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• Orthogonality. Two vectors are orthogonal if:

ψϕ = ∫ t max ψ()t ϕ()t dt = 0 (7.66)
t min

• Basis vectors. Any function in Hilbert space can be expanded in a
linear combination of the basis vectors {u }, such that:
n
ϕ= cu (7.67)
∑ n n
n
and such that the elements of the basis vectors obey the orthonor-
mality relations:

uu =δ (7.68)
m n m n,

• Decomposition rule. To find the c ’s, we follow the same procedure
n
adopted for finite-dimensional vector spaces; that is, take the inner
product of the expansion in Eq. (7.67) with the bra u . We obtain,
m
using the orthonormality relations [Eq. (7.68)], the following:

u ϕ = ∑ c u u n ∑ c δ = c (7.69)
=
m n m n mn, m
n n
Said differently, c is the projection of the ket ϕ onto the bra u .
m
m
• The norm as a function of the components. The norm of a vector can
be expressed as a function of its components. Using Eqs. (7.67) and
(7.68), we obtain:

δ
ϕ 2 = ϕ ϕ = ∑ ∑ n m n m = ∑ ∑ n mn m, = ∑ c n 2 (7.70)
cc
cc u u
n m n m n
Said differently, the norm square of a vector is equal to the sum of
the magnitude square of the components.

Application 1: The Fourier Series
The theory of Fourier series, as covered in your calculus course, states that a
function that is periodic, with period equal to 1, in some normalized units can
be expanded as a linear combination of the sequence {exp(j2πnt)}, where n is
an integer that goes from minus infinity to plus infinity. The purpose here is
to recast the familiar Fourier series results within the language and notations
of the above formalism.

© 2001 by CRC Press LLC