Let us expand a function f ( r → ) having Fourier transform on a basis of exponential functions

f ( r → ) = ∑ k → c ( k → ) e i k → r → (2.1.1)

where k → belongs to an infinite set of vectors obeying the condition that the scalar product k → r → is dimensionless for the following relation to hold

e i k → r → = 1 ⋅ e i k → r → = e i ( k → r → + 2 π ) (2.1.2)

Under such condition we may write

∫ R 3 e − i k → 0 r → f ( r → ) d r → = ∑ k → c ( k → ) ∫ R 3 e − i k → 0 r → e i k → r → d r → = ( 2 π ) 3 ∑ k → c ( k → ) δ ( k → − k → 0 ) = ( 2 π ) 3 c ( k → 0 ) (2.1.3)

so that we may state the theorem:

“Any function f ( r → ) having Fourier transform may be written under the form

f ( r → ) = ( 2 π ) 3 / 2 ∑ k → f ˜ ( k → ) e i k → r → (2.1.4)

where k → r → is dimensionless and f ˜ ( k → ) is the Fourier transform of f (r→)

f ˜ ( k → ) = F T f ( r ¯ ) = ( 2 π ) − 3 / 2 ∫ R 3 e − i k → r → f ( r → ) d r → (2.1.5)

Now from the well known formulas

f ( x + a ) = e a ∂ x f ( x ) (2.1.6)

F T D x f ( x ) = F T f ′ ( x ) = i k F T f ( x ) (2.1.7)

we get

F T δ ( x − a ) = F T e − a D x δ ( x ) = e − i a k F T δ ( x ) = e − i a k ( 2 π ) − 1 / 2 (2.1.8)

so that by (2.1.4)

δ ( r → − r → 0 ) = ∑ k → e − i k → 0 r → e i k → r → = ∑ k → e i k → ( r → − r → 0 ) (2.1.9)

Consider a particle situated at the position r → 0 and having a mass m and a constant celerity v → 0 . Defining

k → 0 = 2 π λ 0 v → 0 ν 0 = 2 π λ 0 n → (2.1.10)

where λ 0 has the dimension of a length as it must be for k → 0 r → to be dimensionless we see from (2.1.9) that the formula

δ ( k → − k → 0 ) δ ( r → − r → 0 ) = e i k → 0 ( r → − r → 0 ) = exp i ( 2 π λ 0 ( r → − r → 0 ) n → ) (2.1.11)

represents at the same time this particle and a wave. Thank to the property e ± i 2 π = 1 this wave has a wavelength λ 0 and consequently a period

T 0 = λ 0 / v 0 (2.1.12)

The wave function of this particle is then within a multiplicative constant

Ψ 0 ( r → , t ) = A exp i ( k → 0 ( r → − r → 0 ) − 2 π T 0 t ) (2.1.13)

This is the insight of the principle of wave-particle duality conceived by Planck in 1900 [1] and Einstein in 1905 [2]. It constitutes the first quantization of quantum mechanics.

As

k → // v → // p → = m v → (2.2.1)

we may define a universal constant θ having dimension M L 2 T − 1 then link p → with k → by the relation

p → = θ k → = θ 2 π λ v → ν = θ 2 π λ n → (2.2.2)

in order to get the form of the relation between momentum and associated wavelength

p = θ k = θ 2 π λ (2.2.3)

in accordance with the hypothesis proposed in 1923 by de Broglie [3].

The wave function of the considered particle may then be put under the form

Ψ 0 ( r → , t ) = A exp i θ − 1 ( p → 0 ( r → − r → 0 ) − 2 π θ T 0 t ) (2.2.4)

By dimensional consideration we see that the quantity 2 π θ T 0 is an energy that we baptize E 0 and propose to assimilate it with the energy of the quoted particle

E 0 = 2 π θ T 0 (2.2.5)

By comparison with the formulae of Planck-Einstein [1] [2] and de Broglie [3]

E = h T , p = h λ (2.2.6)

we get the identifications

θ = h 2 π = ℏ (2.2.7)

p → = ℏ k → (2.2.8)

and see that k → is the commonly called wave-vector of a wave.

