# Chapter 6: Classical and Quantum-mechanical Probability

Much of introductory quantum mechanics involves finding and understanding the solutions to Schrödinger's wave equation and 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}

# Table of Contents

## 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.

^{1}While 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).

^{2}This 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).