Particles and Waves and the Uncertainty Principle

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.

Thomas A. Brody

Shukri Klinaku

Tarek Yehia

This paper focuses on the complex relationship between Heisenberg's Uncertainty Principle and the nodal structure of wave functions in a variety of quantum systems including the quantum harmonic oscillator, the particle in a 1D box , and the particle on a ring. We argue that the uncertainty in conjugate variables, like location and momentum, is generally a function of the number of nodes. As our investigation reveals, the nature of this influence depends on the system. This paper demonstrates that Heisenberg's Uncertainty Principle is influenced by the nodal structure of wave functions and how the nature of this dependence is system-dependent.

Dewi Wulandari

Most physics books do not reveal clearly how the Heisenberg uncertainty principle was derived. This uncertainty comes from the consequence of the wave-particle duality of matter giving statement that position and momentum cannot be measured in the same time. This article tries to reveal mathematics background behind the expression the Heisenberg uncertainty using supported mathematics background such as Fourier transform, Fourier transform integral, the probability of Gaussian distribution and it ends up with the expression of wave function which describe the localized particle giving relation Heisenberg uncertainty principle.

Antony J. Bourdillon