Appel W. Mathematics for Physics and Physicists

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Justiﬁcation of sinusoidal regime analysis 357

Indeed, a ssume that the input signal u and the output signal v are related

by a convolution equation v(t) = T (t) ∗u(t), where T is either a function or

a tempered distribution. If u is of th e form u(t) = e

2πiνt

, then v(t) = T (t) ∗

2πiνt

is inﬁnitely differentiable (because the exponential is) and Theorem 8.24

yields

v(t) =



T (θ), e

2πiν(t−θ)





T (θ), e

−2πiνθ



2πiνt

which is indeed a complex exponential with the same frequency as the in-

put signal. The transfer function is, for the frequency ν, equal to Z(ν) =



T (θ), e

−2πiνθ



or, in other words, it is the value of the Fourier transform of

the impulse response T :

Z =

T .

The quantity Z(ν) may also be seen as the eigenvalue associated to the eigenvector

t 7→ e

2πiνt

of the convolution operator T .

The same reasoning applies for a real-valued exponential. If the input

signal is of t he type u(t) = e

(where p is a real number), the output signal

can be wr itten

v(t) =

T (θ), e

p(t−θ)



T (θ), e

−pθ



still according to Theorem 8.24. The “transfer function” p 7→



T (θ), e

−pθ



is then of course the Laplace transform of the impulse response T (see Chap-

ter 12).

Remark 13.2 (Case of a resonance) This discussi o n must be slightly nuanced: if Z(ν) is inﬁ-

nite, which means the Fourier transform of the distribution T is not regular at ν, one cannot

conclude that the output signal w ill be sinusoidal.

A simple example is given by the equation of the driven harmonic oscillator:

′′

+ ω

v = e

2πiνt

= u(t).

This equation can be writ ten u = [δ

′′

+ ω

δ] ∗ v, the solution of which, in D

′

, is (see

Theorem 8.33 on p age 239)

v = T ∗ u with T (t) =

H (t) sinωt.

The Fourier transform T (computed more easily by taking the Laplace transform evaluated at

2πiν, and applying the formula (12.1), page 345) is then

Z(ν) =

T (ν) = fp

−4π



ν −

2π



−



ν +

2π



For the values ν = ω/2π and ν = −ω/2π, there is divergence.

The transfer function Z(ν) is computed, ab initio, at the beginning of Chapter 15. This

will exp l ain why imaginary Dirac distributions appear, in relation with the c hoice of a solution

of the equation u = [δ

′′

+ ω

δ] ∗ v; for a solution in D

′

−

, the Dirac spikes would have the

opposite sign. A solut ion which does not involve any Dirac distribution exists also, but it does

not belong to any convolution algebra.

358 Physical applications of the Fourier transform

In fact, this reﬂects th e fact that the solutions of the differential equation are not sinu-

soidal; instead, they are o f the form

v(t) =

2iω

t e

i t

at the resonance frequency. More details are given in Chapter 15.

13.2

Fourier transform of ve c tor ﬁelds:

longitudinal and transverse ﬁelds

The Fourier transform can be extended to an n-dimensional space.

DEFINITION 13.3 The Fourier transform of a function x 7→ f ( x) with

variable x in R

is given by

f (νν

νν) =

···

f ( x) e

−2πiνν

νν·x

All the properties of the Fourier transform extend easily to this case, mutatis

mutandis; for instance, some care is required for scaling:



f ( a x)



|a|



νν



One may also deﬁne, coordinate-wise, the three-dimensional Fourier trans-

form of a vector ﬁeld x 7→ A( x). Using this, it is possible to deﬁne transverse

ﬁelds and longitudinal ﬁelds.

Let x 7→ A( x) be a vector ﬁeld, deﬁned on R

and w ith values in R

This ﬁeld may be decomposed into three components A = (A

, A

), each

of which is a scalar ﬁeld, that is, a function from R

to R. Denote by A

the

Fourier transform of A

, A

that of A

, and A

that of A

. Putting together

these three scalar ﬁelds into a vector ﬁeld k 7→ AA

AA ( k), we obtain the Fourier

transform of the original vector ﬁeld:

AA ( k) =

ZZZ

A( x) e

−2πi k·x

DEFINITION 13.4 Let A( x) be a vector ﬁeld x ∈ R

with values in R

Denote by k 7→ AA

AA ( k) its Fourier transform. For any k ∈ R

, decompose the

vector AA

AA ( k) in two components:

• the ﬁrst is parallel w ith k and is denoted AA

(k); it is given by the

formula

(k)

def

AA ( k) · k

kkk

k ;

Heisenberg uncertainty relations 359

• the second is orthogonal to k and is denoted AA

⊥

(k); it is given by

⊥

(k)

def

= AA

AA ( k) −AA

(k).

