Mehdi Hage-Hassan

Mehdi Hage

On the N-dimensional hydrogen atom in momentum representation

We derive the wave function for N-dimensional hydrogen atom in the momentum representation with the phase factor using the generating function method and Hankel's integral.

1-Introduction

 correctly.

In a previous paper [9-10] we presented a new and elementary method for the determination of the wave function in momentum space for two and three dimensions using the generating function and Hurwitz transformations for a direct integration of the Fourier transform. But Hurwitz transformations are valid only for N = 2, 3, 5 and 9. In this work we present, the generalization for N≥3 by using the technique of generating function and the Hankel's integral of Bessel functions and therefore we determine the wave function in momentum space with the exact phase factor for any order N. We also show that the phase factors of wave functions representing the coordinates and momentum spaces are dependent. The paper is structured as follows. First in the second part we outline the derivation of the wave function of coordinates. We determine the wave function in momentum space in the third part. The fourth part is devoted to a comment on the phase's factors.

The N-dimensional hydrogen atom in representation space

.

The wave function of hydrogen atom in representation space

The Schrödinger equation of the hydrogen atom in N-dimensions space [START_REF] Dehesa | Fisher information of D-dimensional hydrogenic systems in position and momentum spaces[END_REF] is

) ( ) ( 2 2 2 r E r r e N r r h Ψ = Ψ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - Δ -μ (2.1)
Where μ is the reduced mass.

We write ) , , , (

With , 2 , , 1 , 0 , 0 - = ≤ ≤ ≤ ≤ N j j j K π θ π θ π ϕ 2 0 ≤ ≤ .
And atomic unit are used through the text.

The method of separation of variables is used by many authors [START_REF] Louck | [END_REF][14][15] for the resolution of the Schrödinger equation and we write only the solution:

( ) ( )

) ( ) ( ) ( ) ( } { , 2 2 1 2 2 / , } { , , } { , , r l N l l n r l N l n r l l n l n Y r L e r N Y r R r Ω = Ω = Ψ - + - - - μ λ μ μ ω ω ω r (2.3) ) 2 / ) 3 ( ( 2 2 , )! 3 )( 2 / ) 3 ( ( 2 )! 1 ( 2 / 1 , - + = = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ - + + - + - - = N n N l n N n l n N n l n δ ω 2.2 The radial function ) ( ) ( 1 x L l n α - - Are the Laguerre polynomials ! ) 1 ( )) ( ( 0 2 n n dx x L x e n x + + Γ = ∫ ∝ - α α α (2.4)
The generating function of Laguerre polynomials [START_REF] Erdelyi | Heigher transcendal functions[END_REF][START_REF] Gradshteyn | table of integrals series and products[END_REF] is:

∑ ∝ = - - + - = 0 1 1 ) ( ) 1 ( 1 ) ( n x z z n n e z x L z α α (2.5) ) ( ) 1 ( ) ( ) ( ) ( x L x L dx d k k n k n k k + - - = α α 1 , 2 2 - - = - - + = + l n k n N l k α Consequently 3 - + = N l α We deduce also that ) ( ] ) 1 ( exp[ ) 1 ( ) ( 0 2 2 1 2 1 x L z z z x z z n N l l n n N l l ∑ ∝ = - + - - + + = - - - - (2.6)

The angular function

) ( } { , r l Y Ω μ
Is the Hyperspherical function

1 1 1 ) )(sin (cos 2 1 ) ( 2 1 } { , } { , + + + ∏ - = + - = Ω j j j j j j j N j im l r l C e A Y μ μ α μ μ ϕ μ μ θ θ π (2.7) With ) 2 ( 2 )! )( ( ) ( 1 2 2 1 1 2 1 1 } { , 1 + - - + - = + + + Γ - + + Γ = + ∏ j j j j j j j N j j j l j j A μ μ α π μ μ μ α α α μ α μ And ) , , , ( }) { , ( , , 1 2 1 2 1 2 1 - - = = ≥ ≥ ≥ = - - = N N j l l m l j N μ μ μ μ μ μ α K K . ) (cosθ α n C
Is the Gegenbauer polynomial of degree n and parameter α.

The N-dimensional hydrogen atom in the momentum representation

Using the development of the free wave in space of N-dimensions, the generating functions of Laguerre polynomials and Hankel's integral we determine the generating function of momentum representation and hence the wave function in momentum space.

The Generating function and the momentum representation

The wave function of hydrogen atom in momentum representation is:

r d r e p l n r p i N l n r r r r r ) ( ) 2 ( 1 ) ( } { , , . 2 / } { , , μ μ π Ψ = Ψ ∫ - (3.1) ( ) ( ) r d Y r L r e N r l N l l n l r r p i N l n r r r ) ( ) 2 ( } { , 2 2 1 2 . 2 / , Ω = - + - - - - ∫ μ ω ω ω π (3.2) r d
r Is defined by:

r N N d r dx dx dx r d Ω = = -1 2 1 K r . (3.3) We note that r r p i δ + ⋅ r r
can be regarded as the scalar product of two vectors in Euclidean space Using the development of the wave function of the free particle in N-dimensions [14][15]:

1 2 , ) /( ) ( ) ( ) ( ) 2 ( 1 2 * ] [ ] [ ] [ 2 / . - + = Ω Ω = - ∝ ∑ N l pr pr J Y Y i e N p L r L L l N r p i ν π ν r r (3.4) We find ( ) ( ) (
)

) ( ) ( ) /( ) ( ) ( ) 2 ( 1 ) ( ] [ 1 1 2 0 2 2 1 2 , } { , , . 2 / } { , , p L l N N N l l n r l l n l n r p i N l n Y i dr r pr pr J r L e r N r d r e p Ω - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = Ψ = Ψ - - ∝ - + - - - - ∫ ∫ ν ω μ μ ω ω π r r r r r (3.5) Multiply by ) /( ) /( 1 2 / , N l n l n z N + ω
and do the summation we write first the generating function for the basis and the generating function with ω=constant which is very useful for the determination of wave function in momentum space.

∑ ∝ = + - Ψ = 0 , 2 / , 1 2 ) ( ) ( ) , , ( n l n N l n l n N n p z N p z p G r ω δ (3.6) And ( ) ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = - - ∝ - + - - - ∝ = ∫ ∑ dr r pr pr J r L e r z z p G N N N l l n r l n n 1 1 2 0 2 2 1 2 0 ) /( ) ( ) , , ( ν λ ω δ dr r pr J e z z r N l l 1 0 1 2 1 ) ( ) 1 ( ) ( + ∝ - - + + ∫ - - = ν ν γ 1 2 - + = N l ν (3.7) With δ ω ω γ 2 , 1 1 2 = - + = z z (3.8)

Derivation of the generating function

Using Hankel's integral [START_REF] Gradshteyn | table of integrals series and products[END_REF]:

] 1 ) [Re( , ) ( ) 2 / 3 ( ( ) 2 ( 2 ) ( 2 / 3 2 2 1 0 - > + × + Γ = + + ∝ - ∫ ν γ π ν γ ν ν ν ν γ p p dr r r p J e r
(3.9)

We find the generating function as follows:

= + × + Γ - - + - + + 2 / 3 2 2 1 2 1 ) ( ) 2 / 3 ( ( ) 2 ( 2 ) 1 ( ) ( ν ν γ π ν γ p p z z N l l 2 1 2 2 2 2 1 2 1 1 2 2 ) ) 1 ( ) 1 ( ( ) 1 ( ) ( ) 2 1 ( ) 2 ( ) 1 ( 1 + + + + + - + + + + - - - + + Γ - + = N l N l l N l N l z z p z z N l p z z δ π ω ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + - = + - - - + + + Γ = + + + + + - + 2 2 2 2 2 1 2 2 1 2 1 2 2 1 2 , ) 2 1 ( ) 1 ( ) ( ) ( ) 2 1 ( ) 2 ( 2 δ δ δ π δ p p x z zx z z p N l p N l l N l N l (3.10)
This function is the generalization of the generating function (4.10) of the paper [START_REF] Hage-Hassan | On the hydrogen wave function in Momentum-space, Clifford Algebra and the Generating function of Gegenbauer polynomials[END_REF], by a minus sign due to the derivation of (4.9).

The Wave function in the momentum representation

We have [START_REF] Hage-Hassan | On the hydrogen wave function in Momentum-space, Clifford Algebra and the Generating function of Gegenbauer polynomials[END_REF] [ ]

0 0 ) , , ( ! 1 ) , , ( ! 1 =
+ + + + - - = + - - - 1 2 1 3 2 1 1 1 2 1 2 2 1 )] ( ) ( [ ) 1 ( ) 2 1 ( ) 1 ( ) ( n N l l n N l l n n l N l l x C x C z z zx z z (3.12)
And the formula [START_REF] Erdelyi | Heigher transcendal functions[END_REF] )] (

C x C x C n n n n α α α α α - + - + - - = + ) ( )[ 1 ( ) ( ) ( ) ( 1 ) ( 1 ) 1 ( 1 x
(3.13) We find that the expression (3.11) may be written as This hyperphysical function may be written in term of the components of the vector defined by the angles ) , , ,

n k K - + - = θ θ θ π K r 2 2 ( ' 1 1 k
and the length This vector may be derived from K r by rotation about the vector perpendicular to the space N E with angle of rotation

θ π + - 2 .
We can also determine the representation {p} by this method if the potential has an additional term 2 / r A .

Appendix: a remark on the phase factor

We show by an example that inattention error are repeated in many textbooks of quantum mechanics [START_REF] Chpolski | Physique atomique" Edition Mir[END_REF][START_REF] Landau | Mécanique Quantique[END_REF][START_REF] Greiner | Quantum mechanics (An Introduction)[END_REF] and work [START_REF] Aquilanti | The d-dimensional hydrogen atom: Hyperspherical harmonics as momentum space orbital and alternative sturmian basis sets[END_REF][START_REF] Dehesa | Fisher information of D-dimensional hydrogenic systems in position and momentum spaces[END_REF] on the phase factor of the basis of the representation {p}of the harmonic oscillator and the hydrogen atom.

In quantum mechanics the Schrödinger equation of harmonic oscillator is

) ( ) ( x E x H ψ ψ = (4.1) ) ( 2 1 2 2 2 2 x m p m H x ω + = h i p x x = ] , [ (4.2)
4.1 In the representation {x} we find

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = - + x m H e n m x n x m n n h h h ω π ω ψ ω 2 2 2 1 2 ) ! 2 ( ) ( (4.3)
The well known generating function is

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = - + p m H e n m p n p m n n h h h ω ω π ψ ω 1 ) ! 2 1 ( ) ( ' 2 2 1 2 1 2 (4.4)

4.2.1

The phase factor is can not be determined in this case it is chosen equal to one.

