QuantumPhysics.dvi

A more complicated example is provided by spaces of square integrable complex functions

on some intervalS(or all of) the real lineR, defined by

L^2 (S)≡{f:S→C; (f,f)<∞} (f,g)≡

∫

S

dxf(x)∗g(x) (3.20)

These spaces will be ubiquitous in quantum mechanics. The Fourier transform gives a

convenient way of describing L^2 (S). For example, on an interval S = [−πℓ,+πℓ] with

periodic boundary conditions (or equivalently a circe of radiusℓ), we have

f(x) =

∑

m∈Z

fm

eimx/ℓ

√

2 πℓ

(f,g) =

∑

m∈Z

fm∗gm (3.21)

wheregmare the Fourier components ofg. This shows thatL^2 andL^2 (S) are isomorphic.

The basis forL^2 (S) used here corresponds to

|m〉∼

eimx/ℓ

√

2 πℓ

(3.22)

and is orthonormal in view of the relation,

∫+πℓ

−πℓ

dx

(

eimx/ℓ

)∗

einx/ℓ= 2πℓδm,n (3.23)

which is a standard relation of Fourier analysis.

The Hilbert spaceL^2 (R) must be handled with additional care. Fourier analysis (as well

as the limitℓ→ ∞of the above example) suggests a basis given by exponentials exp(ikx)

withk∈R. The problem is that these basis vectors are not square integrableand form a

basis which isnot countable. Nonetheless, countable bases forL^2 (R) do exist. A familiar

example is provided by the orthonormal basis of wave functions forthe harmonic oscillator,

Hn(x)e−x

(^2) / 2

whereHn(x) are the Hermite polynomials. Equivalently, one can work with a

non-orthonormal but simpler basis given byxne−x

(^2) / 2

for alln= 0, 1 , 2 ,···,∞.

In the physical world, space is not infinite. Using the entire real lineRis an idealiza-

tion, which often simplifies problems, such as in the thermodynamic limit. In practice, one

may resort to approximatingRby a large interval [−πℓ,+πℓ], subject to certain boundary

conditions, and then take the limit.

Alternatively, we shall later on learn how to deal with such non-countable bases.