The longitudinal component of the ﬁeld A is given by the inverse Fourier

transform of the ﬁeld k 7→AA

(k):

( x)

def

= F

−1



AA ( k) · k

|k|



Its transverse component is then

⊥

( x)

def

= F

−1



⊥

(k)



= A( x) − A

( x).

DEFINITION 13.5 A vector ﬁeld is transverse if its longitudinal component

is zero, and is longitudinal if its transverse component is zero.

From the differentiation theorem, we can der ive the well-known result:

PROPOSITION 13.6 A transverse ﬁeld satisﬁes the relation

∇ · A( x) = 0 at any x ∈ R

that is, its divergence is identically zero.

It is important to notice that the property of “being transverse” or “being

longitudinal” is by no means a local property of the vector ﬁeld (it is not suf-

ﬁcient to know the value of the ﬁeld at a given point, or in the neighborhood

of a point), but a property that d epends on the value of the ﬁeld at all points

in space (see Exercise 13.1 on page 375).

Remark 13.7 A monochromatic ﬁeld, with wave vector k, is transverse if it is perpendicular

to k at any k; it is easy to have an intuitive idea of this. A general transverse ﬁeld i s then a

sum of monochromatic transverse ﬁelds.

13.3

Heisenberg unce rtainty relations

In nonrelativistic quantum mechanics, the state of a spinless particle is

represented by a complex-valued function ψ which is square integrable and

of C

class. More precisely, for any t ∈ R, the function ψ(·, t) : r 7→ ψ( r, t)

is in L

). This function represents the probability amplitude of the particle

at the given point, that is, the probability density a ssociated wit h the presence

360 Physical applications of the Fourier transform

of the particle at point r in space and at time t is |ψ( r, t)|

. T he function

ψ(·, t) is normalized (in the sense of the L

norm) to have norm 1:

for any t ∈ R,



ψ(·, t)



ZZZ



ψ( r, t)



r = 1.

DEFINITION 13.8 The wave function in position representation is the

square-integrable function thus deﬁned. The vector space H of the wave

functions, with the usual scalar product, is a Hilbert space.

DEFINITION 13.9 The position operator along the O x-axis is denoted X ,

and similarly Y and Z are the position operators along the O y- and O z-

axes, where R

def

= (X , Y , Z). The operator X is deﬁned by its action on wave

functions given by

X ψ = ϕ with ϕ(·, t) : r 7−→ x ψ( r, t),

where we write generically r = (x, y, z).

Be careful not to make the (class ical!) mistake of stating tha t ψ and X ψ

are proportional, with proportionality constant equal to x!

It is customary to denote



ψ(t)



the function ψ(·, t), considered as a vector

in the vector space of wave functions.

The scalar product of two vectors is

then given by



ψ(t)



ϕ(t)



def

ZZZ

ψ( r, t) ϕ( r, t) d

(recall the chosen convention for the hermitian product in quantum mechan-

ics: it is linear on the right and semilinear on the left), and the position operator

acts as follows:



ψ(t)





ϕ(t)



with ϕ( r, t) = x ψ( r, t).

With each vector |ψ〉 is associated a linear form on the space H by means

of the scalar product, which is denoted 〈ψ| ∈ H

∗

. Thus, the notation 〈ϕ|ψ〉

can be interpreted in two (equivalent) ways: as the scalar product of ϕ with ψ

(in the sense of the product in L

), or as the action of the linear form 〈ψ| on

the vector |ϕ〉 (see Chapter 14 for a thorough discussion of this notation).

Only “physical” wave functions belong to this space. Some useful fu nct ions, such as

monochromatic waves r 7→ e

i k·r

do not; they are the so-called “generalized kets,” whi ch are

discussed in Chapter 14, in particular, in Section 14.2.d.

In fact



ψ(t)



is a vector in an abstract Hilbert space H . However, there exists an isometry

between H and H = L

), that is, the vector



ψ(t)



can be represented by the function

r 7→ ψ( r, t): this is the position representation. B u t, si nce the Fourier transform is itself

an isometry of L

) into itself, the vector



ψ(t)



may also be represented by the f u nction

p 7→

ψ( p, t), which is the Fourier transform of the previous function. This is called the

momentum representation of the wave vector.