.2 the direct calculation of the representation {p} by the Fourier transformation

Using the generating function gives So we deduce that we are not free to choice the phase factors of representations {x} and {p} in an independent manner.

5 )

 5 This result is incorrect.

4. 4

 4 But if we perform the calculation with )

Finally we derive the wave function in the momentum representation

-- (3.14)


Meng Wu

Xuhui Wang

Boniface Nkonga

Bernard Mourrain

Gang Xu

Qian Ni

Yuan Liu

Flux-aligned Quad Mesh Generation in Magnetohydrodynamic Simulation

Flux-aligned mesh generation plays an important role in the magnetohydrodynamic (MHD) simulation of Tokamak plasmas.In this paper, we present the existence theory of flux-aligned meshes by generalized Morse theory to the situation in Tokamak simulation. A high-order algorithm is developed to validate the theory by generating flux-aligned quad meshes with the same topologies as the typical flux contours in JOREK for Tokamak configuration MAST.

Introduction

Mesh generation has been investigated for many years and plays a fundamental role in many fields, such as geometric processing and the finite element method. There is a rich literature on two-dimensional mesh generation. A thorough survey can be found in [Bommes et al.(2012)] [Armstrong et al.(2015)] and other recent works [Chen et al.(2019)] [ Lei et al.(2020)] [Xiao et al.(2020)]. For quad-mesh generation, there is a specified requirement in Tokamak experiments [Guillard et al.(2018)]. In the following, we briefly present the mesh generation background in Tokamak simulation.

The understanding of magnetohydrodynamic (MHD) instability is essential for the optimization of magnetically confined plasma in Tokamak. Numerical simulation plays an important role in the investigation of the nonlinear behavior of these instabilities and the interpretation of experimental observations [START_REF] Czarny | [END_REF]]. MHD instability in Tokamak plasma is believed to be the following quasi bidimensional [Guillard et al.(2018)]: the fluctuation scales in the direction parallel to the magnetic field are typically 3 to 4 orders of magnitude larger than those in the transverse plane. It is important to consider a sufficiently fine spatial resolution in the transverse direction that makes the use of grids/meshes, which are adapted to the magnetic flux surfaces in each 2D poloidal plane particularly attractive, i.e., fluxaligned quad mesh generation is important for further MHD simulation in Tokamak plasma. This has led to a number of state-of-the-art numerical codes in various fieldor flux-aligned grid techniques [Schneider et al.(2006)] [Dudson and Leddy(2017)] [Guillard et al.(2018)] [START_REF] Tamain | [END_REF]] [Chang et al.(2009)] [Stegmeir et al.(2018)] [Pamela et al.(2019)]. However, research on the existence of flux-aligned blockstructured meshes is rare. Theoretical analysis helps to improve the robustness of existing codes. Classical Morse theory [Spivak and Wells(1963)] is a natural way to build a flux-aligned block-structured mesh generation algorithm [Guillard et al.(2018)]. Regarding the application of MHD simulation, generally, a flux function is not a Morse function.

(1) Regarding the distribution of critical points, Morse functions are functions with special isolated critical points. The shape of the poloidal cross-section of the averaged flux surfaces in the poloidal planes is governed by magnetic equilibrium, i.e., given by solving the Grad-Shafranov equation. The distribution of isolated critical points of the Grad-Shafranov equation solution is more complex than that of the Morse functions [START_REF] Alessandrini | [END_REF]] [Arango and Gómez(2011)] [Cheeger et al.(2015)].

(2) The smoothness of Morse functions is at least C 2 on physical domains. In MHD simulation, the numerical solutions of the Grad-Shafranov equation are C 1 spline surfaces [START_REF] Jorek-Team | JOREK-Team, Jorek is a non-linear extended mhd code for toroidal x-point geometries[END_REF]] [START_REF] Czarny | [END_REF]]. A flux is a C 1 spline surface defined over a mesh with an irregular topology, such as the structure illustrated in Figure 1.

In this paper, we assume that the input of flux-aligned quad mesh generation is Figure 1: A quad mesh with an irregular topology at the X-point.

a physical domain and a flux surface with a set of isocurves. Flux surfaces F (x, y)s are generally C 1 functions with C 2 piecewise on a physical domain Ω. Numerical solutions of the MHD simulation [START_REF] Czarny | [END_REF]] are in this class of flux functions. In addition, following [Guillard et al.(2018)], we suppose F (x, y)| ∂Ω is a constant. The contributions of this paper are as follows:

(1) We present the existence theory of a flux-aligned quad mesh with any given isocurves.

(2) We design a high-order algorithm to verify the flux-aligned quad mesh generation theory.

(3) Numerical experiments on generating flux-aligned quad meshes of interest by simulating MHD in the Tokamak are given.

The structure of the remainder of this paper is listed as follows. In Section 2, we discuss how to decompose physical domains such that there is a flux-aligned quad mesh with any given isocurves. In Section 3, we present a framework for a high-order algorithm to generate flux-aligned quad meshes. In Section 4, numerical experiments on the MHD simulation of Tokamak plasma are conducted in Section 2. Then, we conclude this paper in Section 5 by indicating directions for future work, and the proofs of lemmas are collected as an appendix in Section 6.

Flux-aligned quad meshes existence theory

In this section, we consider a flux-aligned parameterization, which is the continuous form of flux-aligned quad meshes. This concept contains all the possible of isocurves. It allows us to follow any given level set, which is concretely defined as follows.

Definition 2.1. P is called a flux-aligned parameterization with respect to a flux function F (x, y), if P : Σ -→ Ω such that the s-curves of P(s, t) are iso-curves of F (x, y). s-curves and t-curves are perpendicular to each other, where an s(or t)-curve is P(s, t), s is a constant and

Σ = [s 0 , s 1 ] × [t 0 , t 1 ].
When we generate a flux-aligned quad mesh with a given iso-curve set, suitable s-parameters and t-parameters are formed according to the isocurves. Thus, we consider the existence of a flux-aligned parameterization that is well defined along common edges between adjacent subdomains. Generally, a flux-aligned parameterization does not exist globally. In Section 2.1, we present how to decompose a physical domain into subdomains and demonstrate the existence of a flux-aligned parameterization on each subdomain in Section 2.2.

Critical point-based domain decomposition

In this section, a domain decomposition approach is presented to generate a fluxaligned parameterization on a physical domain. The necessity of domain decomposition is explained in Example 2.1. Then, the details of the decomposition approach are presented.

Example 2.1.

(a)

F (x, y) = (2 -x 2 -y 2 )(1 -x 2 )(1 -y 2 ) is a flux function, where (x, y) ∈ Ω, Ω = {(x, y) ∈ R 2 |0.5 ≤ F (x, y) ≤ 1.8} shown in
3: F (x, y) = xy(1 -x 2 )(1 -y 2 ) in Example 2.1(b)
In the following, the critical point-based domain decomposition method is presented.

Definition 2.2. Let F (x, y) be a C 1 function. p ∈ Ω is called a critical point of F (x, y) if ∂F ∂x | p = ∂F ∂y | p = 0.
In Example 2.1 (a), there is no critical point. In Example 2.1 (b), the central point (0, 0) is a critical point. We note that there is no flux-aligned parameterization. In the following, flux functions are always supposed to be C 1 and C 2 piecewise, which includes numerical solutions of MHD simulation in Tokamaks by JOREK [START_REF] Jorek-Team | JOREK-Team, Jorek is a non-linear extended mhd code for toroidal x-point geometries[END_REF]] [START_REF] Czarny | [END_REF]].

Definition 2.3. On a physical domain Ω, let A = {p i } p i ∈I be all the critical points of F (x, y) and

V = {F (p i )|p i ∈ A}.
Ω can be decomposed by curves

C = {(x, y) : F (x, y) = c i , c i ∈ V} ∪ {∂Ω},
into connected components called subdomains. This type of decomposition is called critical point-based domain decomposition.

Remark 2.1. Finiteness Hypothesis: Regarding the computation, there is a finiteness hypothesis. We list the structures in the decomposition process, and we assume that their numbers are finite in this paper.

1. Connected components in C, i.e., curves and isolated points in C;

2. Subdomains of critical point-based domain decomposition;

3. Nonsmooth points of the curves in C.

In the following, we illustrate the critical point-based decompositions of Example 2.1(b) in Example 2.2 and explain the subdomains, isolated points in C and nonsmooth points of curves in C.

Example 2.2. Let F (x, y) = xy(1 -x 2 )(1 -y 2 ), (x, y) ∈ [-1, 1] × [-1, 1].
A ={(0, 0), (0, 1), (0, -1), (

√ 3 3 , √ 3 3 ), ( √ 3 3 , - √ 3 3 ), (- √ 3 3 , √ 3 3 ), (- √ 3 3 , - √ 3 3 ), (0, 1), (0, -1), (1, 1), (-1, 1), (1, -1), (-1, -1)}; V ={0, 4 27 , - 4 27 }, then C = {(x, y) : x = 0, or y = 0, or x = 1, or x = -1, or y = 1, or y = -1} ∪ {( √ 3 3 , √ 3 3 ), ( √ 3 3 , - √ 3 3 ), (- √ 3 3 , √ 3 3 ), (- √ 3 3 , - √ 3 3 )}.
In Figure 4(a), the critical point-based decomposition is obtained by removing C from Ω. Notably, there are

(1) 4 isolated points:

( √ 3 3 , √ 3 3 ), (- √ 3 3 , √ 3 3 ), (- √ 3 3 , √ 3
3 ), and (-

√ 3 3 , - √ 3 3 );
(2) 9 nonsmooth points: (1, 1), (1, -1), (-1, 1), (-1, -1), (0, 1), (0, -1), (0, 0), (-1, 0), and(1, 0); (3) Ω is divided into 4 subdomains denoted as Ω 1 , Ω 2 , Ω 3 , and Ω 4 . In the next section, the theoretical analysis of the existence of a flux-aligned parameterization on each subdomain is presented.

Existence of a flux-aligned parameterization on each subdomain

In the following, we list several lemmas for constructing flux-aligned parameterizations in Theorem 2.6.

(1) Based on the implicit function theorem, we can obtain the following lemma.

Lemma 2.1. Let c not be a critical value of F (x, y), i.e., c / ∈ V. Then, C = {(x, y) ∈ Ω : F (x, y) = c} consists of a set of C 1 loops without self-intersection.

