Introduction to Quantum Mechanics 2

3Eigenvalues and Eigenvectors of an Observable

General objective

The general objective is to know the properties of the eigenvalues and eigenvectors of an observable.

Specific objectives

On completing this chapter, the reader should be able to:

• – define a representation;
• – represent a ket and a bra;
• – represent an operator;
• – represent the adjoint of an operator;
• – recognize a Hermitian matrix;
• – determine the properties of the eigenvalues of an observable;
• – determine the properties of the eigenvectors of an observable;
• – distinguish between a simple eigenvalue and a degenerate eigenvalue;
• – use the characteristic equation;
• – know the properties of the eigenvectors and eigenvalues of a Hermitian operator;
• – establish the evolution equation of the mean value of an observable;
• – define a complete set of commuting observables (CSCO);
• – know the properties of conservative systems;
• – integrate Schrödinger’s equation applied to conservative systems;
• – establish Ehrenfest’s theorem.

Specific objectives

• – Matrix calculus.
• – Observable.
• – Hamiltonian.
• – Properties of the space of states.

3.1. Representation

3.1.1. Definition

In quantum mechanics, the passage from vector calculus introduced in the space of states to matrix calculus in the same space is based on the choice of a representation. For this purpose, a discrete or continuous orthonormal basis is chosen, in which:

• – kets and bras are represented by numbers (their components on the basis vectors);
• – operators are represented ...