Heisenberg uncertainty relations 361

DEFINITION 13.10 The average position on the x-axis of a particle charac-

terized by the wave function ψ( r, t) is the quantity

〈x〉(t)

def



ψ(t)



X ψ(t)





ψ(t)



ψ(t)



ZZZ



ψ( r, t)



In addition, to every wave function ψ( r, t) is associated its Fourier trans-

form with respect to the space variable r, namely,

Ψ( p, t)

def

ZZZ

ψ( r, t) e

−i p·r/ }h

(2π }h)

This function represents the probab ility amplitude for ﬁnding the particle

with momentum p. Of course, this function is also normalized to have norm

1, since for every t ∈ R we have



Ψ(t)



Ψ(t)



ZZZ



Ψ( p, t)



p =

ZZZ



ψ( r, t)



r = 1

by the Parseval-Plancherel identit y.

DEFINITION 13.11 The wave function in momentum representation is the

Fourier transform Ψ( p, t) of the wave function ψ( r, t) of a particle.

DEFINITION 13.12 The momentum operator P

def

= −i }h∇ = (P

, P

) is

the linear operator a cting on differentiable functions by

ψ = ϕ



or P

|ψ〉 = |ϕ〉



with ϕ(·, t) : r 7−→ −i }h

∂ψ

∂ x

( r, t).

The average momentum along the x-direction of a particle character ized by

the wave function ψ( r, t) is

〈p

〉(t)

def



ψ(t)



ψ(t)



= −i }h

ZZZ

ψ( r, t)

∂ψ

∂ x

( r, t) d

From the known properties of the Fourier transform, we s ee that it is

possible to extend the deﬁnition of the operator P

on wave f unctions “in

momentum representation” by

def

= F



−1

[Ψ]



which gives explicitly, by the differentiation th eorem,

Ψ = Φ with Φ(·, t) : p 7−→ p

Ψ( p, t).

362 Physical applications of the Fourier transform

DEFINITION 13.13 The position uncertainty of a particle characterized by

the wave function ψ( r, t) is the quantity ∆x d eﬁned by the following equiva-

lent e xpressions:

(∆x)

def



x −〈x〉







−〈x〉

= 〈X ψ|X ψ〉−〈ψ|X ψ〉

ZZZ



x −〈x〉





ψ( r, t)



r = kX ψ −〈x〉ψk

DEFINITION 13.14 The momentum u nce rtainty of a particle characterized

by the wave function ψ( r, t) is the quantity ∆ p

deﬁned by the following

equivalent expressions:

(∆p

)





−〈p

〉

= 〈P ψ|P ψ〉−〈ψ|P ψ〉

ZZZ



∂ψ

∂ x

( r, t)



r −〈p

〉

ZZZ



Ψ( p, t)



p −〈p

〉

= kP

ψ −〈p

〉ψk

Note that





may also be expressed as the average value of the operator

= −}h

∂

∂ x

in the state |ψ〉, since an easy integration by parts reveals that





def







= −}h

ZZZ

ψ( r, t)

∂

∂ x

( r, t) d

ZZZ



∂ψ

∂ x

( r, t)



To perform this integration by parts, it is necessary that the derivative of ψ

decay fast enough, which is the case when 〈∆p

〉 exists.

Remark 13.15 The average values and the uncertainties should be compared with the expecta-

tion value and standard deviation of a random variable (see Chapter 20).

We are going to show now t hat there exists a relation between ∆x and

∆p

. To star t with, we have the following lemma:

LEMMA 13.16 Let ψ be a wave function.

i) If the wave function is translated by the vector a ∈ R

by putting

( r, t)

def

= ψ( r − a, t),

then neither ∆x nor ∆ p

is changed.

ii) Similarly, if a translation by the quantity k ∈ R

is performed in momentum

space, or, what amounts to the same thing, if the wave function in position

representation is multiplied by a phase factor by deﬁning

Heisenberg uncertainty relations 363

( r, t)

def

= ψ( r, t) e

i k·r / }h

then neither ∆x nor ∆p

is changed.

iii) Consequently, it is possible to deﬁne a wave function

ψ( r, t) which is centered

in p osition and momentum (i.e., such that the average position and the average

momentum are zero) and has the same uncertainties as the original function

ψ( r, t) with respect to position a nd momentum.