(2) For a curve in Lemma 2.1, there is a parameterization.

Lemma 2.2. Let C be a curve of C defined in Lemma 2.1. Then, there is a param- eterization ϕ of C, ϕ :[s 0 , s 1 ] -→ C s → (x, y), such that dϕ ds = (-F y , F x ) ||∇F || 2 , or dϕ ds = (F y , -F x ) ||∇F || 2 . ( 1
)
(3) For the values of F (x, y) on the boundaries of a subdomain Ω i , we have the following lemma.

Lemma 2.3. There are only two different values of F (x, y) on ∂Ω i and they are a boundary value or a critical value of F (x, y).

For instance, in Example 2.2, there are four subdomains {Ω i : i = 1, 2, 3, 4}, where

F | ∂Ω 1 = {0, - 4 27 }, F | ∂Ω 2 = {0, 4 27 }, F | ∂Ω 3 = {0, 4 27 }, F | ∂Ω 4 = {0, - 4 27 }.
Here, 0, 4 27 , and -4 27 are the critical values of F (x, y); and 0 is also the boundary value of F (x, y). Based on Lemma 2.3, we present the following definition.

Definition 2.4. Let c i,1 , c i,2 (c i,1 < c i,2 ) be the values of F (x, y) on ∂Ω i . Define Ω i,ϵ := {(x, y) ∈ Ω i : c i,1 + ϵ ≤ F (x, y) ≤ c i,2 -ϵ} where ϵ is small enough such that c i,1 + ϵ < c i,2 -ϵ.
By Definition 2.4, Ω i,ϵ ⊂ Ω i . The following gives an example of Ω i,ϵ .

Example 2.3. Let us consider F (x, y) in Example 2.2. F | ∂Ω 1 = {-4/27, 0}. Ω 1,ϵ = {(x, y) ∈ Ω 1 : - 4 27 + 1 1000 ≤ F (x, y) ≤ 0 - 1 1000 },
where ϵ = 1 1000

. Ω 1,ϵ is shown in Figure 4(b), where we can find that (i) there is no critical point in Ω 1,ϵ ;

(ii) Ω 1,ϵ is homeomorphic to an annulus, and its boundaries consist of two closed curves defined by F (x, y) = -4 27 + 1 1000 and F (x, y) = 0 -1 1000 .

(4) This phenomenon can be explained by the following lemma.

Lemma 2.4. Ω i,ϵ is homeomorphic to an annulus, and its boundary consists of two closed curves defined by F (x, y) = c i,1 + ϵ and F (x, y) = c i,2 -ϵ, where c i,1 , c i,2 and ϵ are defined in Definition 2.4.

The proofs of Lemmas 2.2, 2.3 and 2.4 are given in the Appendix. Now, we prepare to prove the existence of flux-aligned parameterizations on a subdomain. First, we prove that there is a flux-aligned parameterization on Ω i,ϵ in Lemma 2.5, and then a flux-aligned parameterization is constructed on Ω i in Theorem 2.6.

Lemma 2.5. For Ω i,ϵ , there is a flux-aligned parameterization P i,ϵ

P i,ϵ :Σ i,ϵ -→ Ω i,ϵ , where Σ i,ϵ = [s 0 , s 1 ] × [0, T p ]. Proof. Based on Lemma 2.4, ∂Ω i,ϵ = C 1 ∪C 2 , where C 1 and C 2 are boundary curves and F (C 1 ) = c i,1 + ϵ, F (C 2 ) = c i,2 -ϵ, where c i,1 + ϵ < c i,2 -ϵ. By Lemma 2.2, there is a parameterization of C 1 ϕ : [s 0 , s 1 ] -→ C 1 , such that dϕ ds = (-F y , F x ) ||∇F || 2 . Moreover, ∀p s ∈ C 1 , consider    dψ dt = ∇F ||∇F || 2 , ψ(0) = p s .
(2) According to the Picard-Lindelöf theorem [Teschl(2012)], there is a unique solution

ψ ps : [0, t ps ] -→ R 2 that satisfies (2), and ψ ps (t ps ) is a boundary point of Ω i,ϵ , F (ψ ps (t ps )) > F (ψ ps (0)) because ∇F ̸ = 0, i.e., ψ ps (t ps ) ∈ C 2 . Define P i,ϵ (s, t) = ψ ϕ(s) (t), (s, t) ∈ Σ i,ϵ (3)
where

Σ i,ϵ = ∪ s∈[s 0 ,s 1 ] {s} × [0, t ps ] ⊂ R 2 .
According to the definition of ψ,

∂P i,ϵ ∂t = ∇F ||∇F || 2 .
Moreover, for a fixed s,

∂F (P i,ϵ (s, t)) ∂t = ∇F • ∂P i,ϵ ∂t = 1.
Thus,

F (P i,ϵ (s, t)) = t + h(s), (4)
where h(s) is a function of s. According to the definition of P i,ϵ (s, t),

P i,ϵ (s, 0) = ψ ϕ(s) (0) = ϕ(s).
Compared to (4),

F (P i,ϵ (s, 0)) = F (C 1 ) = c i,1 + ϵ, that is, h(s) = c i,1 + ϵ, and
F (P i,ϵ (s, t)) = t + c i,1 + ϵ (5)
i.e., for a fixed t 0 , F (P i,ϵ (s, t 0 )) is a const t 0 + c i,1 + ϵ. According to this result,

∂F (P i,ϵ ) ∂s = ∇F • ∂P i,ϵ ∂s = 0.
Thus,

∂P i,ϵ ∂t ⊥ ∂P i,ϵ ∂s . (6)
According to (5),

F (P i,ϵ (s, t ps )) = t ps + c i,1 + ϵ.
Moreover, P i,ϵ (s, t ps ) = ψ ϕ(s) (t ps ) ∈ C 2 , i.e.,

F (P i,ϵ (s, t ps )) = c i,2 -ϵ. Thus ∀s ∈ [s 0 , s 1 ], t ps = c i,2 -c i,1 -2ϵ := T p ,
and Σ i,ϵ = [s 0 , s 1 ] × [0, T p ]. Thus, based on ( 5) and ( 6), P i,ϵ is a flux-aligned parameterization of Ω i,ϵ .

Remark 2.2. In the proof of Lemma 2.5, C 1 can be replaced by any isocurve on Ω i,ϵ . For instance, take c = (c i,1 + c i,2 )/2 and c i,1 + ϵ < c < c i,2 -ϵ, where ϵ (> 0) is sufficiently small. Considering the curve C ⊂ Ω i,ϵ such that F (C) = c, according to Lemma 2.2, there is a unique parameterization of C such that

ϕ : [s 0 , s 1 ] -→ C and dϕ ds = (-F y , F x ) ||∇F || 2 .
ϕ can be instead the one in Lemma 2.5, and there are constants T 0 , T 1 > 0 such that

P i,ϵ : [s 0 , s 1 ] × [-T 0 , T 1 ] -→ Ω i,ϵ ,
and P i,ϵ (s, 0) = ϕ(s).

We are ready to prove the existence of a flux-aligned parameterization on a subdomain.

Theorem 2.6. There is a flux-aligned parameterization P i ,

P i :Σ i -→ Ω i (s, t) → (x, y),
where there are s 0 , s 1 , t 0 , and t 1 (> 0) such that

Σ i = [s 0 , s 1 ] × [t 0 , t 1 ].
Proof. According to the definition of Ω i,ϵ , for ϵ > 0, Ω i,ϵ ⊂ Ω i and

Ω i = ∪ ϵ>0 Ω i,ϵ .
For

ϵ 1 > ϵ 2 > 0, Ω i,ϵ 1 ⊂ Ω i,ϵ 2 . Select ϵ 0 > 0 such that ∅ ̸ = Ω i,ϵ 0 ⊂ Ω i . According to Lemma 2.4, ∂Ω i,ϵ 0 = C 1 ∪ C 2 . Suppose ∂Ω i = C 1 ∪ C 2 and F (C i ) = c i (i = 1, 2). Take c = (c 1 + c 2 )/2, where C = {(x, y) ∈ Ω i : F (x, y) = c}. Suppose that ϵ n = ϵ 0 /2 n , where n = 0, 1, • • • . Then, C ⊂ Ω i,ϵn . Based on Lemma 2.5, P i,ϵn (s, t) : [s 0 , s 1 ] × [-t 1,n , t 2,n ] -→ Ω i,ϵn . Since Ω i,ϵn ⊂ Ω i,ϵ n+1 , t 1,n+1 ≥ t 1,n , t 2,n+1 ≥ t 2,n . Let t 1 = lim n→∞ t 1,n , t 2 = lim n→∞ t 2,n . Define P i : Σ i -→ Ω i as P i (s, t) = P i,ϵn (s, t), when (s, t) ∈ [s 0 , s 1 ] × [-t 1,n , t 2,n ] ⊂ Σ i , where Σ i = [s 0 , s 1 ] × (-t 1 , t 2 ) ⊂ R 2 .
According to the uniqueness of Picard-Lindelöf theorem, P i,ϵ n+1 | Ω i,ϵn = P i,ϵn . Thus P i is well-defined. Moreover, according to Lemma 2.5,

∂P i ∂t = ∇F ||∇F || 2 , ∂P i ∂s ⊥ ∂P i ∂t ,
In addition, for t ∈ (-t 1 , t 2 ), F (P i (s, t 0 )) is a constant, i.e., P i (s, t 0 ) is a flux-aligned parameterization of Ω i .

Remark 2.3. (1) In Theorem 2.6, the regularity of flux functions is C 1 with C 2 piecewise, which is a weaker requirement of the regularity of Morse functions. This type of flux function includes numerical solutions of the MHD simulation of Tokamak plasma. The flux-aligned quad mesh generation algorithm [Guillard et al.(2018)] is a special case presented here.

(2) Consider the boundary of Ω i in Theorem 2.6. Although ∂Ω i,ϵ are smooth curves, there may be nonsmooth points and isolated points on ∂Ω i . For example, ∂Ω 2 has 4 nonsmooth points (0, 0), (1, 0), (0, 1), and (1, 1) and an isolated point ( √ 3/3, √ 3/3) in Figure 4(a). Flux-aligned parameterization degenerates at nonsmooth points and isolated points. This degeneration will cause singularity of the flux-aligned quad-mesh generation. Thus, it is reasonable that there are algorithms to eliminate degeneration. In fact, these codes have been designed. For example, edge plasma codes (SOLPS, FBGKI, TOKAM3X, and SOLEDGE) and GYSELA exclude a central part of the core plasma around the magnetic axis. The boundary of this central part will be considered a constant flux surface.

