Quantum Mechanics for Mathematicians

For a choice of orthonormal basis{ej}, i.e., satisfying

〈ej,ek〉=δjk

a useful choice of label is the indexj, so

|j〉=ej

Because of orthonormality, coefficients of vectors|α〉with respect to the basis
{ej}are
〈j|α〉

and the expansion of a vector in terms of the basis is written

|α〉=

∑n

j=1

|j〉〈j|α〉 (4.4)

Similarly, for elements〈α|∈V∗,

〈α|=

∑n

j=1

〈α|j〉〈j|

The column vector expression for|α〉is thus











〈 1 |α〉

〈 2 |α〉

..

.

〈n|α〉











and the row vector form of〈α|is
(
〈α| 1 〉 〈α| 2 〉 ... 〈α|n〉

)

=

(

〈 1 |α〉 〈 2 |α〉 ... 〈n|α〉

)

The inner product is the usual matrix product

〈α|β〉=

(

〈α| 1 〉 〈α| 2 〉 ... 〈α|n〉

)











〈 1 |β〉

〈 2 |β〉

..

.

〈n|β〉











IfLis a linear operatorL:V →V, then with respect to the basis{ej}it
becomes a matrix with matrix elements

Lkj=〈k|L|j〉

The expansion 4.4 of a vector|α〉in terms of the basis can be interpreted as
multiplication by the identity operator

1 =

∑n

j=1

|j〉〈j|