Chapter 6: Classical and Quantum-mechanical Probability

Introduction

Much of quantum mechanics involves finding and understanding the solutions to the Schrödinger wave equation and by applying Born's probabilistic interpretation to these solutions.¹ We begin by first reviewing some of the basic properties of classical probability distributions before discussing quantum-mechanical probability and expectation values.²

Sections

Section 6.1: Probability Distributions and Statistics.
Section 6.2: Classical Probability Distributions for Moving Particles.
Section 6.3: Exploring Classical Probability Distributions.
Section 6.4: Probability and Wave Functions.
Section 6.5: Exploring Wave Functions and Probability.
Section 6.6: Exploring Wave Functions and Expectation Values.

Problems

Problem 6.1: Rank the animations by the acceleration of the balls given a relative probability distribution.
Problem 6.2: Which of the 9 wave functions could be a localized wave function?
Problem 6.3: Calculate uncertainties of position and momentum for a particle in a box.
Problem 6.4: Calculate uncertainties of position and momentum for a particle in a harmonic oscillator.
Problem 6.5: Calculate expectation values of x for an arbitrary state.

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¹While there are at least eight additional formulations, we will primarily focus on Schrödinger's. For all nine, see D. F. Styer, et al., "Nine Formulations of Quantum Mechanics," Am. J. Phys. 70, 288-297 (2002).²This initial focus is suggested in L. Bao and E. Redish, "Understanding Probabilistic Interpretations of Physical Systems: A Prerequisite to Learning Quantum Physics,'' Am. J. Phys. 70, 210-217 (2002).