Proof. The average position of the translated wave function ψ

〈x〉

ZZZ



( r, t)



r =

ZZZ

(x −a

)



ψ( r, t)



r = 〈x〉−a

The position uncertainty is then

(∆x)





−〈x〉

ZZZ



ψ( r − a, t)



r −



〈x〉− a



ZZZ

(x −a

)



ψ( r, t)



r −



〈x〉

−2a

〈x〉+ a







−2a

〈x〉+ a

−



〈x〉

−2a

〈x〉+ a







−〈x〉

= (∆x)

The momentum uncertainty of |ψ〉 is of course unchanged.

The proof is identical for the translates in momentum space.

Finally, the centered wave functio n is, for a given t, equal to

ψ( r, t)

def

= ψ



r − 〈r 〉, t



i〈p〉·r/ }h

THEOREM 13.17 (Heisenberg uncertainty relations) Let ψ be a wave function

for which the average position and avera ge momentum, as well as the position and

momentum uncertainties, are deﬁned. Then we have

∆x ·∆ p

(13.1)

(and similarly for y and z).

Moreover, equality holds only if ψ is a gaussian function.

Proof. Translating the wave function as in Lemma 13.16, we may assume that 〈x〉 = 0

and 〈p

〉 = 0. Moreover, to simplify notation, we omit the dependency on t. We have

(∆x)

ZZZ



ψ( r)



r and (∆p

)

= }h

ZZZ



∂ψ

∂ x

( r )



The Cauchy-Schwarz inequality for functions in L

states that



ZZZ

f ( r ) g( r ) d



ZZZ



f ( r )



r ·

ZZZ



g( r )



Applying this inequality to f ( r) = x ψ( r) and g( r) =

∂ψ

∂ x

( r ), we obtain



ZZZ

x ψ( r)

∂ψ

∂ x

( r ) d



ZZZ



ψ( r)



r ·

ZZZ



∂ψ

∂ x

( r)



(∆x)

(∆p

)

. (∗)

364 Physical applications of the Fourier transform

But, using an integration by parts, since x is real and the boundary terms cancel (see

Remark 13.19 concerning this point), we can write

ZZZ

x ψ( r)

∂ψ

∂ x

( r) d

r = −

ZZZ



ψ( r)



r −

ZZZ

x ψ( r)

∂ψ

∂ x

( r ) d

or 2 Re

ZZZ

x ψ( r)

∂ψ

∂ x

( r) d



= −

ZZZ



ψ( r)



r = −1.

Since |z| ¾ |Re(z)| for all z ∈ C, t he left-hand side of (∗) satisﬁes



ZZZ

x ψ( r)

∂ψ

∂ x

( r ) d



ZZZ

x ψ( r)

∂ψ

∂ x

( r ) d



This immediately implies (13.1).

The Cauchy-Schwarz inequality is an equality only i f the two f u nctions considered

are proportional; but it is easy to see that X ψ and ∂ψ/∂ x can only be propor tional if

ψ is a gaussian as a function of x (this is a simple differential equation to solve).

If the usual deﬁnition of the Fourier transform is kept (with the factor 2π

and without }h), t he following equivalent theorem is obta ined:

THEOREM 13.18 (Uncertainty relation) Let f ∈ L

(R) be a function such that

the second moment

|f (x)|

dx and

′

(x)|

dx both exist. Deﬁne





def

kf k



f (x)



dx,





def

kf k



f (ν)



dν =

4π

kf k



′

(x)



dx.

Then we have









16π

with equality if and only if f is a gaussian.

Remark 13.19 A point that was left slightly fuzzy above requires explanation: the cancellation

of the boundary terms during the integration by parts. Denote by H the space of square

integrable functi ons ψ, such that X ψ and P

ψ are also square integrable. This space, with the

norm N deﬁned by

N (ψ)

= kψk

+ kX ψk

+ kP ψk

ZZZ



ψ( r)



r +

ZZZ



ψ( r)



r +

ZZZ



Ψ( p)



is a Hilbert space.

If ψ ∈ H is of C

∞

class with bounded support with respect to the variable x, the proof

above (and in particular the integration by parts w ith the boundary terms vanishing) is correct.

A cl assical result of analysis (proved using convolution with a Dirac sequence) is that the

space of C

∞

functions with bounded support is dense in H . By continuity, the preceding

result then holds for all ψ ∈ H .

Remark 13.20 Another approach of the uncertainty relations is p roposed in Exercise 14.6 on

page 404.

