QuantumPhysics.dvi

ofN(the set of positive integers). The Gramm-Schmidt procedure goes as follows. We start

with the first vector and normalize it,

| 1 〉=|u 1 〉/||u 1 || (3.8)

To construct the second vector| 2 〉of the orthonormal basis, we project|u 2 〉onto the space

perpendicular to| 1 〉, and normalize the resulting vector,

|v 2 〉=|u 2 〉−| 1 〉〈 1 |u 2 〉 | 2 〉=|v 2 〉/||v 2 || (3.9)

This process may be continued recursively as follows. Given the firstmorthonormal vectors

{| 1 〉,| 2 〉,···,|m〉}, one constructs|m+ 1〉by the same process,

|vm+1〉=|um+1〉−

∑m

i=1

|i〉〈i|um+1〉 |m+ 1〉=|vm+1〉/||vm+1|| (3.10)

When the dimension ofHis finite, this process terminates. Separability of the Hilbert space

Hin the infinite-dimensional case guarantees that even in this case, the process will converge.

3.1.3 Decomposition of an arbitrary vector

Therefore in any finite-dimensional or separable infinite dimensionalHilbert space, we may

decompose any vector|φ〉onto a countable orthonormal basis,

|φ〉=

∑

n

cn|n〉 (3.11)

wherecn are complex numbers given by〈n|φ〉. It is often convenient to leave the range

fornunspecified, so that we can simultaneously deal with the case of finite and infinite

dimensions. Since by definition of the Hermitian inner product, we have〈φ|n〉=〈n|φ〉∗, we

immediately see that the bra〈φ|dual to the ket|φ〉has the following decomposition,

〈φ|=

∑

n

c∗n〈n| (3.12)

while the norm is given by

||φ||^2 =〈φ|φ〉=

∑

n

|cn|^2 (3.13)

In finite dimension, this norm is automatically finite for finite values ofcn, but this is not so

in infinite dimension. Vectors inHmust have finite norm; the requirement of completeness

is necessary to guarantee that sequences with finite norm indeed converge to a vector inH

with finite norm. In particular, completeness means that if||φ||<∞, then the sequence

|φN〉=

∑N

n=1

cn|n〉 (3.14)

converges to|φ〉and may be used to approximate|φ〉to arbitrary precision asN grows.

More precisely, for anyǫ >0, there exists anN such that||φ−φN||< ǫ.