Dirac Notation We Will Introduce the Notion of a (Finite-Dimensional

Dirac Notation

We will introduce the notion of a (ﬁnite-dimensional) linear vector space, V, informally. For a rigorous discussion you can look at P. R. Halmos, “Finite-dimensional Vector Spaces”, for example. A linear vector space, V, is a set of vectors with an abstract vector denoted by |vi (and read ‘ket vee’). This notation introduced by Paul Adrien Maurice Dirac(1902-1984) is elegant and extremely useful and it is imperative that you master it.1 The space is endowed with the operation of addition(+) for each pair of vectors and multiplication by ‘scalars’ (which belong to the ﬁeld of complex numbers, C, in our case):

Vector addition is commutative: |ui + |vi = |vi + |ui.

Vector addition is associative: |ui + ( |vi + |wi ) = ( |ui + |vi ) + |wi .

Scalar multiplication is distributive in

(a) the scalars : ( c1 + c2 ) |vi = c1|vi + c2|vi c1 c2 ∈ C (b) the vectors : c ( |ui + |vi ) = c |ui + c |vi c ∈ C

Scalar multiplication is associative: c1( c2|vi ) = (c1c2) |vi.

There exists a null vector, denoted by |0i such that |vi + |0i = |vi. One can show uniqueness of the null vector and that |0i = 0|vi for any vector |vi.

For every ket |vi there exists a vector denoted by | − vi such that |vi + | − vi = |0i. One can show that the inverse is unique; we also have |vi + (−1)|vi = 0|vi = |0i. So we have denoted (−1) |vi by | − vi.

Roughly speaking, we can summarize the axioms by saying that all the normal operations with which you are familiar while studying ordinary vectors and scalars are legal.

One can endow the linear vector space with an inner product (the generalization of the dot product) to make it an inner product space. The inner product is a complex number denoted by hu|vi. This is represented by the bracket symbol and hence, the term bra for hu| and ket for |vi. The inner product has the following properties:

(P1) hu|vi = hv|ui∗. Since we have deﬁned the vector space over complex numbers this is different from the more familiar case of a vector space over real numbers in which case the dot product is real and the order is immaterial.

1The seminal book by Dirac, The Principles of Quantum Mechanics published in 1930 introduced the formalism of quantum mechanics in general and this notation in particular. This is not an easy book to read!

(P3) Clearly hv|vi is real and is deﬁned to be non-negative: hv|vi ≥ 0 and equals zero if and only if |vi = |0i.

Our definition of the inner product has tacitly used the definition of a bra, hu|. The bra vectors are in one-to-one correspondence with ket vectors. With every |vi we associate a unique hv|. So the bras share the same linear vector space structure as the kets. The one key property which makes the definition of the scalar product well-defined as observed above is that the bra associated with the ket |wi ∗ ∗ |wi = c1|ui + c2|vi is hw| = c1hu| + c2hv| .

A set of l vectors |v1i, |v2i,..., |vli is said to be linearly independent if

Xl cj |vji = 0 implies cj = 0, for every j . (1) j=1

Exercise: In ordinary three-dimensional space write down three vectors which are mutually orthogonal. Are they linearly independent? Give a set of three linearly independent vectors which are not orthogonal.

A set of vectors is said to constitute a basis if it is linearly independent and spans the space, i.e., every vector |vi in the space can be expressed as a linear combination of the elements of the basis set (with complex coeﬃcients.) A linear space is said to be n-dimensional if and only if it has a basis of n vectors. Clearly then in an n-dimensional vector space one can express any vector |vi as a linear combination of a set of n linearly independent vectors; if this were not possible we would have n + 1 linearly independent vectors contradicting our assumption. The basis vectors can be made mutually orthogonal and normalized to unity. We will use the notation |eji for j = 1, 2, ··· n for one such set of n orthonormal vectors:

hei|eji = δij (2) where δij is the Kronecker delta function which is one if the two indices are the same and zero otherwise.

2 Every vector(ket) |vi can be expanded in terms of the orthonormal basis as Xn |vi = vj |eji . (3) j=1

Take the inner product of the above equation with hei|; one obtains, upon using the orthonormality 2 of the basis vectors the useful result vi = hei|vi . Similarly, the corresponding bra vector can be expressed in terms of the dual basis: Xn Xn ∗ ∗ hv| = hej| vj = vj hej| . j=1 j=1

∗ Note that sometimes the complex number vj is placed before the bra as in the last expression. It ∗ represents the same linear combination of the basis bra vectors multiplied by the complex numbers vj .

One concrete realization of this formalism is obtained by thinking of the abstract vectors in a specific basis and associating a column vector with a ket with the elements being complex numbers. Clearly we can add such column vectors (assuming all of them have the same number of components), multiply them by complex numbers such that they obey the axioms. Now one can think of a bra associated with each ket as simply the complex-conjugated transpose (sometimes referred to as the adjoint) of the column vector: it is a row vector with each element being the complex conjugate of the corresponding element of the row. Explicitly, with |vi in an n-dimensional space we associate a column vector (in a particular basis)   v1    v2     v  |vi →  3  (4)  .   .  vn where vj are complex numbers. Note that if one chooses a different basis the same abstract vector, |vi is represented by a column vector with different entries. The corresponding bra is given by

∗ ∗ ∗ ∗ hv| → (v1, v2, v3, ··· vn) . (5) This concrete identiﬁcation is helpful in keeping track of the concepts of bras and kets. In more mathematical parlance, the space of bras and space of kets are dual to each other. The dot product is given by   v1    v2    Xn ∗ ∗ ∗ ∗  v  ∗ hu|vi = (u1, u2, u3, ··· un)  3  = uj vj (6)  .   .  j=1 vn It is obvious, for example, that hu|vi = hv|ui∗. The various properties of the inner product are also clear in this representation. Note that |vi hu| is a very diﬀerent beast from hu|vi.

2Note apart from the unfamiliar notation this is nothing more than what you know from vector algebra:

Given ~a = ax ˆi + ay ˆj + az kˆ then ay = ˆj · ~a generalized to n-dimensions and complex vectors.

3 It is clear from the representation that we have employed that it is an n × n matrix, i.e., it is an operator! Please write out the operator explicitly. Once more, the advantage of the formal notation is that many general results can be proved compactly and without explicitly writing vectors and operators in a basis.

The basis vectors (of the chosen basis) are then given by     1 0      0   1       0   0      |e1i =  0  |e2i =  0  etc. (7)          0   0  . . . .

The jth element of |eji is 1 while all the other elements are 0. It is easy to check that these n unit vectors constitute an orthonormal basis. The representation of any complex vector in terms of the basis vectors is obvious.

Linear Operators in Dirac notation

The linear operators themselves form a linear space in that the sum of two operators Aˆ and Bˆ is deﬁned by ( Aˆ + Bˆ ) |ui = Aˆ |ui + Bˆ |ui and multiplication by a complex scalar is deﬁned by the action of c Aˆ on any ket as follows:

(cAˆ) |vi = c( Aˆ|vi ) . (8) so that we have

Cˆ = c1Aˆ + c2Bˆ ⇒ Cˆ|vi = c1 Aˆ|vi + c2 Bˆ|vi for all |vi . (9)