A-Dimensional Complex Vector Spaces

We need to generalize the ideas of three-dimensional vector algebra to an iV-dimensional space in which the vectors can be complex. We will use the powerful notation introduced by Dirac, which expresses our results in an exceedingly concise and simple manner. In analogy to the basis ej in three dimensions, we consider N basis vectors denoted by the symbol i , i = 1, 2. N, which are called ket vectors or simply kets. We assume this basis is complete so that any ket vector a can be written as 

After we specify a basis, we can completely describe our vector a by giving its N components a,-, i = 1, 2. iV with respect to the basis i . Just as before, we arrange these numbers in a column matrix a as 

Now we introduce an abstract bra vector a whose matrix representation is a The scalar product between a bra a and a ket b is defined as 

For this to be identical to our definition (1.44) of the scalar product we must have that 

In analogy to Eq. (1.11), we define an operator ( ) as an entity which when acting on a ket a converts it into a ket fe . 

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