In the following, we present the methods to eliminate degeneration at isolated points and nonsmooth points based on the present codes.

Isolated Point Elimination:

The flux-aligned parameterizations degenerate at isolated points. That is, for this case, following what [START_REF] Huysmans | [END_REF]] has done, we remove an isolated point p I using an isocurve with F (x, y) = c + ϵ, where c = F (p I ) and ϵ is a small constant. We adopt the approach to eliminate degeneration at isolated points in Figure 5(b).

Nonsmooth Point Elimination: We introduce gradient integral curves by taking nonsmooth points as their initial points to split subdomains into regular subdomians. We adopt the approach to eliminate degeneration at the nonsmooth points shown in Figure 5(c).

Definition 2.5. The subdomain with F (x, y) = c + ϵ as its boundary is called a central subdomain. In addition to central subdomains, others are called regular subdomains. Remark 2.4. A compatible flux-aligned quad-mesh on a global computational domain can be obtained by choosing appropriate parameters on adjacent subdomains.

Framework of the flux-aligned quad-mesh generation algorithm

In this section, the framework of the flux-aligned quad mesh generation algorithm is presented in Section 3.1, and an example is illustrated in Section 3.2.

Algorithm framework

The flux-aligned quad mesh generation algorithm is divided into two subalgorithms.

The first subalgorithm is used to decompose a given physical domain into subdomians. The second subalgorithm is used to generate an order of these subdomains. Following this order, a conformal quad mesh is obtained. In the following, we present the framework of this algorithm.

Algorithm: Flux-aligned quad-mesh generation algorithm

Input: A physical domain Ω and a flux function F (x, y) Output: A flux-aligned quad-mesh defined on Ω Subalgorithm 1: Domain decomposition algorithm Subalgorithm 2: Quad mesh generation algorithm

The flux-aligned quad mesh generation algorithm is presented in the following. To execute this algorithm, we need to generate two basic curves.

1. Gradient Integral Curve Generation: Computing gradient integral curves, it is defined by

dX dt = ∇F ||∇F || 2 , X(0) = p 0 , (7)
where p 0 ∈ Ω i .

For a regular subdomain, choose p j on an isocurve boundary as an initial value, where j = 1, 2, • • • , n i . We can solve ( 7) and obtain the gradient integral curves 6(a).

X 1 (t), X 2 (t), • • • , X n i (t) shown in Figure

Isocurve Generation:

Choose q k on a gradient integral curve boundary X 1 (t) (or X n i (t)). Based on X 1 (t) (or X n i (t)) and q k , we compute the parameter t k for each q k , where k = 1, 2, • • • , m i . Then, X j (t k ) and X j+1 (t k ) are connected, where

j = 1, 2, • • • , n i -1 and k = 1, 2, • • • , m i as shown in
: Ω r 1 , Ω r 2 , • • • , Ω r n ; central subdomains: Ω c 1 , Ω c 2 , • • • , Ω c
: Ω r 1 , Ω r 2 , • • • , Ω r n ; central subdomains: Ω c 1 , Ω c 2 , • • • , Ω c m .
Output: A flux-aligned quad-mesh defined on Ω. 1: Ordering subdomains: Order regular subdomains in two ways (a gradient-induced graph and an iso-curve induced graph), and then order central subdomains. 2: Flux-aligned quad mesh generation on regular subdomains: Based on the gradient-induced graph and isocurves induced graph, we generate gradient integal curves and isocurves on each subdomain. 3: Quad mesh generation on central subdomains: The algorithm was developed in [Wu et al.(2017)].

Thus, it can induce different flux aligned quad meshes. The mesh distribution of flux aligned quad meshes is determined by flux functions and the initial points p j and q k .

Example of generating flux-aligned quad-meshes

In this section, an example is used to explain the flux-aligned quad-mesh generation algorithm.

Example 3.1. Subalgorithm 2. Quad mesh generation algorithm:

Input: Ω = [-1, 1] × [-1, 1], F (x, y) = xy(1 -x 2 )(1 -y 2 ).
(1) Ordering: In Figure 7 (b), we assign an index to each subdomain.

Gradient induced Graph (G-Graph): Following the gradient direction, the relationship graph is presented as 3 -→ 9; 2 -→ 8; 7 -→ 13; 10 -→ 16; 1; 4; 5; 6; 12; 11; 15; 14.

Isocurves induced Graph (I-Graph): Following the iso-curve direction, the relationship graph is presented as (2) Flux-aligned quad-mesh generation on regular subdomains: For each subgraph in the G-Graph, initial points are given following the gradient direction. The gradient integral curves are computed. For instance, with the given initial points on an isocurve edge, curves can be computed, as shown in Figure 8 (3) Quad mesh generation on central subdomains: For each central subdomain, initial points are given by the endpoints of the gradient integral curves. Take the 17th subdomain as an example. After generating aligned quad-meshes on regular subdomains, there are several points on its boundary, as shown in Figure 9(a). Using the algorithm in [Wu et al.(2017)], as shown in Figure 9(b), a quad mesh can be generated and it is compatible with boundary points, i.e., the quad-mesh is conformal. Please see an enlarged view of the central subdomain in Figure 10. In Figure 9(c), all the quad mesh generation central subdomains are generated.

1 -→ 2 -→ 3 -→ 4; 5 -→ 6 -→ 7 -→ 8; 9 -→ 10 -→ 11 -→ 12; 13 -→ 14 -→ 15 -→ 16. -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 (a) -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 (b) -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 (c) -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 (d)

Runge-Kutta based algorithm and numerical experiments

In this section, to validate the theory in this paper, a specific algorithm is developed to solve gradient integral curves in Section 4.1. This algorithm is based on the high-order Runge-Kutta method to solve (7) numerically. Then, in Section 4.2, this algorithm is applied to generate flux-aligned quad meshes using the generation algorithm framework in Section 3.

and obtain a new point p ′ 0 ∈ Ω i such that ∇F | p ′ 0 ̸ = 0. According to Theorem 4.1, p ′ 0 is on X j . Thus it is used to instead of p 0 as an initial value and solves

dX j dt = ∇F ||∇F || 2 , X j (0) = p ′ 0 ,

Numerical Experiments

In In this paper, by introducing flux-aligned parameterizations, we develop a theory to generate a flux-aligned quad mesh with any given isocurves for application to MHD simulation and design a high-order algorithm to test the theory via numerical experiments on the MHD simulation of Tokamak plasma. In this paper, we demonstrate the existence of flux-aligned quad meshes. It is a theoretical supplement for the flux-aligned quad mesh generation algorithm in [Guillard et al.(2018)]. This research will contribute to improving the robustness of existing codes by knowing more information about flux-aligned parameterizations. Using the theory of the flux-aligned quad mesh generation algorithm, eliminating isolated and nonsmooth points cannot be avoided to generate nondegenerate analysis-suitable quad meshes. However, these tricks have effects on the robustness of the current algorithms. In the future, we will research a robust and efficient algorithm for generating flux-aligned quad meshes by modifying a flux function locally to avoid eliminating isolated and nonsmooth points.

6 Appendix Lemma 2.2 Proof:

Based on Lemma 2.1, C is a regular loop without self-intersection, where C = C 0 ∪C 1 • • •∪C n such that C n ∩C 0 = p n,0 and C i ∩C i+1 = p i,i+1 . C i (i = 1, 2, • • • , n-1) are C 2 -curves and C 1 -joint at p i,i+1 and p n,0 . On C i , according to the Picard-Lindelöf Theorem, there is a unique solution First, we prove that the number of critical values and the boundary value is more than 2 in (1). Then, in (2), we prove the number is less than 3.

ϕ i,i+1 : [s i , s i+1 ] -→ R 2 such that dϕ i,i+1 ds = (-F y , F x ) ||∇F || 2 , ϕ i,i+1 (s i ) = p i,i+1 . (8
(1). Ωi is a bounded closed set, and F reaches C 1 continuous on Ωi . Then, on Ωi , F is reached the maximum and minimum values denoted as M and m, respectively, where M > m because F is not a constant function on Ω i and Ω i ⊂ Ωi . The maximum and minimum values are the critical values or the boundary values. Thus, the number of critical values and the boundary values is more than 2. Moreover, there is no critical value on Ω i ; then, these critical values and boundary values are reached at ∂Ω i because Ωi = Ω i ∪ ∂Ω i .

(2). In this part, we will prove that the number is less than 3 by contradiction. Suppose there are at least 3 critical values or boundary values of Ω on Ωi denoted as c 1 , c 2 and c 3 . Without loss of generality, c 1 < c 2 < c 3 . By the analysis in (1), there are p 1 , p 3 ∈ ∂Ω i such that F (p 1 ) = c 1 and F (p 3 ) = c 3 . Moreover, F is a continuous function on Ωi = Ω i ∪ ∂Ω i . Thus, for ϵ > 0, there are p 1,δ , p 3,δ ∈ Ω i such that

F (p 1,δ ) < c 1 + ϵ < c 2 < c 3 -ϵ < F (p 3,δ ). ( 9
)
Ω i is a connected set; then, it is a path-connected set. Thus, there is a curve Γ ⊂ Ω i whose endpoints are p 1,δ and p 3,δ . Considering F | Γ , it is continuous on Γ. Thus, there is a p 2 ∈ Γ ⊂ Ω i such that F (p 2 ) = c 2 . There is a contradiction between p 2 ∈ Γ and the definition of Ωi .

Lemma 2.4 Proof:

It is easy to prove that Ω i,ϵ is compact. Because there is no critical point in Ω i,ϵ , based on Theorem 3.1 in [Spivak and Wells(1963)], Ω i,ϵ is homeomorphic to an annulus or a cup of some annuluses. Moreover, ∇F | Ω i ̸ = 0, thus Ω i,ϵ is homeomorphic to an annulus. ∂Ω i,ϵ are consist of two loops with F (x, y) = c i,1 + ϵ and F (x, y) = c i,2 + ϵ.

  Figure 2(a). In Figure 2(b), the isocurves can be contained by a flux-aligned parameterization.

