Examples of quantum integrals

TL;DR

This paper introduces various types of quantum integrals, including expectation calculations for quantum coin flips and measures, along with quantum analogs of the fundamental theorem of calculus.

Contribution

It presents new methods for quantum integration, including centering, variable change formulas, and three distinct quantum integral types with theoretical properties.

Findings

Quantum expectation of coin flips derived

Quantum integrals for destructive pairs analyzed

Quantum fundamental theorem of calculus established

Abstract

We first consider a method of centering and a change of variable formula for a quantum integral. We then present three types of quantum integrals. The first considers the expectation of the number of heads in $n$ flips of a "quantum coin". The next computes quantum integrals for destructive pairs examples. The last computes quantum integrals for a (Lebesgue)^2 quantum measure. For this last type we prove some quantum counterparts of the fundamental theorem of calculus.