2 This de¬nition applies to vectors in Hilbert space. See Chapter 14.

320 Fourier transforms

where the second line follows from the ¬rst one by partial integration. Choos-

ing for k0 the value for which the last form is a minimum: k0 = k , we

obtain

2

(v, v) = σk . (12.32)

Thus, (12.26) becomes

∞

u— v dx|2

≥|

22

σx σk

’∞

∞ 2

—

≥ Re u v dx

’∞

∞ 2

1 — —

(x ’ x )(f f + f f ) dx

=

4 ’∞

∞ 2

1 d

(x ’ x ) (f — f ) dx

=

4 dx

’∞

∞ 2

1 1

—

= f f dx =. (12.33)

4 4

’∞

Hence σx σk ≥ 1 .

2

12.5 Examples of functions and transforms

In the following we choose three examples of real one-dimensional symmet-

ric functions f (x) that represent a con¬nement in real space with di¬erent

shape functions. All functions have expectations zero and are quadratically

normalized, meaning that

∞

f 2 (x) dx = 1. (12.34)

’∞

This implies that their Fourier transforms are also normalized and that the

expectation of k is also zero. We shall look at the width of the functions in

real and reciprocal space.

12.5.1 Square pulse

The square pulse and its Fourier transform are given in Fig. 12.1. The

equations are:

1 a

f (x) = √ , |x| < (12.35)

2

a

a

|x| ≥

= 0,

2

12.5 Examples of functions and transforms 321

2

1

f(x) F(k)

1

0

’a/2 0

a/2

’2 ’1 0 1 2 ’4 ’2 0 2 4

Figure 12.1 Square pulse f (x) with width a (x in units a/2, f in units a’1/2 ) and

its transform F (k) (k in units 2π/a, F in units (a/2π)1/2 ).

1 1

f(x) F(k)

0 0

’a a

’2 ’1 0 1 2 -3 -2 -1 0 1 2 3

Figure 12.2 Triangular pulse f (x) with width 2a (x in units a, f in units (2a/3)’1/2 )

and its transform F (k) (k in units 2π/a, F in units (3a/4π)1/2 ).

x =0 (12.36)

12

σx = x2 =

2

a (12.37)

12

a sin φ

φ = 1 ka

F (k) = , (12.38)

2

2π φ

k =0 (12.39)

σk = k 2 = ∞

2

(12.40)

σx σ k = ∞ (12.41)

12.5.2 Triangular pulse

The triangular pulse is a convolution of the square pulse with itself. See Fig.

12.2. The equations are:

1 3

(a ’ |x|), |x| < a

f (x) = (12.42)

a 2a

|x| ≥ a

= 0,

x =0 (12.43)

322 Fourier transforms

1 1

f(x) F(k)

0 0

’4 ’2 0 2 4 ’4 ’2 0 2 4

Figure 12.3 Gaussian pulse f (x) with variance σ (x in units σ,√f in units

√

(σ 2π)’1/2 and its transform F (k) (k in units 1/2σ, F in units (2σ/ 2π)’1/2 ).

12

σx = x2 =

2

a (12.44)