# 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 ...

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