Figure 2 :

 2 Figure 2: Isocurves of F (x, y) and Ω in Example 2.1(a)

Figure

  Figure 3: F (x, y) = xy(1 -x 2 )(1 -y 2 ) in Example 2.1(b)

  Figure 4: (a): Critical point-based decomposition in Example 2.2; (b): Ω 1,ϵ (in green)

Figure 5 :

 5 Figure 5: Elimination of isolated points and nonsmooth points on [-1, 0] × [-1, 0] with xy(1 -x 2 )(1 -y 2 ) as the flux function

  Figure 6(b).

  j

Figure 6 :

 6 Figure 6: The gradient integral curves and a flux-aligned quad-mesh on a regular subdomain Ω i The first and second subalgorithms are presented in the following. The details of the first subalgorithm are explained in Section 2.1.

m 1 :

 1 The critical point-based decomposition 2: Isolated point elimination 3: Nonsmooth point elimination Remark 3.1. The mesh generation speed and distribution are important for application in MHD simulation. In this paper, aligned parameterizations are continuous.Subalgorithm 2: Quad mesh generation algorithm Input: F (x, y); regular subdomains

Subalgorithm 1 .

 1 Domain decomposition algorithm: (1) Critical point-based decomposition: Based on the computation in Example 2.2, we present the critical point-based decomposition and obtain the 4 connected open set Ω 1 , Ω 2 , Ω 3 and Ω 4 shown in Figure 4(a).

Figure 7 :

 7 Figure 7: Isolated point elimination (a), nonsmooth point elimination and the indices of subdomains (b) (2) Isolated point elimination: There are 4 isolated points (-√ 3/3, √ 3/3), ( √ 3/3, √ 3/3), (-√ 3/3, -√ 3/3), and ( √ 3/3, -√ 3/3). For an isolated point, we remove this point

Figure 8 :

 8 Figure 8: (a) 3rd and 9th subdomains; (b) initial points and gradient integral curves; (c) gradient integral curves; (d) for each subgraph in the I-Graph, initial points are given following the isocurves direction. Iso-curves are computed.

  (b), For each subgraph, we use the same method to generate integral curves, as shown in Figure8(c). Then,

Figure 9 :

 9 Figure 9: A conformal quad-mesh generated on a central subdomain

  this section, flux-aligned quad-meshes are generated. Their contours of single-null discharge, symmetric double-null discharge and nonsymmetric double-null discharge have the same topology as the typical flux contours in the MAST device in Figure 11. For simulation, the quad-mesh is aligned to equilibrium flux surfaces for the three regions of the core, the scrape-off layer (SOL) and the private region. Alignment along open flux surfaces in the SOL is important to accurately treat the fast parallel transport of energy along the magnetic field lines. Simulation of single-null discharge (SNDs) in MAST with the single-null contours of JOREK in Figure 11 (a). The double-null quad-meshes of JOREK are generated by poloidal flux contours for the three standard double-null cases: the upper double-null in Figure 11(b), the symmetric double-null in Figure 11(c) and the lower double-null in Figure 11(d).

Figure 11 :

 11 Figure 11: Typical flux contours for single-null discharge and double-null discharges in the MAST device [Pamela et al.(2013)].

Figure 12 :

 12 Figure 12: A single-null discharge flux contour experiment

  x , F y ) • (-F y , F x )/||∇F || 2 = 0. Thus ϕ i,i+1 ([s i , s i+1 ]) is a part of C. Then we can define ϕ : [s 0 , s n ] -→ C, by joining ϕ| [s i ,s i+1 ] = ϕ i,i+1together. We can consider dϕ ds = (F y , -F x ) ||∇F || 2 similarly. Lemma 2.3 Proof:

Runge-Kutta based gradient integral curve generation algorithm

If ∇F | p 0 ̸ = 0, the Runge-Kutta method can be used directly. Because of the initial points on ∂Ω i , ∇F | p 0 might be zero. There is a theorem obtained by reparameterization to solve this problem.

Theorem 4.1. Let X 1 and X 2 be curves in R 3 . They satisfy

Then X 1 and X 2 share the same locus with each other.

The Runge-Kutta based gradient curve generation algorithm is presented in the following.

(

using the Runge-Kutta method.

(2) If ∇F | p 0 = 0, solve dX ′ dt = ∇F, X ′ (0) = p 0 ,


M Arminjon
email: mayeul.arminjon@hmg.inpg.fr

On the relation Hamiltonian -wave equation, and on non-spreading solutions of Schrödinger's equation

Keywords: 03, 40, Kf -Waves and wave propagation: general mathematical aspects 03, 65, -w -Quantum mechanics

According to Schrödinger's ideas, classical dynamics of point particles should correspond to the « geometrical optics » limit of a linear wave equation, in just the same way as ray optics is the limit of wave optics. It is shown that the « geometrical optics » analogy leads to the correspondence between a classical Hamiltonian H and a linear wave equation in a natural and general way. In particular, the correspondence is unambiguous also in the case where H contains mixed terms involving momentum and position. This is obtained through a theory of the dispersion relations, which leads to properly define the group velocity and to enlighten its role. Using this latter notion, it is shown that, for a quite general class of potentials, « momentum states » can be defined in a physically more satisfying way than by assuming plane waves. These momentum states are solutions of the time-dependent Schrödinger equation and their amplitude functions move rigidly on a well-defined trajectory. In the case of a spatially uniform force field, such momentum states must have a singularity and the trajectory is defined by Newton's second law in the given force field.

Introduction

De Broglie found that the analogy between the variational principles which arise in geometrical optics and in Hamiltonian dynamics had its origin in a wave-particle duality valid for material particles as well as for electromagnetic radiation. Elaborating on de Broglie's ideas, Schrödinger proposed that classical dynamics of point particles should correspond to the « geometrical optics » approximation of a linear wave equation, in just the same way as ray optics is a limiting approximation of wave optics [1]. In quantum mechanics, the correspondence between a classical Hamiltonian and a quantum wave equation, first obtained by Schrödinger for the nonrelativistic Hamiltonian of oneor several mass particles, is introduced in a general axiomatic way.

The first part of the present work is an attempt to understand this correspondence in the light of the modern theory of (classical) waves. More precisely, the correspondence between a linear wave equation and a classical Hamiltonian (algebraic in the momentum p) will be analysed in the framework of the theory of dispersion relations in a linear medium. This theory is exposed e.g. by Whitham [START_REF] Whitham | Linear and Non-Linear Waves[END_REF], but some definitions and results will have to be extended here. It will be shown that the above correspondence is naturally and most generally justified under the assumption that the classical Hamiltonian should describe the trajectories corresponding to the « geometrical optics » limit of the considered wave equation. Thus, the wave equation deduced from a classical Hamiltonian is a priori no more « quantum » than is the d'Alembert equation for the classical electromagnetic potential. This correspondence is one-to-one also in the general case where the Hamiltonian contains mixed terms involving momentum p and position x (and time). Contrary to the usual elementary presentation of this notion, the dispersion relation relevant to the definition of the group velocity does not depend on a particular wave packet, but instead on the considered wave equation. The crucial result which justifies the general correspondence Hamiltonian-wave equation is that, with any linear wave equation, one may associate a Hamiltonian dynamical system that rules the motion of the wave vector; in this system, the relevant velocity is the group velocity and the Hamiltonian is none other than the dispersion associated with the wave equation. In the « geometrical optics » limit, the wave function obeys simultaneously the dispersion equation and the wave equation, which is not true for general solutions of the wave equation

In the second part of this paper, the foregoing results will be used to reflect on what could be an appropriate definition of a « momentum state » for a quantum object, if one considers that a quantum object is really defined by a time-dependent wave function obeying a precise wave equation. From this viewpoint, an obvious sufficient condition allowing one to say that « the momentum is well-defined » is that (i) the group velocity should be spatially uniform. This property is true for a plane wave in the case of the usual Schrödinger equation, of course, but it is shared with a plenty of different wave functions. There are additional properties which should be satisfied by the wave function of a « momentum state » for a single non-relativistic particle:

(ii) it should obey the time-dependent Schrödinger equation and, furthermore, (iii) it should be spatially limited in the minimum sense that the amplitude should vanish at infinity -a property obviously wrong for plane waves. It will be shown that, for a quite general spatial dependence of the potential, there indeed exist wave functions with just these properties. If the spatial dependence of the potential is unchanged with time, up to a translation and a linear function of the position, then the group velocity remains spatially uniform as the time goes. As a consequence, the amplitude function of any such wave function moves rigidly, thus, in particular, it does not spread out at all, and it follows a trajectory defined by integration of the group velocity. Non-spreading solutions of Schrödinger's equation have been found recently by Barashenkov & Rodrigues [3], but the amplitude functions of their solutions have a cylindrical symmetry, hence a such solution can hardly be interpreted as describing a moving quantum object localized in space. Moreover, in order to better understand the transition to classical mechanics, it is interesting to define particular momentum states, which may be called « classical momentum states », by imposing the additional condition that (iv) the wave function obeys the dispersion relation exactly. It will be shown that properties (i) to (iv), taken together, lead to the unavoidable conclusion that: (a) classical momentum states can exist only in so far as the spatial variation of the force field is negligible, i.e., the potential must be linear in x;

(b) in that case, they do exist, for whatever time-dependence of the force field;

(c) they must have a singularity, whose motion obeys Newton's second law.

In contrast, general momentum states do not necessarily have a singularity. We finish the paper by briefly commenting the meaning of these results in relation to the two different attempts, made by de Broglie and Schrödinger, to obtain a realistic interpretation of QM.

Dispersion relations and the correspondence Hamiltonian -wave equation

Dispersion equation and wave equation in a (classical) linear medium

As everyone knows, a general wave  may be defined as the product of a « smoothly varying » amplitude A(t, x) and an oscillating part ((t, x)), with  a periodic scalar function of the dimensionless real variable , the latter or « phase » being itself a function of the time t and the position x. The « position » x belongs to an N-dimensional configuration space M, which may be the physical space (N = 3). In physics, is usually the complex exponential: () = exp(i) (or its real part, cos), but other « wave profiles » do also occur. The notion of a smoothly varying amplitude means that the wave structure may be recognized: to the very least, the relative amplitude variation A/A should not exceed the order of  /T , with T the period of the wave profile . Now if we have a physical law for a given wave phenomenon, it will lead to a partial differential equation for the wave function . Let us assume that  is a scalar and that this equation is linear:

(2.1) P  a 0(X)  + a1 µ (X) Dµ + ... +   0     ... N n an µ 0 ...µ N (X) (D0) µ 0...(D N) µ N  = 0,
where X, with coordinates x µ (0    N), is the relevant point of the product « time » × configuration space (extended configuration space R×M), and Dµ = ,  is the partial derivative. Quite often, the order of the equation is n = 2:

(2.2) P  a 0(X)  + a1  (X) , + a2  (X) , , = 0.
Given the phase function (X ) of a general wave, one defines the wave covector K =  = (-, k), i.e. K = , . In the linear case, it is appropriate to consider a sinusoidal wave profile: ()= exp(i), hence , = iK. Let us consider a wave function  = A exp(i), with A constant, which is an « elementary wave » at the point X considered, i.e., such that K = 0 at point X. The necessary and sufficient condition for such wave function to obey Eq. (2.2) at point X is:

(2.3) (X, K)  X(K)  a 0(X) + ia1  (X) K  a2  (X) K K= 0,
the obvious generalization to Eq. (2.1) being (under the condition that all derivatives of the K 's, up to the order n -1, vanish at the point X considered):

(2.4) (X, K)  X(K)  a 0(X) + ia1 µ (X) K + ... + i n   0     ... N n
an µ 0 ...µ N (X) K0 µ 0 ...KN µ N = 0.

