The inner product or dot product is an essential element in QM. It allows one to project out the aspects of a physical state that are of interest. If one wants to know where the particle is then the QM question becomesè What are the amplitudes for the eigenstates of interest. This discussion will focus on the inner product and the rules for manipulating inner products.

INNER PRODUCT

is a general state in ket-space. All possible physical states are vectors in this space.

The space is spanned by an infinite set of basis vectors . Where j can be a discrete index {j} or a continuous index {such as x}

is a vector in bra-space. This the dual space to ket-space. There is a one to one correspondence between the two spaces. Because of the one-to-one relationship the same principles apply to the bra-space. There are corresponding basis vectors the dual to the basis vectors in ket space.

For QM we require a specific property for inner products:

INNER PRODUCT RULE

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

This is required because we would like to interpret projects on states as probabilities:

for fixed j; This is the probability of measuring the value j for a particular observable. It is guaranteed to be real by nature of the inner product rule.

Normalizing the state to one is also valid when the inner product rule holds.

OPERATORS

Operators are transformation on vectors in the space to new vectors in the space.

A new vector results when the operator acts on . The action of the operator must be defined for all vectors in the space. {A good example of operators acting on vector spaces is the rotation of a vectors in 3-d [], a new vector may be generated through rotation. Rotation is an example of an operator that acts on vectors in our coordinate space [position, velocity, force].}

In general operators can be well behaved in that they have an inverse and are linear. They may possess the property of commutation i.e. the order of operation is irrelevant or non-commutation i.e. a change in the order produces a different vector. Of course one can define operations that are not well behaved. In QM one uses operators that have the requisite properties to ensure the correct nature of the theory. [Hermitian, unitary …]

In order to incorporate operators into the inner product formalism one must define the action of the operator wrt the inner product.

alternate definition DEF 1	book definition DEF 2
Given an operator that act on the ket-space on requires that there exist and operator that acts on the bra space and that this operator produce the dual vector in bra-space to the one produced in ket-space. Here the two operators are define wrt different spaces. [Curly brackets are used to specifically identify the operator and the vector on which it acts]	Given an operator that act on the ket-space one requires that there exist and operator that acts on ket space such that the dual of this new vector doesn’t change the inner product. Here the two operators are defined wrt ket space. [ means: find and use in the inner product. Notation is important and an operator inside a bra implies using that operator in the ket space and they transforming the result to the bra space
These are two different ways to define the adjoint but they become equivalent if one also requires that operators can act either to the right or to the left.
[Dirac pgs 24,25]. If is some ket vector that can be part of inner product as above then can be interpreted as a new bra vector under the action of . The inner product must not change and this defines how the operator must act on both spaces.

To understand the direction of the operation we return torotaions. 2-d rotations suffice to illustrate the point.

rotation through the angle .

Consider the standard dot product of two vectors.

Now imagine that we want to rotate a vector C and then take the dot product. I can write this in a similar notation as

rotate C through an angle θ and then take the dot product with A.

I can preserve the dot product but allow the rotation to occur on either C or A. So I define what I mean to rotate A by demanding that the dot product doesn’t change. In this case if I rotate A by –θ (minus theta) the dot product remains the same.

NOTE: A and C are not effected in the same manner.

Notice that if I use matrix multiplication

You can multiply to the left or right. The result of the rotation matrix on x,y is a rotation through θ while a,b times the matrix results in a rotation in the opposite direction - θ.

We see that the notation using row and column vectors and 2x2 matrices allow for matrix multiplication either to the left or to the right. You can see that the effect of a matrix on a column is different than on a row and that the difference when the matrix represents a rotation corresponds to a change in sign for θ.

The matrix result above is obtained using the Dirac notation by substituting a complete set of basis vectors.

another example multiplication by a constant

Multipliation to the

to the rightè

to the left è

OPERATORS produce different results when acting left or right.

IF

THEN in general.

a completely different vector to preserve the inner product.

SUMMARY

1		general equation for operator
2		a general equation for adjoint operator
3		DEF 1(eqv equ in bra as ket) DEF 2
4		DEF 1 (adj prod same vector in bra as the op in ket) DEF 2

The key point is that an operator and its adjoint are defined so that one can predict the action of either in bra or ket space. This leads to the 4 equations above.

interesting relationships