_{1}

^{*}

Finding that in the formula of expansion of a function into a series of wave-like functions the coefficients are its Fourier transforms, if existed, we deduce mathematically all the principles and hypothesis that illustrated physicists utilized to build quantum mechanics a century ago, beginning with the duality particle-wave principle of Planck and including the Schr ödinger equations. By the way, we find a simple Fourier transform relation between Dirac momentum and position bras and a useful permutation relation between operators in phase and Hilbert spaces. Moreover, from the found particle-wave duality formula we prove and obtain again essentially by mathematical analysis all the laws of wave optics concerning reflections, refractions, polarizations, diffractions by one or many identical 3D objects with various forms and dimensions.

From the find that a function f ( r → ) may be expanded into a series of functions e i k → r → with coefficients equal to ( 2 π ) 3 / 2 multiplies the Fourier transform f ˜ ( k → ) of f ( r → ) we arrive to obtain that a particle moving with celerity v → 0 , momentum p → 0 creates a wave, confirming the wave-particle duality principle conceived by Planck and Einstein in 1900-1905. Moreover we obtain that p 0 is inversely proportional to the wavelength of this wave conformed with the hypothesis of de Broglie and that the particle’s energy is proportional to the wave’s frequency conformed with the proposition of Planck. The coefficient of proportionality is then identifiable with the Planck’s constant h.

The Exclusion principle of Pauli may be explained by the assimilation of two particles having the same momentum and the same position with only one having double momentum so that the de Broglie wavelength is divided by two which is a paradox.

From the fact that δ ( p → − p → 0 ) represents the momentum-representation of the state | p → 0 〉 and ( 2 π ) − 3 2 e i ℏ − 1 p → 0 r → its position-representation we obtain the relation 〈 k → | = F T 〈 r → | where p → = ℏ k → . These relations lead to the canonical commutation relations [ r ^ j , p ^ l ] = i ℏ δ j l I ^ , E = i ℏ ∂ t of Born which in turn lead to the well known Schrӧdinger equations. Utilizing the relation 〈 k → | = F T 〈 r → | we see also that the Heisenberg’s incertitude relation Δ x Δ p > ℏ / 2 is a matter of Fourier transform relation between the rectangular function a − 1 ( H ( k + a ) − H ( k − a ) ) and the function sin ( a x ) / ( a x ) , H ( x ) being the Heaviside function.

Consider an atom having a discrete spectrum of states each having a value of energy E j . It is represented by 〈 E | α 〉 = ∑ j = 1 N δ ( E − E j ) . By searching the maximum values of | 〈 t | α 〉 | 2 we see that from time to time there have emission/absorption of a wave having frequency ν j k = h − 1 ( E k − E j ) conformed with the theory of Bohr. Besides we obtain permutation relations between functions of creation and annihilation operators in second quantization.

By the same formula giving quantum mechanics’ principles we realize that the product of a wave e i k → 0 r → and an object described by a function f ( r → ) is a sum over e i k → r → with coefficients equal to ( 2 π ) 3 / 2 f ˜ ( k → − k → 0 ) . This opens a simple way to calculate the amplitude of diffraction of a wave by a 3D object such as a semi-space which leads to the Descartes, Snell’s laws, Fresnel equations, then by a set of identical objects having different geometric forms such as plane which leads to the Braag’s formula, pyramid, sphere, etc.

Details of the finds are explained successively in the following paragraphs.

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 [

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 [

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 [

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 [

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 [

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 [

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 [

( − 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

A state

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

The function

We may then write that

Because

Similarly because the couple

Consider a state

By Fourier transform we get

so that

By (2.8.3) we see that the probability for observing

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 [

Let

We have

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

So, let

Now from (2.9.3)

so that by recursion we get

From (2.9.5) we can’t sum over

so that if

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

Consider a set of objects centered at the points

and get a useful formula giving the amplitude of diffraction in some direction

The semi space under the plane Oxy is described by the object function

From the theorem (2.1.4) we see that

so that there are diffracted waves only for

Equations (3.3.3) gives the Descartes law of reflection [

Now, let

The amplitudes

, (3.3.5)

Remarking that the Fourier transform of a Heaviside function

we get

In order to calculate the coefficients μ, ν we will make use of the law of conservation of energies. The incoming density of energy at the interface Oxy is proportional to

The above equations and the formula

lead by (3.2.3) to the following

· Taken

· Taken

· Taken

From (3.3.12) we find again the Brewster’s condition for total polarization

The equation of a sphere centered at O and having radius R as shown in

Its Fourier transform is invariant in a rotation around the origin so that

As conclusion we see that in a diffraction by a sphere the amplitude of diffraction is inversely proportional to

Let

For example, for

From (3.2.3) we obtain for example the amplitudes of diffraction of a plane wave by parallel planes perpendicular to Oz at the points

The maximum amplitudes of diffraction correspond, because

The formula (3.3.19) is identical with the Braag’s formula [

Someone has said that “Physics is the studies of Nature, how matter and radiation behave, move and interact thorough space and time. Mathematics, on the other hand, is logical deductive reasoning based on initial assumption. There are many different systems of mathematics that can describe the same physical phenomenon.” Accordingly this work which improves and completes a previous work [

May this work brings closer students to modern physics!

The author acknowledges Prof. Geneste J.P. for reading and appreciating this work at the World conference on quantum mechanics and nuclear engineering holt in 2019 September at Paris. He thanks warmly the reviewer for giving many judicious remarks and for judging this work as meaningful. He thanks Dr. Feltus Chr. at Luxembourg LIST for laborious writing assistance. He dedicates this work to the Ho Chi Minh-city University of Natural Sciences and the Université libre de Bruxelles where he was formed in the past.

The author declares no conflicts of interest regarding the publication of this paper.

Si, D.T. (2019) Principles of Quantum Mechanics and Laws of Wave Optics from One Mathematical Formula. Applied Mathematics, 10, 892-906. https://doi.org/10.4236/am.2019.1011064