We shall name Eq. (2.3) (or (2.4)) the dispersion equation of Eq. (2.2) (or (2.1)). If one makes any coordinate change x '  = x '  (x  ) in Eq. (2.2), the coefficient a0(X) is left unchanged and the a1  (X) series transforms like a vector. However, the a2  (X) series transforms like a contravariant second-order tensor, if and only if one makes an « infinitesimally linear » coordinate change, i.e., if  2 x '  /x  x  = 0 at the point considered. Hence, the left-hand side of the dispersion equation

(2.
3) is an invariant scalar only under infinitesimally linear coordinate transformations.

Similarly, the condition K (X ) = 0, used to derive (2.3) from (2.2), is covariant only under infinitesimally linear coordinate transformations. This means that, at each point X of the extended configuration space R×M, a linear group of privileged coordinates must be available among the local coordinates valid in a neighborhood of X. Hence, R×M must be equipped with a more particular structure than just that of a differentiable manifold. At this point, the special role of the time coordinate does not appear compelling. A pseudo-Riemannian metric  on R×M is hence enough: we then select the set of the locally geodesic coordinate systems (LGCS) at X for [ (X ) = 0 for all ,  and ], two of which exchange indeed by an infinitesimally linear transformation. Thus, in order to evaluate the dispersion equation (2.3) at X, we can take any

LGCS and we get the polynomial function X of covector K; this function does not depend on the

LGCS. Hence, the function (of X and K, thus a function defined on the « cotangent bundle », T*(RM), to RM) is well-defined. In the sequel we shall need the « projection time » t as a preferred time coordinate (up to a constant factor: x 0 = t ).

Similarly, for an equation of order n  3, the dispersion equation (2.4) is an invariant only under coordinate changes whose all derivatives, up to the order n, are zero at the point X considered. The condition that all derivatives of the K 's, up to the order n - for a real wave equation, only odd derivatives, or only even derivatives, are allowed [START_REF] Whitham | Linear and Non-Linear Waves[END_REF]. However, one may mix odd and even derivatives, provided one accepts complex coefficients in the wave equation -as for Schrödinger's equation. Now we state the essential result of this Section: the correspondence between the linear operator P and the function  is one-to-one, in other words one may uniquely pass from the wave equation to the dispersion equation and conversely. This is undoubtedly true, since the data of either P or  is equivalent to the data of the set of the scalar functions a 0, a1  , a2  , and so on, of the point X in RM. The important point is that this result is true also in the general case with variable coefficients, in which a 0, a1  , a2  , ..., do depend on X. In the case with constant coefficients, the inverse correspondence, from (2.4) to (2.1), amounts to the substitution (2.5)

K    1 i D .
This remains true in the case with variable coefficients, provided one orders each monomial in the dispersion equation as in (2.3) or (2.4), i.e., « X before K ». This ordering is the natural ordering for a polynomial in K with coefficients that are functions of X. It corresponds to the ordering in Eq. (2.1) or (2.2), which also is the natural ordering for the general linear differential equation: the (variable) coefficient comes before the differentiation.

Group velocity and Hamiltonian motion

As mentioned above, we now need the preferred « projection » time t, with X = (t, x), and we seek to compute the frequency  = K0 as a function of the « spatial » part k of K:  = (k1, ..., kN) = (k). When extracting the real roots of the polynomial equationX(K) = 0, considered as an equation for the unknown K0 with the data k1, ..., kN, one may follow the different roots W1, ..., Wn (at most n) as functions of X. It is assumed in the following that one such particular root has been identified, and we have thus:

(2.6)

 = W(k1, ..., kN ; X ) = W(k ; X ) = W(k, x, t ),
which is called « the » dispersion relation, it being kept in mind that several different such relations may in general be extracted from the unique dispersion equation [START_REF] Whitham | Linear and Non-Linear Waves[END_REF]. Of course,W, which will be called the dispersion, is in general not a polynomial function of k at fixed X. Now let us assume that some wave function of the general form (2.7)

(t, x) = A(t, x) exp(i(t, x)),
is such that the corresponding wave covector obeys exactly the dispersion relation (2.6) (we do not assume here that  is an exact solution of the wave equation (2.1), and indeed this is in general incompatible with our assumption). One defines the group velocity for this wave function as the « spatial » vector with components (2.8)

C j = C j (k, x, t ) =     W k t j k x , ,
(Latin indices will be reserved for « spatial » components). This is the natural generalization [START_REF] Whitham | Linear and Non-Linear Waves[END_REF] of the usual notion of group velocity in a uniform medium. From the definitions K = ,  and  =  K0, and from the symmetry of the second derivatives of , one gets (2.9)

     x k t j j   0,     k x k x i j j i  ,
and since here  (t, x) = W(k(t, x), x, t ) by hypothese, Eq. (2.9)1 may be rewritten as [START_REF] Whitham | Linear and Non-Linear Waves[END_REF] (2.10)

      W x C k x k t j i j i j    0.
This is a hyperbolic equation for the (spatial) wave covector k, showing that k propagates with the group velocity. It is immediate to verify that it can be put in the following characteristic form

[2]:

(2.11)

d d k t W x j j     on the curve d d x t C W k j j j     .
Thus, the motion of the wave vector in a general nonuniform medium is governed by a

Hamiltonian dynamical system, the Hamiltonian being the dispersionW. To the author's knowledge, this crucial result has been overlooked in the literature on quantum mechanics.

We also point out that, to our knowledge, the present rigorous definition of the dispersion equation, and its rigorous correspondence with the wave equation, are new for the case of variable coefficients (i.e., a « non-uniform medium »): Whitham [START_REF] Whitham | Linear and Non-Linear Waves[END_REF] gives the definition and the result only in the case of a uniform medium and says (on p. 382) that, « for a slightly nonuniform medium, it would appear reasonable to find the dispersion relation first for constant values of the parameters of the medium and then reinsert their dependence on x, t. » Our method to extend the definitions and the result to the general case, viz. considering an « elementary wave at point X », leads us to discuss the covariance of the equations in a more general framework than that of absolute time plus Cartesian coordinates of the Euclidean space. Of course, the nonuniform case is essential for quantum mechanics since a non-uniform external field is always present in practice. It is important to note that (especially in the non-uniform case), a solution of the wave equation does not in general obey the dispersion equation, and vice versa.

The quantum correspondence and its (non-)ambiguity

Let us consider a classical system of point particles with « position » x (in the configuration space M), its dynamics being given by a Hamiltonian system with Hamiltonian H and conjugate momentum p (a « spatial » covector, though only under infinitesimally linear space coordinate changes, as one may check by studying the transformation of Eq. (2.12)1):

(2.12)

d d p t H x j j     , d d x t H p j j    .
Following de Broglie and Schrödinger, let us imagine that those microscopic objects, which we initially described as point particles, actually have a spatial structure made of some (unknown)

waves; and that this classical Hamiltonian dynamics only describes the « skeleton » of the wave motion, in precisely the same way as geometrical optics describes the trajectories of « light rays », which constitute the skeleton of the light wave propagation pattern. Thus, we expect that the dynamics (2.12) should be that approximation of the wave motion which becomes exact at the « nil wave length » limit where, in the neighborhood of any point X in the extended configuration space, the wave may be considered as a plane wave, that is

(2.13) A  Const. and   k.x   t = K x 
in Eq. (2.7); of course, condition (2.13)2 means that the higher-order derivatives of  can be neglected: K,   0, and so on. We seek for the linear wave equation ( here we merely want that their projections in the extended configuration space RM are the same. If we consider the Lagrangians (x,  x , t ) and L(x,  x , t ) respectively associated with W and H by the Legendre transformation, the condition is that the extremals of the respective action integrals must be the same. The obvious, simplest way to ensure that this is true, is to impose that  and L are equal up to a multiplicative constant. We denote this constant by  ... Thus, we are lead to admit that L =  , whence by Legendre transform:

(2.14)

H =  W, or E =  ,
and for the canonical conjugate momenta:

(2.15)

    L x x    , or p = k.
The a priori interpretation of the correspondence (

    H t m V t p x p x , , ,   2 2 .
In this case, the data of H (or W ) already represents the dispersion equation: more exactly, it is the simplest (lowest degree) dispersion equation that has W as one possible dispersion. Therefore, the correspondence (2.5) gives unambiguously the wave equation. This is also true if the coefficients of the polynomial depend on x and t. Note that the composition of the two correspondences (E, p) (, k) [Eqs. (2.14) and (2.15)] and (2.5) gives the « quantum correspondence » :

(2.18)

E t p x j j     i i       , .
From Eq. (2.17), one thus gets Schrödinger's equation:

(2.19) H i            2 2m V t  .
Or (ii) H is not a polynomial in p, but some polynomial function of H is itself a polynomial in p.

In that case, we take the dispersion equation as that polynomial function  which has the lowest degree, which is necessarily unique up to a constant factor, and we thus obtain uniquely the linear wave equation with the lowest possible order. In practice, the Hamiltonian H will then appear precisely as involved in such polynomial function: this is the case for the relativistic Hamiltonian of, for instance, a free particle, which appears in the equation To be complete, there is still the case that (iii) H is not an algebraic function of p, that is, no polynomial function of H becomes a polynomial in p. This means simply that no linear wave equation can be associated with the given classical Hamiltonian by this correspondence.

Comments

The usual correspondence (2.18), between a classical Hamiltonian and a « quantum » wave equation, may thus be arrived at by a seemingly new method, based on the dispersion relation in a linear medium. The interest one may find to this method is that it gives a rather rigorous basis to the essence of wave mechanics: the idea that classical dynamics of point particles should correspond to the « geometrical optics » limit of a linear wave equation. It has been shown that this idea leads, modulo the study of dispersion relations, to the correspondence (2.18) in a natural and general way. The method is not limited to the case of a single particle: it works for any Hamiltonian system. Moreover, the method also indicates how to make the correspondence unambiguous in the case where the classical Hamiltonian contains mixed terms involving both the momentum p and the (configuration-)space-time position X =(t, x). The rule to obtain the wave operator unambiguously is simple: just put the function of X as a multiplying coefficient before the monomial in p -as the dispersion equation has to be a polynomial in p at fixed X. This is important in the case with a gravitational field, for a curved space-time means that, for example, the kinetic energy becomes a sum of such mixed terms (since it involves the metric tensor, that depends on X ).

