The concepts and techniques discussed in the previous chapter are intended to prepare you to cross the bridge between the mathematics of vectors and functions and the expected results of measurements of quantum observables such as position, momentum, and energy. In quantum mechanics, every physical observable is associated with a linear “operator” that can be used to determine possible measurement outcomes and their probabilities for a given quantum state.
This chapter begins with an introduction to operators, eigenvectors, and eigenfunctions in Section 2.1, followed by an explanation of the use of Dirac notation with operators in Section 2.2. Hermitian operators and their importance are discussed in Section 2.3, and projection operators are introduced in Section 2.4. The calculation of expectation values is the subject of Section 2.5, and as in every chapter, you’ll find a series of problems to test your understanding in the final section.
Operators, Eigenvectors, and Eigenfunctions
If you’ve heard the phrase “quantum operator” and you’re wondering “What exactly is an operator?,” you’ll be happy to learn that an operator is simply an instruction to perform a certain process on a number, vector, or function. You’ve undoubtedly seen operators before, although you may not have called them that. But you know that the symbol “ √ ” is an instruction to take the square root of whatever appears under the roof of the symbol, and “d( )/dx” tells you to take the first derivative with respect to x of whatever appears inside the parentheses.
The operators you’ll encounter in quantum mechanics are called “linear” because applying them to a sum of vectors or functions gives the same result as applying them to the individual vectors or functions and then summing the results.