inner product: produces a complex number from vectors in the Hilbert space
	ket vector
	bra vector
bra-ket	There are two distinct spaces. There is a on-to-one correspondence so that for every ket there is an associated bra and vice versa.
	The spaces have the necessary relationship to fulfill this inner product rule.
	Columns [kets] and rows [bras] that have coefficients that are complex conjugates fulfill this relationship. [Matrix arithmetic]
	You can find an orthonormal basis to span the spaces.
	Operators have adjoints or associated dagger operators.
	The relationship maintains the inner product rule. Below we show two ways to characterize the relationship between and .
	The operators as matrices are related by the complex conjugate transpose.
	With a complete set of vectors ei or x one can construct an identity operator that can be inserted into vector equations without changing the meaning of the equation.
	Any quantum state can be represented wrt a given basis as a sum over all the basis vectors with each vector assigned an amplitude that can be complex. If the basis in continuous the sum is represented by an integral and the amplitude becomes a function.
	Using the inner product you can project out an amplitude.
	Using the above formalism you can derive the Fourier transform relationship between amplitudes wrt the x and p basis
	General operator equation. An operator takes a vector into another vector.
	Using a complete set of states one can write operators as functions that express the amount of each component present in the new vector. is analogous to a matrix.
	When the basis is discrete the same result appears as the more obvious matrix math.
	Hermetian or self adjoint operators. Observables have Hermetian op. These are real symmetric or imaginary antisymmetric “matrices”.
	Unitary operators

Notation

1. Curly brackets show the direction of operation

2. Including the operator inside the bra or ket implies operation on the ket space.

The book defines all operations on the ket vectors. I prefer to define the adjoint by its impact on the bra space.

The inner product rule is:

DEF1

The adjoint is defined wrt the bra-space but this space has an algebra based on the one-to-one relationship to the ket space and the requirements imposed on the space due to the inner product rule. How the vectors are related in the bra space is determined by the inner product.

DEF2

This leads to

One can restrict all operators to the ket space. So the adjoint is the operator that will be used on the ket space if the inner product rule is applied. Since braèket and vice versa the operator in the equation must now operate on a new ket vector and the impact is a different operation ie the adjoint.

The two approaches yield

1
Operators that are related because they produce the same vector but in opposite spaces.
2
Operators that work on the ket vectors when the product rule is used to switch roles

In principle this is a different result. However one can extend either definition by defining the operator to act in either direction. Because the one form of the operator can be expressed as an amount that relates a basis vector to a basis state you can see that the coefficients can be used to build a vector in either space first and the inner product rule will hold.

Observations:

• Operators that are applied left or right have a different effect on the associated bra and ket spaces. Thus rotation, as we understand it as a process, rotates one space clockwise and the other counterclockwise. [This difference is somewhat obscured by the matrix formalism that generates this difference automatically. Sometimes the math of applying a matrix in either direction appears to be the same ie just matrix multiplication by the same matrix on both spaces. However the row-column multiplication is different and a row times a matrix is the same as a column times the transpose of that matrix.]
• If an operator is simply a multiplicative constant then the effect of this operator to the right or left is simply the obvious multiplication. One might say that multiplication by a number must be the same operation in both spaces but when viewed through the on-to-one nature of the spaces, multiplication produces a different result:

So it is trivial to extend the meaning of the operation either left or right in this case, the result is not the same because a different vector is created in each space.

• Hermetian operators are sometimes called real symmetric. Clearly if is the complex conjugate transpose of , then a real symmetric matrix is Hermetian. However a purely imaginary antisymmetric matrix is also Hermetian. Thus the derivative operator as described as a matrix (see 4-momentum.doc)is purely antisymmetric and is Hermetian. Real symmetric or imaginary antisymmetric operators are Hermetian.
• A Hermetian operator therefore behaves identically on the spaces.
• The simplest Hermetian operator is a real multiplicative constant.

Rotate a vector keep the basis fixed and examine the vector after a rotation.

The operator can be expressed as a matrix or a set of numbers that relate the amount of jth component that become ith component.

You can similarly develop a representation of an operator wrt the position basis. The result should be

The operator has a representation with respect to the x-basis.

Using the p-basis

Relating amplitudes between x-basis and the p-basis