We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions $f$ on $\R^d$ which may be written as $P(x)\exp (Ax,x)$ , with $A$ a real symmetric definite positive matrix, are characterized by integrability conditions on the product $f(x)\hat{f}(y)$ . We also give the best constant in uncertainty principles of Gelf'and Shilov type. We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambiguity function or the Wigner transform). We complete the paper with a sharp version of Heisenberg's inequality for this transform.We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions $f$ on $\R^d$ which may be written as $P(x)\exp (Ax,x)$, with $A$ a real symmetric definite positive matrix, are characterized by integrability conditions on the product $f(x)\hat{f}(y)$. We also give the best constant in uncertainty principles of Gelf'and Shilov type. We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambiguity function or the Wigner transform). We complete the paper with a sharp version of Heisenberg's inequality for this transform.

arXiv:math/0102111Feb 14, 20010 citations

Visualization of the Uncertainty Principle from the Particle-Centric View: Pre-Wave Function and Post-Wave Function

Clara Snow, Michael G. Messina

The Uncertainty Principle is usually discussed twice in a Quantum Chemistry course. The first discussion is a “pre-wave function” explanation that occurs before students know about wave functions, but do know the de Broglie hypothesis. The second description of the Uncertainty Principle is “post-wave function”, i.e., after students have learned about wave functions, operators, and expectation values. The more common pre-wave function explanations of the Uncertainty Principle can lead students to believe that the Uncertainty Principle sets limits upon what can be measured in the lab. This is the “measurement view” of the Uncertainty Principle. The Uncertainty Principle says something much deeper than this; particles do not possess some properties solely because of their wave nature. This is the “particle-centric view” of the Uncertainty Principle. We show that the Uncertainty Principle can be explained from the preferred particle-centric view when students only know about the de Broglie wavelength and basic statistics. To accompany this pre-wave function discussion of the Uncertainty Principle, we have designed an interactive Module that students can use to visualize the Uncertainty Principle from the particle viewpoint. We also describe a way to explain the Uncertainty Principle, which is complementary to the pre-wave function discussion, after the students know about wave functions, using the same elementary statistical analysis used in the pre-wave function presentation. We have designed a second interactive Module that students can use to visualize the Uncertainty Principle when particles are described by complex wave functions.The Uncertainty Principle is usually discussed twice in a Quantum Chemistry course. The first discussion is a “pre-wave function” explanation that occurs before students know about wave functions, but do know the de Broglie hypothesis. The second description of the Uncertainty Principle is “post-wave function”, i.e., after students have learned about wave functions, operators, and expectation values. The more common pre-wave function explanations of the Uncertainty Principle can lead students to believe that the Uncertainty Principle sets limits upon what can be measured in the lab. This is the “measurement view” of the Uncertainty Principle. The Uncertainty Principle says something much deeper than this; particles do not possess some properties solely because of their wave nature. This is the “particle-centric view” of the Uncertainty Principle. We show that the Uncertainty Principle can be explained from the preferred particle-centric view when students only know about the de Broglie wavelength and basic statistics. To accompany this pre-wave function discussion of the Uncertainty Principle, we have designed an interactive Module that students can use to visualize the Uncertainty Principle from the particle viewpoint. We also describe a way to explain the Uncertainty Principle, which is complementary to the pre-wave function discussion, after the students know about wave functions, using the same elementary statistical analysis used in the pre-wave function presentation. We have designed a second interactive Module that students can use to visualize the Uncertainty Principle when particles are described by complex wave functions.

Journal of Chemical EducationDec 16, 20230 citations

Uncertainty and quantum mechanics

Tim Freegarde

Cambridge University Press eBooksNov 8, 20120 citations

Stronger Schrödinger-like uncertainty relations

Qiu-Cheng Song, Cong‐Feng Qiao

arXiv:1504.01137Physics Letters AJul 12, 201649 citations

Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms

Aline Bonami, Bruno Demange, Philippe Jaming

We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions f on \mathbb{R}^d which may be written as P(x)\exp (-\langle Ax, x\rangle) , with A a real symmetric definite positive matrix, are characterized by integrability conditions on the product f(x) \widehat{f}(y) . We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambiguity function or the Wigner transform). We complete the paper with a sharp version of Heisenberg's inequality for this transform.We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions f on \mathbb{R}^d which may be written as P(x)\exp (-\langle Ax, x\rangle) , with A a real symmetric definite positive matrix, are characterized by integrability conditions on the product f(x) \widehat{f}(y) . We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambiguity function or the Wigner transform). We complete the paper with a sharp version of Heisenberg's inequality for this transform.

Revista Matemática IberoamericanaApr 30, 2003169 citations

We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions

f

\R^d

which may be written as

P(x)\exp (Ax,x)

, with

A

a real symmetric definite positive matrix, are characterized by integrability conditions on the product

f(x)\hat{f}(y)

. We also give the best constant in uncertainty principles of Gelf'and Shilov type. We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambiguity function or the Wigner transform). We complete the paper with a sharp version of Heisenberg's inequality for this transform.