From now all we say that k → and r → are Fourier transform reciprocal as so as 2 π T = ℏ − 1 E and the time t. The Planck constant h was measured by Millikan [4] in 1916. The best current value for h is 6.62607004 × 10 − 34 m 2 ⋅ kg / sec and is officially utilized from the date 20-05-2019 on to define the value of the kilogram.

A consequence of the relation (2.2.4) and the de Broglie hypothesis (2.2.6) we see that if two particles have the same value of momentum p → and the same position they may be assimilated to one particle with momentum 2 p → so that the dual wave must have its wavelength divided by 2. This leads to a paradox and confirms the Exclusion principle of Pauli [5]. For photons with momentum p = h ν / c too small, two times of it is quasi equal to it so that there is no paradox, i.e. many photons may occupy one position.

In a Hilbert space of Dirac kets and bras let according to (2.1.13)

〈 r → | k → 0 〉 = ( 2 π ) − 3 / 2 exp ( i k → 0 r → ) (2.4.1)

be the position-representation of a state having a definite wave-vector k → 0 .

From the formula

F T e i k → 0 r → = ( 2 π ) − 3 / 2 ∫ R 3 e − i ( k → − k → 0 ) r → d r → = ( 2 π ) 3 / 2 δ ( k → − k → 0 ) (2.4.2)

and (2.4.1) we have

F T 〈 r → | k → 0 〉 = F T ( 2 π ) − 3 / 2 e i k → 0 r → = δ ( k → − k → 0 ) = 〈 k → | k → 0 〉 (2.4.3)

so that, because k → 0 is arbitrary, we get the interesting relation

〈 k → | = F T 〈 r → | (2.4.4)

which gives precision to the latent idea in many researchers that there exists somehow a Fourier relation between momentum and position:

“In quantum mechanics the wave-vector bra 〈 k → | is the Fourier transform of the position bra 〈 r → | ”.

From (2.4.4) we get the relation between momentum-representation and position-representation of a state

〈 k → | Ψ 〉 = F T 〈 r → | Ψ 〉 (2.4.5)

In the Hilbert space of states besides X ^ and P ^ x let us formally define another operator D ^ x

by the relation

D ^ x X ^ − X ^ D ^ x ≡ I ^ (2.5.1)

where I ^ is the identity operator.

Now, in the space of functions let X ⌣ be the operator of multiplication by x and D ⌣ x the derivative operator

X ⌣ f ( x ) = x f ( x ) ; D ⌣ x f ( x ) = f ′ ( x ) (2.5.2)

verifying

[ D ⌣ x , X ⌣ ] ≡ D ⌣ x X ⌣ − X ⌣ D ⌣ x ≡ I ⌣ (2.5.3)

We must be attentive on the fact that the operators X ⌣ , D ⌣ x , P ⌣ x , D ⌣ p act on functions and X ^ , D ^ x , P ^ x , D ^ p x act on bras and kets.

From (2.5.1), (2.5.3) we get

〈 x | D ^ x X ^ − X ^ D ^ x | x ' 0 〉 = ( x ' 0 − x ) 〈 x | D ^ x | x 0 〉 = δ ( x − x ' 0 ) (2.5.4)

D ⌣ x ( X ⌣ − x 0 ) δ ( x − x ' 0 ) = 0 (2.5.5)

( D ⌣ x X ⌣ − X ⌣ D ⌣ x ) δ ( x − x ' 0 ) = ( x 0 − x ) D ⌣ x δ ( x − x ' 0 ) = δ ( x − x ' 0 ) (2.5.6)

so that

〈 x | D ^ x | x 0 〉 = D ⌣ x 〈 x | x 0 〉 (2.5.7)

Besides we have also

〈 x | X ^ | x 0 〉 = x δ ( x − x 0 ) = X ⌣ 〈 x | x 0 〉 (2.5.8)

so that, as x 0 is arbitrary,

〈 x | D ^ x ≡ D ⌣ x 〈 x | ; 〈 x | X ^ ≡ X ⌣   〈 x | (2.5.9)

The above relations associated with (2.4.4) and

F T x f ( x ) = ( 2 π ) − 1 / 2 ∫ − ∞ ∞ i ∂ k e − i k x f ( x ) d x = i ∂ k F ( x ) (2.5.9)

lead to

〈 k | X ^ | k 0 〉 = F T 〈 x | X ^ | k 0 〉 = F T x 〈 x | k 0 〉 = i ∂ k F T 〈 x | k 0 〉 = i ∂ k 〈 k | k 0 〉 = 〈 k | i D ^ k | k 0 〉 (2.5.10)

i.e.

