The content of this chapter may serve as yet another supplemental topic to meet the needs and interests beyond those of a usual course curriculum. Here we shall present an over-simplified, but hopefully totally transparent, description of some of the fundamental ideas and concepts of quantum mechanics, using a pure linear algebra formalism.

Vectors in ℂnand Dirac bracket

Consider the vector space ℂn, consisting of column vectors, and use {e1, …, en} to denote the standard basis of ℂn. For u, υ ∈ ℂn with

recall that the Hermitian scalar product is given by

so that {e1, …, en} is a unitary basis, satisfying (ei, ej) = δij, i, j = 1, …, n.

In quantum mechanics, it is customary to rewrite the scalar product (9.1.2) in a bracket form, 〈u|v〉. Then it was Dirac who suggested to view 〈u|υ〉 as the scalar pairing of a ‘bra’ vector 〈u| and a ‘ket’ vector |υ〉, representing the row vector u† and the column vector υ. Thus we may use |e1〉, …, |en〉 to denote the standard basis vectors of ℂn and represent the vector u in ℂn as

Therefore the bra-counterpart of |u〉 is simply given as

As a consequence, the orthonormal condition regarding the basis {e1, …, en} becomes

and the Hermitian scalar product of the vectors |u〉 and |υ〉 assumes the form

For the vector |u〉 given in (9.1.3), we find that

ai = 〈ei|u〉, i = 1, …, n.

Now rewriting |u〉 as

and inserting (9.1.7) into (9.1.8), we obtain

which suggests that the strange-looking ‘quantity’, should naturally be identified as the identity mapping or matrix,

which readily follows from the associativity property of matrix multiplication. Similarly, we have

Thus (9.1.10) can be applied to both bra and ket vectors symmetrically and what it expresses is simply the fact that |e1〉, …,|en form an orthonormal basis of ℂn.