Adjoint & Inner product

The meaning of the adjoint definitions are completed by the inner product rule:

The constraints of the inner product help define the characteristics of the bra space. One can introduce the adjoint and then see what characteristics it must have to satisfy the above relationship.

Definition 1: In terms of an operator equation that relates two kets, there must be a relationship that relates the associated bra vectors. The operator that relates these corresponding bra vectors is the adjoint of the original. Given as shown below, there must be an that relates the same vectors in the bra space.

This definition however doesn’t guarantee that the above property of the inner product is valid. So using the definition the requirement on the adjoint is

Expressing this wrt a basis:

Requirements on the inner product and the definition of the adjoint require the dagger or adjoint to be the complex conjugate transpose.

Definition 2 (book, book notation): 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.

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.

Definition 2 (book, my notation): The book’s definition of the adjoint but using standard notation.

Either definition to define the adjoint should produce the identical result and therfore must be equivalent.

Equivalence start with def. 1.

First

Let the operator be replaced by another operator

The inner product rule

If there is an operator def

then

therefore

(Since the adjoint of the original is needed for the bra space and since the original is then .

Chapter 4

The Dirac notation that uses BRA and KET vectors is a very transparent way to see many of the quantum state features. If you read, for example, the The New Physics chapter called Conceptual Foundations of Quantum Mechanics by Abner Shimony, you will see how important vector spaces are to the formulation of Quantum. Below is a brief summary of some of these ideas.

general quantum state. It is shown as a KET which indicates that it is a general vector. Quantum states are vectors in Hilbert space. Given any set of specific states we can generate general quantum states by adding states together with different coefficients as shown in what follows (standard vector addition).

specific quantum state labeled by some observables. It will be important to have a notation that allows us to specify some special vectors. The arguments within the KET can signify eigenvalues which then define the vector as a state which is a solution to the eigenequations for an observable or a set of observables.

If one can measure the energy and the angular momentum for a quantum system then the observable energy and the z-component of angular momentum might suffice to define the state. Therefore

and the eigenvalues of energy and angular momentum could label certain specific quantum states.

A general state might then be

a linear combination of some of these specific states. (Any sum of vectors must indeed be a vector).

If the eigenvalues are continuous then the sum becomes an integral

.

Introduce a “DOT” product or an inner product. This is a multiplication rule that turns vectors into numbers. Because we would like to have the “length” of our vectors be real and because imaginary coefficients are allowed the inner product for this space is a bit more complicated. For each of the vectors discussed above, KET, we introduce a BRA. There is one BRA associated with any KET and vice versa. More general formulations of vector products often discuss DUAL spaces. One imagines two versions of the same vector. The versions differ in just the correct way so that the vector nature disappears in the inner product.

general KET so there must be a corresponding BRA

BRA that exists as a companion to the above KET.

Multiplying the two together

The inner product will provide us with a definition for length. One can then introduce vectors that are required to have unit length. Here are a few that we will use:

where the Kronecker delta or the Dirac delta is used to require that the states are normalized to 1 and also reflect the fact the states are orthogonal. One must prove that the states are indeed orthogonal (and the above states are orthogonal) but the normalization is a criteria used to define what we mean by the above states. They all are required to have a length of one.

Since a vector can be written as a sum with coefficients there must be a rule to what happens to the BRA vector associated with a sum of KET vectors.

So each coefficient changes to the complex conjugate when one produces the BRA from the KET.

Thus

This property ensures that the inner product of a vector with itself is real.

Examine the equation

This is the general operator equation. Now we want to find <Ω|. The BRA associated with the KET |Ω>. The question is what happens to the operator.

The book uses this notation for the operator that transforms the vector Ψ in the BRA space to the vector Ω in the BRA space. The book implies that you use the operator directly in the ket space and then find the associated bra.

A person might be tempted to write the expression in this form although this is not an accepted notation. This is perhaps the more natural way to start but as we will find it becomes confusing.

This equation is also an accepted notation. Here the operator must act to the left.

The original operator must be exchanged for the adjoint operator. The adjoint operator has the correct character to work in the BRA space and it transforms the BRA to the BRA vector that corresponds to the original resulting KET vector. O “dagger” , the adjoint of O, operates on BRA vectors to give new BRA vectors. An action of O “dagger” in the BRA space is similar to the operation of O in the KET space.

Since any vector in either space can be represented as a sum of basis vectors, operators can be represented as matrices. The matrix determines how much basis vector |i> is transformed into a |j> vector. This can be encapsulated in a general expression for any operator.

In this format one can operate in either direction as follows:

in this notation any < | > is complex number. So the top sum is

This is just a set of coefficients multiplying a set of KET vectors. In introducing this notation for the operator we have actually introduced a second type of multiplication. This type of multiplication is call the outer product. It results when BRA and KET vectors are combined as | >< |. One might remember that matrix multiplication as a similar quality.

Column matrices are like KET vectors. Row matrices are like BRA vectors. 2x2 matrices are like operators.

There is a special group of operators that satisfy the relation

These are Hermetian operators and all observables have corresponding Hermetian operators. This property ensures that the eigenvalues and therefore the possible values for the observables are real.