X ^ ≡ i D ^ k ≡ i ℏ D ^ p (2.5.11)

Similarly by repeating the reasoning with P ^ x , D ^ p x we get

P ^ x = − i ℏ D ^ x (2.5.12)

Extension to 3D space gives

r ^ ≡ i ∇ ^ k ≡ i ℏ ∇ ^ p (2.5.13)

and finally the commutation relations

[ r ^ j , p ^ l ] = − i ℏ [ r ^ j , ∇ ^ l ] = i ℏ δ j l I ^ (2.5.14)

which have been called quantum conditions and postulated by Born in 1925 [6].

Similarly from the fact that 2 π T = ℏ − 1 E and t are Fourier reciprocal we have

E = i ℏ ∂ t (2.5.15)

From the relations (2.5.6) we may also get an important proposition:

“The eigenvalue equation

A ( X ^ , P ^ ) | α 〉 = a | α 〉

of an arbitrary operator A ( X ^ , P ^ ) leads to the differential equation for the function 〈 x | α 〉

〈 x | A ( X ^ , P ^ ) | α 〉 = A ( X ⌣ , − i ℏ D ⌣ x ) 〈 x | α 〉 = a 〈 x | α 〉 (2.6.1)

For example, with

A ( X ^ , P ^ ) ≡ 1 2 m P ^ 2 + V ( X ^ ) (2.6.2)

we obtain the well known time independent Schrödinger equation [7]

( − ℏ 2 2 m D ⌣ x 2 + V ( x ) ) Ψ ( x ) = E Ψ ( x ) (2.6.3)

As 2 π T = ℏ − 1 E and t are Fourier transform reciprocal we get the time dependent Schrödinger equation

Let be the function equal to zero for and to for as illustrated by Figure 1.

A state where there is incertitude on the wave-number k

corresponds to the momentum-representation

Utilizing the Heaviside function we may write

Thank to (2.1.6), (2.1.7) and the property

we get by Fourier transform of (2.7.3)

so that by (2.7.2)

The graph of has the form (Figure 2).

The function has maximum value for, vanishes for . It and its squared are equal nearly to half of their maxima for or.

We may then write that

Because the relation (2.7.6) is conformed with the uncertainty principle announced by Heisenberg [8] and proven somehow by Kennard [9] in 1927.

Similarly because the couple are reciprocal so as we get

Consider a state which has many stable values for its energy and suppose that is the sum of individual states each of them having only one value of energy or one frequency

By Fourier transform we get

so that

By (2.8.3) we see that the probability for observing at the instant t is maximal for

In other word we see that from time to time there may have emission/absorption of waves with frequencies

This result accords with the theory on the constitution of atoms and molecules of Bohr [10] in 1913.

Let be two operators obeying the condition

We have

because at each time we change AB into BA we must add.

So, let be an entire function and its derivative function we clearly have

Now from (2.9.3)

so that by recursion we get

From (2.9.5) we can’t sum over because of the mixed coefficient under the summation. After thinking we replace (2.9.5) with the following formula

so that if is an entire function we get the fundamental identity between operators obeying the sole condition

and its dual

For examples we have successively

Defining the creation and the annihilation operators by

we get from (2.9.8), (2.9.9), (2.9.10),

Closing this paragraph we propose from (2.9.6) the new version of the Newton’s binomial formula

Consider an object occupied a limited domain D in space and represented by the object function which may be discontinuous

From the formula (2.1.4) we see that the coexistence of a wave and this object may be represented by

Equation (3.1.2) gives rise to the main theorem in wave optics

“The amplitude of diffraction of a wave into a wave by the form of an object is equal to multiplies the Fourier transform of the object function calculated for the deviation of the wave-vector”.

Consider a set of objects centered at the points. Utilizing (2.1.6), (2.1.7) we have

and get a useful formula giving the amplitude of diffraction in some direction of a plane wave by a set of identical objects