**3**

**MATRICES AND OPERATORS**

An *operator* is a mathematical rule that can be applied to a function to transform it into another function. Operators are often indicated by placing a caret above the symbol used to denote the operator. For example, we can define the derivative operator as

We can apply the derivative operator to a function, say, *f* (*x*) = *x* cos *x*:

This idea can be extended to vector spaces. In this case an operator *A* is a mathematical rule that transforms a ket |*ψ*〉 into another ket that we will call |*ϕ*〉*:*

Operators can also act on bras. The result is another bra:

In many cases of interest, an operator transforms vectors into other vectors that belong to the *same* space.

We say that an operator is *linear* if the following relationship holds given complex numbers *α* and *β* and state vectors |*ψ*_{1}〉 and |*ψ*_{2}〉:

More generally, a linear operator acts on a state vector as follows:

The simplest ...

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