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 |ψ1〉 and |ψ2〉:


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

The simplest ...

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