Thus, a classical Hamiltonian (even in an implicit form such as Eq. (2.20)) gives unambiguously a linear wave equation for the more complex wave structure supposed to underly the approximate classical behaviour of a system of particles (the latter corresponding to the nil wave length limit). However, if one admits that it is a wave structure which is the fundamental behaviour, then one must expect, of course, that in some cases no classical Hamiltonian will be relevant, i.e., one must expect that the simple correspondence (2.18) is not always a sufficient tool to obtain a correct wave equation -an example is Dirac's equation.

Momentum states and non-dispersive solutions of Schrödinger's equation

Definition of momentum states: a generalization of plane waves

In QM, it is admitted that a (single) quantum particle has a well-defined momentum p, if and only if its wave function (in the physical space, which coincides, in the case of a single particle, with the configuration space) is that of a plane wave. Moreover, only the spatial dependence of the wave function is envisaged, so that the « wave function » (at a given time) is A exp(i k.x).

Thus, the first wave function encountered in QM has constant modulus and, in particular, it is not square-integrable, although for the case of one quantum « particle » obeying Schrödinger's equation, this is a necessary requirement. Actually, it is easy to show that a plane wave cannot obey the time-dependent Schrödinger equation unless the potential V is a constant number (thus unless the « particle » is in a free motion); this will be proved in passing below. One may ask if the wave function of a moving particle could be defined as a function of time also, and if its spatial extension could be limited in some sense.

Thus, we consider a wave function of the general form (2.7), and we ask under which particular conditions it could be taken to represent a quantum object having, at any given time t, a well-defined momentum p(t). The velocity of the particle is related to the group velocity of the associated wave (Eq. (2.11)), indeed it must coincide with the latter in the « geometrical optics » limit since, in this limit, Eqs. (2.11) and (2.12) must give the same trajectories. Therefore, an obvious sufficient requirement is that (i) the group velocity C must be a function of t only, thus a given vector at a given time, which will represent the velocity of the particle. However, C depends on the dispersion W (Eq. ( 2

(t, x) = k(t ).x  f (t ).
Note that, under condition (i)', the frequency is not necessarily uniform, indeed governed by the relevant Hamiltonian, P -(q/c)A = m  x is the « true » momentum, where P is the canonically conjugate momentum, to which applies the relation P = k). Hence, k = k(t ) is then incompatible with C = C(t ) unless A = A(t) (i.e., no magnetic field).

(3.3)

 = f '(t)  k'(t).x.
Obviously, condition (3.2) is fulfilled by a plenty of possible wave functions, since no restriction is imposed on the amplitude A. But since we recognized that our definition of a momentum state depends on the wave equation, we have one more reason to impose on the wave function  the condition that (ii)  should obey the wave equation -which is anyway a necessary requirement 2 . Furthermore, we assume from now on that the amplitude A, as well as the phase , is real. This is the obvious necessary condition under which the writing of the complex function  in the form (2.7) is (practically) unambiguous: otherwise, the relation between  and its phase  would be totally ambiguous. (The restriction that A also should be real does not play a role in the study of dispersion relations, since one then considers constant amplitudes.) After an easy algebra, the Schrödinger equation (2.19) for the wave function (2.7), with the phase (3.2), is then found equivalent to the two real equations:

(3.4) HA m A VA m A              2 2 2 2   p , (p(t )  k (t )), (3.5) d d A t A t A       C. 0.
Equation (3.5) shows that the amplitude A undergoes a mere translation with the uniform group velocity C(t). In precise terms: for any trajectory t x(t) =t (x0) (with x0 the position at t = 0) such that, any time t, dx/dt = C(t ), then A is a constant:

(3.6) A(t, x0+ C( ) s s t d 0  ) = A(0, x0).
Since these trajectories are all identical but for a constant space translation, one may say that the quantum object follows a well-defined trajectory, provided both equations (3.4) and (3.5) are satisfied in the whole space and for a non-zero interval of time. Let us ask whether this is possible.

Existence of momentum states for a general spatial dependence of the potential

In addition to the amplitude function A(t, x), the unknowns in Eqs. (3.4) and (3.5) are the wave vector k(t) determining the group velocity C(t), Eq. (3.1), and the scalar function f (t) which, together with k(t ), determines the phase , Eq. (3.2). Due to this unusual set of unknowns, the discussion is a bit subtle. In order to establish for which kind of potentials V(t, x) one may find solutions to these equations, we consider the initial value problem: we assume the initial data A0(x) = A(0, x) (for all x), k(0) and f (0). If one knows the dependence k = k(t ), Eq. (3.5) is completely solved by Eq. (3.6) and the initial data A0(x) for all x. Now Eq. ( 3.4) at t =0 may be seen as restricting the possible initial data for a given initial potential V0(x) =V(0, x):

(3.7)   H 1 0 2 0 0 0 0 2 2 2 0 0 0 2 A m A V A A f m             k' . x k ( ) , ' ( ) ( )  
.

Hence, an easy way to find a licit initial data A0 for a quite general initial potential V0, is to assume the additional data k'(0) (amounting to the data of the initial acceleration!) and to determine A0 and µ (hence f '(0)) as an eigenfunction and the corresponding eigenvalue for the problem (3.7).

This is simply an eigenvalue problem for the modified Hamiltonian H1, deduced from the true Hamiltonian operator H by adding the linear term k' . x ( ) 0 to V0, thus by adding a uniform force field, just as in the Stark effect. If V0 is a « typical » potential of QM, it is hence clear that there should be solutions to this problem, and this is indeed the case for a nil potential as well as for the Coulomb potential [5, pp. 440-442]. In order to get a well-defined problem with welldefined solutions, one still needs to precise the boundary conditions for A0 in the eigenvalue problem (3.7)1. As is usual in QM, we may content ourselves by imposing the condition that (iii) A0 vanishes at infinity (A0(x)  0 as |x| ). Note that, conversely, if one starts from a licit initial data A0 and µ, Eq. (3.7)1 determines the initial acceleration of the particle, a0 = k'( ) 0 /m.

In other words, there is no undetermination for a0, but the mere fact that, in a given arbitrary potential, the initial amplitude of the wave function cannot be arbitrary for a momentum state.

Also note that one still may consider (3.7)1 as defining a particular potential, the now arbitrary initial amplitude function A0 being given.

Having shown that the restriction (3.7) imposed to the initial data A0, furthermore assigned to vanish at infinity, may be satisfied for a quite general initial potential V0, we now have to ask if Eqs. (3.4) and (3.6) may hold as the time goes. Thus, the amplitude A(t, x) is known from Eq. (3.6) if the time evolution C(t), or equivalently k(t), is known. Inserting (3.6) into (3.4), we get

(3.8)   V t t f t m t A A s s t ( , ) ( ) '( ) ( ) ( ) x k' . x k x C                     2 2 0 0 0 2  d , whence (3.9)     V t V s s t f t f m t t ( , ) ( ) ( ) ( ) '( ) '( ) ( ) ( ) x x C k' k' . x k k                    0 0 2 2 0 0 2 0 d   .
Therefore, up to its translation that must follow the motion of the quantum object, the initial potential must change only by a time-dependent linear function,

C        0 0 d + u(t).x + U(t). (3.10) V t V s s t ( , ) ( ) x x
Conversely, if this is the case, then, by rewriting (3.10) in the form (3.9), the data of this linear function (together with the initial data A0(x) which fixes k'(0), see above) determines uniquely k'(t) for all t ; hence, accounting for the initial data k(0), it determines k(t) and C(t) for all t. Thus, it also determines f '(0) -f '(t) for all t. We see that f '(0) must be added to the initial data,

although, for a given potential, the data f '(0) is subject to the constraint that µ (Eq. (3.7)2) is an eigenvalue of the problem (3.7)1. Thus, if the time evolution of the potential obeys Eq. (3.9), we define A(t, x) by Eq. (3.6). It is then almost obvious, and easy to check, that Eq. (3.4) continues to be verified for t > 0 .

In summary, there indeed exist momentum states, defined as wave functions that (i) have a uniform group velocity, (ii) obey the time-dependent Schrödinger equation and (iii) have a vanishing amplitude at infinity, provided that the initial potential changes only through a uniform translation determined by a time-dependent linear function of the position, Eq. (3.10). The arbitrary time-dependence of that linear function determines the time evolution of the (spatially uniform) momentum and that of the frequency (which is spatially non-uniform, except for the case where the momentum is conserved, i.e. the case u(t)  0 in Eq. (3.10), plus k'(0)=0). The amplitude function of these momentum states moves rigidly with the motion. The evolution equation of the momentum is got by applying the gradient operator to Eq. (3.8):

(3.11) d d d p k' x x C t t V t m A A s s t                    ( ) ( , ) ( ) 2 0 0 0 2  ,
it gives, by construction, a uniform evolution if Eq. (3.9) is satisfied. One recognizes the second term in the r.h.s. of Eq. (3.11) as a particular case of the well-known « quantum potential » (see e.g. Vigier [START_REF] Vigier | Structure des Micro-Objets dans l'Interprétation Causale de la Théorie des Quanta[END_REF]).

Thus, there exists a very general class of non-spreading solutions of Schrödinger's equation, each of which represents a « localized object » in the sense that the amplitude vanishes at infinity, and with a well-defined momentum. This is in striking contradiction with the standard presentation of quantum mechanics, in particular with the usual exegesis of the uncertainty principle, though it does not actually contradict either the latter principle or QM in general (see the discussion in § 3.4). Note that condition (3.10), imposed to the timedependence of the potential, is less artificial than it may appear at first sight. First, it is satisfied by any time-independent potential (in this case, our result is not very interesting since any real solution of the Schrödinger equation for stationary states yields a wave function with similar properties, once it is multiplied by the sinusoidal time-dependence in order to get a solution of the time-dependent Schrödinger equation). More importantly, condition (3.10) is satisfied by any quantum object (1) assumed to remain at rest with respect to a moving object [START_REF] Whitham | Linear and Non-Linear Waves[END_REF] with which it interacts, provided the interaction potential (21) is independent of the motion of the system (1)+(2) (which may be approximately true), and provided the action on (1) of the exterior of the system (1)+( 2) is a spatially uniform force field. An obvious subcase is the case where (1) is actually "alone", it is hence the potential corresponding to a spatially uniform force field. It is worth to study that case separately.

