The inner product rule:

For any two general states ,

where is a corresponding bra for every ket

This requires a one-to-one invertible map between the two spaces. There is however a difference between the two spaces. This is difference is required by the inner product rule. An example of the difference is

If

then

Because there are two spaces, bra and ket space. The question arises, How does a general operator impact the bra and the ket space.

Since the starting relationship is (We introduced operators by their action on the ket space.)

And because there must be states , . Then the fact that they are related by an operator in ket space means there must be relationship in bra space. However since the spaces are different there may be a corresponding but different map between general vectors , as there is between , .

To start with we will use curly brackets to show upon which vector the operators operates and the dagger operator is defined as the related operator that maps bra’s the same way as its associated operator maps kets.

The inner product rule then becomes

.

Because the operators that we are considering are linear operators and that one can show that the map that details the action of an operator on a basis sufficiently defines the general action of a linear operator, we will be able to express an operator as:

This representation of an operator generates the matrix multiplication rules with which the student should be familiar. If one accepts this notation as correct based on the arguments then the action of an operator in an inner product is either to the right or the left. Thus we have shown (once the above form is accepted)

Either direction is allowed.

Thus

The book uses an alternate formulation.

Given that the operator operates on the ket vector then operates on the ket vector such that the inner product is unchanged.

where : The operator is included inside the bra or the ket. This implies that the operator transforms the ket into a new ket and then if a bra vector is needed the associated bra vector is chosen. So the book defines the action of all operators on the ket space only.

For a ket vector: operates on a ket to get a new ket. This is equivalent to the standard notation.

and when an operator is included inside a bra: is the operator that you use first on the ket space and then find the associated bra vector. So first you apply the given operator on the ket space. The result is then used to find the associated bra vector.

Putting these two vectors into an inner product

The book chooses to define the action of the operator inside the bra and ket with respect to the ket space so that the direct action of the operators on the bra space is undefined and the direction of the operators is explicit (only consider the operation on kets). Standard notation develops the relationship between the operator and its action to the right and to the left and then defines a more transparent relationship between the operator and its adjoint. These are equivalent formulations. Either is acceptable and either will provide the student with the necessary mathematical structure to solve QM problems.