Analytic signals 365

13.4

Analytic signals

It is customary, in electricity or optics, for instance, to work with complex

sinusoidal signals of the type e

iωt

, for which only the real part has physical

meaning. This step from a real signal f

(R)

(t) = cos(ωt) to a complex signal

f (t) = e

iωt

may be generalized in the case of a nonmonochromatic signal.

First of all, notice that the real signal may be recovered in two different

ways: as the real part of th e associated complex signal, or as the average of the

signal and its complex conjugate.

We can generalize this. We consider a real signal f

(R)

(t) and search for

the associated complex signal. In the case f

(R)

(t) = cos(2πν

t) (with ν

> 0),

the spectrum of f

(R)

contains both frequencies ν

and −ν

. However, this

information is redundant, since the intensity for the frequency −ν

is the same

as that intensity for the frequency ν

, since f

(R)

is real-valued. The complex

signal f (t) = e

2πiνt

associated to the real signal f

(R)

is obtained as follows:

take the spectrum of f

(R)

, keep only the positive frequencies, multiply by two,

then reconstitute the signal.

In a general way, for a real-valued signal f

(R)

(t) for which the Fourier trans-

form

(R)

(ν) is deﬁned, its spectrum is hermitian, that is , we h ave

(R)

(ν) =

(R)

(−ν) for a ll ν ∈ R;

thus, “half” the spectrum

(R)

sufﬁces to recover f

(R)

If the spectrum

(R)

(ν) does not have a singularity at the origin, or at least

if is sufﬁciently regular, we can consider the truncated spectrum

2H (ν) ·

(R)

(ν).

The inverse Fourier transform of this truncated spectrum will give the analytic

signal associated to t he function f

(R)

. Since the inverse Fourier transform of

2H (ν) is

−1



2H (ν)



= F

−1

[1 + sgn ν] = δ(t) + pv

πt

(see Theorem 11.16 on page 304), we deduce tha t the inverse Fourier transform

of 2H (ν) ·

(R)

(ν) is

−1



2H (ν) ·

(R)

(ν)





δ + pv

πt



∗ f

(R)

= f

(R)

(t) −

(R)

′

)

′

− t

′

We make the following deﬁnition:

DEFINITION 13.21 An analytic signal is a function with caus al Fourier trans-

form, t hat is, a function f such that the Fourier transform vanishes for nega-

tive frequencies.

366 Physical applications of t he Fourier tran sform

DEFINITION 13.22 Let f

(R)

be a real signal with sufﬁciently regular Fourier

transform.

The imagina ry signal associated to f

(R)

is the opposite of its

Hilbert transform:

(I)

(t)

def

= −

(R)

′

)

′

− t

′

DEFINITION 13.23 The analytic signal associated to the real signal f

(R)

is the

function

t 7−→ f (t)

def

= f

(R)

(t) + i f

(I)

(t) = f

(R)

(t) −

f (t

′

)

′

− t

′

the spectrum of which is

f (ν) = 2H (ν) ·

(R)

(ν).

The analytic signal associated with f

(R)

is analytic in the sense of Deﬁni-

tion 13.21: it s spectrum is identically zero for frequencies ν < 0. Since

f (ν)

is causal, it has a Laplace transform F (p) deﬁned at least on the right-hand

half-plane of the complex plane. Since the signal f is the inverse Fourier

transform of

f , it is given by

f (x) = F (−2πix).

Hence there is an analytic continuation of f to the upper half-plane,

which can

be obtained using the Laplace transform of t he spectrum:

f (z) = F (−2πiz).

This analytic continuation is holomorphic in this u pper half-plane, and there-

fore satisﬁes the Kramers-Kronig relations — which are neither more nor less

than the relation deﬁning f

(I)

: we’ve made a complete loop!

The fact that f may be c ontinued to an analytic fu nction on the upper half-place

justiﬁes also the terminology “analytic signal.”

All this may be summarized as follows:

THEOREM 13.24 Let f : R → C be a function which has a Fourier transform. If

the spectrum of f is causal, then f may be analytically continued to the complex upper

half-plane.

Conversely, if f may be analytically continued to this half-plane, and if the integral

of f along a half-circle in the upper half-plane tends to 0 as the radius tends to inﬁnity,

then the spectrum of f is causal.

(See [79] for more details.)

In the sense that the product H (ν)

(R)

(ν) exists; this means that

(R)

cannot involve a

singular distribution at the origin 0.

The upper half-plane because of the similitude x 7→ −2πix: the i mage of the upper

half-plane is the right-hand half-plane.