle the dispersions are divergent quantities. This makes (1) and (2) meaningless! (See [a4]–[a6].) A far more satisfactory mathematical formulation of the uncertainty principle has been given by H.J. Landau and H.O. Pollak [a5] in 1961. Amazing as it may be, this formulation has as yet not found its way to the textbooks.

A beautiful survey of the uncertainty principle can be found in [a7], which also contains a wealth of references.

References

[a1]	W. Heisenberg, "The physical principles of quantum theory" , Dover, reprint (1949) (Translated from German)
[a2]	E.H. Kennard, "Zur Quantenmechanik einfacher Bewegungtypen" Z. Physik , 44 (1927) pp. 326–352
[a3]	H.P. Robertson, "The uncertainty principle" Phys. Rev. , 34 (1929) pp. 163–164
[a4]	J. Hilgevoord, J. Uffink, "The mathematical expression of the uncertainty principle" F. Selleri (ed.) A. van der Merwe (ed.) G. Tarozzi (ed.) , Proc. Internat. Conf. Microphysical Reality and Quantum Description (Urbno, Italy, 1985) , Reidel (1988) pp. 91–114
[a5]	J. Hilgevoord, J. Uffink, "A new look on the uncertainty principle" , Internat. School of History of Science: Sixty-two Years of Uncertainty: Historical, Philosophical and Physical Inquiries into the Foundations of Quantum Mechanics (Erice, Sicily) (1989)
[a6]	J. Uffink, J. Hilgevoord, "Uncertainty principle and uncertainty relations" Found. Phys. , 15 (1985) pp. 925
[a7]	J. Uffink, "Measures of uncertainty and the uncertainty principle" , R.U. Utrecht (1990) (Thesis)
[a8]	J.-M. Lévy-Leblond, F. Balibar, "Quantics-rudiments of quantum physics" , North-Holland (1990) (Translated from French)