ˆ

h(k) = (12.16)

’∞

ˆ

where h is an operator acting on f (x) with the property that

ˆ

h exp(ikx) = h(k) exp(ikx). (12.17)

Examples are

‚

ˆ

h = ’i ,

h(k) = k, (12.18)

‚x

318 Fourier transforms

2

ˆ =’ ‚ ,

k2 ,

h(k) = h (12.19)

‚x2

n

ˆ = i’n ‚ .

h(k) = k n , h (12.20)

‚xn

Proof We prove (12.16). Insert the Fourier transforms into (12.16), using

(12.3) and (12.17):

∞ ∞ ∞ ∞

1

—ˆ

dk F — (k )e’ik x F (k)heikx

ˆ

f hf dx = dx dk

2π

’∞ ’∞ ’∞ ’∞

∞ ∞ ∞

1 —

dxei(k’k )x

= dk dk F (k )F (k)h(k)

2π ’∞ ’∞ ’∞

∞

dkF — (k)F (k)h(k) = h(k) .

=

’∞

ˆ

In general, an operator A may be associated with a function A of x and/or

k, and the expectation of A de¬ned as

∞

f — (x)Af (x) dx.

def ˆ

A= (12.21)

’∞

ˆ

An operator A is hermitian if for any two quadratically integrable functions

f (x) and g(x)

—

∞ ∞ ∞

— —

g A— f — dx.

ˆ ˆ ˆ

f Ag dx = g Af dx = (12.22)

’∞ ’∞ ’∞

In particular this means that the expectation of a hermitian operator is

real, as is immediately seen if we apply the hermitian condition to g = f .

Operators that represent physical observables, meaning that expectations

must be real physical quantities, are therefore required to be hermitian. It

also follows that the eigenvalues of hermitian operators are real because

the eigenvalue is the expectation of the operator over the corresponding

eigenfunction (the reader should check this).

12.4 Uncertainty relations

If we de¬ne the variances in x- and k-space as

def

σx = (x ’ x )2 ,

2

(12.23)

def

σk = (k ’ k )2 ,

2

(12.24)

12.4 Uncertainty relations 319

we can prove that for any normalized function f (x) the product of the square

root of these two variances (their standard deviations) is not less than one

half:

σx σ k ≥ 1 . (12.25)

2

This is the basis of the Heisenberg uncertainty relations for conjugate vari-

ables.

Proof The proof1 starts with the Schwarz inequality for the scalar products

of any two vectors u and v:

(u, u)(v, v) ≥ (u, v)(v, u) = |(u, v)|2 (12.26)

which is valid with the de¬nition of a scalar product of functions.2

∞

u— v dx.

def

(u, v) = (12.27)

’∞

This the reader can prove by observing that (u ’ cv, u ’ cv) ≥ 0 for any

choice of the complex constant c, and then inserting c = (v, u)/(v, v). We

make the following choices for u and v:

u = (x ’ x )f (x) exp(ik0 x) (12.28)

d

v= [f (x) exp(ik0 x)], (12.29)

dx

where k0 is an arbitrary constant, to be determined later. The two terms

on the left-hand side of (12.26) can be worked out as follows:

∞

(x ’ x )2 f — f dx = σx

2

(u, u) = (12.30)

’∞

and

∞

d— d

(v, v) = f exp(’ik0 x) f exp(ik0 x) dx

dx dx

’∞

∞

d2

—

=’ f exp(’ik0 x) f exp(ik0 x) dx

dx2

’∞

∞ ∞

—

f — f dx + k0

=’ f f dx ’ 2ik0 2

’∞ ’∞

’ 2k0 k +

2 2

=k k0 , (12.31)

1 See, e.g., Kyrala (1967). Gasiorowicz (2003) derives a more general form of the uncertainty

relations, relating the product of the standard deviations of two observables A and B to their

commutator: σA σb ≥ 1 | i[A, B] |.

ˆˆ