« Classical » momentum states

In view of Eq. (3.4), we may wish to impose to a momentum state the additional requirement that (iv) the dispersion relation is satisfied:

(3.12)     p 2 2 0 m V .
One might guess that those particular momentum states may have something do with the classical limit of the wave equation because, as we have seen, the very construction of wave mechanics is based on the assumption according to which the classical limit corresponds to a situation (the geometrical optics) in which the wave equation and the dispersion equation can be verified simultaneously. However, since the geometrical optics is a limit, one might expect that, in situations relevant to this limit, these two equations cannot be verified exactly. This is not the case, as we shall now see. The requirement (3.12) implies that Eq. (3.4) is equivalent to the Laplace equation:

(3.13) A = 0.

Moreover, Eq. (3.12), in which we recall that p = p(t )  k (t ), implies also that V is a linear function of x: since  is given by Eq. (3.3), we get indeed from (3.12):

(3.14) V(t, x) =    f t t m '( ) ( )   k' . x k 2 2 2 .
Conversely, if we know that A = 0, then, unless A(t, x) = 0, Eq. (3.4) is equivalent to Eq.

(3.12) and thus implies Eq. (3.14). In particular, a plane wave (A = Const, k = Const,  = Const) can be a solution of the Schrödinger equation only if the potential V is a constant, as announced. This result, plus the fact that a plane wave has a constant modulus over the space, seems to imply that a plane wave cannot represent a correct description of the objective state of a single quantum object having a well-defined momentum. The more general result expressed by Eq. (3.14) is that the dispersion relation cannot be verified for a momentum state which is an exact solution of the Schrödinger equation, unless the potential V is linear in space, i.e., unless the force field F is uniform. In contrast to the case of a general momentum state (Eq. (3.11)), the evolution of p is purely classical if the potential has this form (3.14):

(3.15) d d d d p k F t t V t      ( ).
Hence, momentum states satisfying the additional condition (iv) will be called « classical momentum states ». We thus find that, for the usual Schrödinger equation, a classical momentum state can exist only over space and time intervals such that the traversed force field may be considered as spatially uniform.

Thus, considering a uniform force field F(t), let us study such a classical momentum state precisely. Given the initial momentum p0, its time evolution is completely determined by Newton's second law (3.15). Due to Eq. (3.1) with k = k(t), the momentum may really be considered as uniformly distributed in the wave (as for any momentum state). The dispersion relation (3.12), i.e., « E =  », determines the frequency .Note that it depends on space and time. The spatial variation of A is determined by the Laplace equation (3.13). Here, we wish that (a) A is defined in the whole space (except perhaps at a point singularity) and harmonic, and (b) A tends towards zero as r  |x| . First, these conditions imply that there is a ball |x|  r 1 such that A is not bounded in this ball, since otherwise it would be harmonic except perhaps at one point, and bounded in the whole space, hence [7, p. 305] it would be harmonic in the whole space; being harmonic and bounded in the whole space, it would be a constant, hence zero by condition (b). Hence, A is not harmonic at one point of the ball |x|r 1 (since otherwise it would be continuous, hence bounded, in this ball), which we choose as the origin. Thus, A has a point singularity at x = 0. Now if we impose the subsidiary condition that A is not more singular than 1/r, i.e., rA is bounded as r  0, then we get by the same theorem [7, p. 305] that there is a constant R such that A -R/r is a harmonic function  on the whole space. But  is bounded on the whole space since it is bounded in every bounded domain and since, as A and 1/r, it tends towards zero as r  .

Hence,  is a constant , which must be zero. Thus, conditions (a) and (b) imply that A = R/r with R an arbitrary constant and r  |x| after the appropriate choice of the origin x = 0, unless one would accept that A be more singular than 1/r . Condition (b) gives A  R/r as r  , independently of the latter condition [7, p. 314]. Unfortunately, R/r is not squareintegrable. Note that, if we replace condition (b) by the less severe condition that (b)' A is bounded as r  , just the same arguments lead to A =  + R/r. Now let us consider, for R and  > 0, the function

(3.16) A R  (x) =  + R/r if 0 < r  R/, A R (x) = 0 if r  R/.
From a physical point of view, this continuous function is undistinguishable from R/r if  is small enough, and it has a finite square integral. Of course, A does not satisfy A = 0 on the sphere r = R/, but A vanishes from that sphere and, in traversing the sphere, its derivatives undergo an arbitrarily small discontinuity if  is small enough. We note finally that different amplitudes functions A may be found if one accepts that A has, for instance, a onedimensional singularity (such as a circle). However, as long as one demands that A vanish at infinity, a singularity is unavoidable since A must be a harmonic function. 3 3.4 Comments on the physical interpretation of the momentum states Thus, in a uniform force field -V = F(t), the Schrödinger equation admits the wave functions  should be considered as a small « cut-off ». In replacing A R  by R/r, one gets classical momentum states that satisfy Schrödinger's equation everywhere except at the singularity and that vanish at spatial infinity, though without having a finite square integral. The fact that the evolution of the momentum is governed by Newton's second law seems to make the transition from classical mechanics to wave mechanics quite transparent: classical mechanics would envisage as a point particle what is in fact a spatially extended wave. And since a « classical momentum state » remains so only in a uniform force field, classical mechanics would be valid only in the case that the force fields vary slowly in space.

Moreover, we have now definitely proved by a class of analytical examples that, contrary to the standard presentation of QM, a « quantum particle » (as defined by a squareintegrable wave function obeying Schrödinger's equation) may have a finite and constant spatial extension and a well-defined momentum. Furthermore, the amplitude of the wave function increases as 1/r towards a point singularity which moves according to Newton's second law. 4 These classical momentum states look like an exact realization of de Broglie's ideas according to which a quantum object would involve both a « real » wave and a singularity whose motion is « guided » by the wave (see e.g. Vigier [START_REF] Vigier | Structure des Micro-Objets dans l'Interprétation Causale de la Théorie des Quanta[END_REF]). Note, however, that 4 If one accepts a one-dimensional singularity, the behaviour near this singularity will have to be determined, e.g. it is not guaranteed in advance that a square-integrable function may be obtained. Anyhow, the amplitude will also undergo a pure translation with the uniform group velocity obeying Newton's second law (Eqs. (3.1, 3.6, 3.15)).

in the case of a general momentum state, the amplitude function A0 is defined by a rather generic eigenvalue problem of quantum mechanics (Eq. (3.7)1) and so, in general, should not exhibit any singularity, even with the additional condition that A0 vanishes at infinity. In our opinion, if one wishes to interpret the present results in the framework of physical realism (and if, moreover, one accepts that quantum mechanics is an exact description), they are more likely to sustain Schrödinger's conception according to which the « matter waves » are the true reality: for Schrödinger, « ... a particle is not a permanent entity », and « ...[one should] hold on to the wave aspect throughout the whole process. » [START_REF] Schrödinger | Physicien et Penseur[END_REF] A question arises then: how is this result compatible with Heisenberg's uncertainty

principle? The answer is, obviously, that the latter applies (as a consequence of the QM postulates regarding the « collapse of the wave function ») to the outputs of measurements of momentum and position. Whereas the present result applies to the objective description of a « quantum particle » (wave function) in a spatially uniform force field, thus excluding a necessarily perturbating action of measurement. This result is hence compatible with the axioms of QM if they are taken in the operational sense, and it has no immediate consequence on the possible predictions that one can make in QM. Moreover, we emphasize that what is called here a « momentum state » is not an eigenfunction of the « momentum operator » in the direction of the current wave vector. We also note that, since the spatial extension of a « classical momentum state » is arbitrary, one still may say in a restricted sense that « the particle has not a well-defined position ». (This is no surprise if one considers that there is indeed no point particle but an extended wave.) Another one might consider, on the contrary, that « the position of the particle » is perfectly defined at any time in the case of a classical momentum state, as the position a(t ) of the singularity. However, for a general momentum state and, even more generally, for a generic solution of the time-dependent Schrödinger equation, there is no singularity which might indicate the position of a point particle.

  p)] 2 -p 2 c 2 -m 2 c 4 = 0. For the latter dispersion equation (up to the correspondence (2.14) and (2.15)), the wave equation obtained by the correspondence (2.5) is, of course, the Klein-Gordon equation, with the d'Alembert equation as the case m = 0. The latter case shows that the approach based on the « geometrical optics » limit does work in the genuine case! It also shows that the quantum correspondence gives merely the wave equation for anyone scalar component, and tells us nothing about the scalar or tensor character of the physically relevant wave field.

  But the exact conditions A = Const, K,  = 0, and so on, are the conditions under which the substitution of (2.7) into (2.1) gives the dispersion equation (2.4).

	Hence, the wave covector K of an exact solution (2.7) to the wave equation (2.1) satisfies the
	dispersion equation (2.4) to an approximation which becomes exact in the « geometrical optics »
	limit where the dynamics (2.12) itself becomes exact.
	Now, a regular solution of the dispersion equation (2.4) should correspond to a unique
	« branch », i.e., it should satisfy one and only one among the different possible dispersion
	relations (2.6), and we assume that it is indeed the case. Therefore, in the « geometrical optics »
	limit, the spatial wave covector k [associated with a wave function (2.7) obeying the wave
	equation (2.1)] obeys also the « Hamiltonian dynamics » (2.11). The only difference with a true

2.1) for which the

Hamiltonian dynamics would represent just that approximation. Assume that the wave function , of the form (2.7), is the relevant solution, in a situation where the Hamiltonian dynamics is a good approximation, to that exact wave equation: in view of what we have just said,  will satisfy the approximate conditions (2.13).

Hamiltonian dynamics is that, in Eq. (2.11), a continuous distribution of Hamiltonian trajectories is involved. However, to account for the fact that (in its domain of relevance), the classical approximation (2.12) successfully considers point particles, we expect that, at least in some cases, the wave pattern is made of spatially concentrated « wave packets », in the exterior of which the amplitude is negligibly small. Obviously, we are lead to admit that the Hamiltonian systems characterized respectively by the dispersion W (Eq. (2.11)) and by the classical Hamiltonian H (Eq. (2.12)) have exactly the same set of possible trajectories x = x(t ). This can be achieved by a canonical transformation (x, k, t, W )  (x, p, t, H ) (thus with the « position » and time variables being left unchanged) only if k = p and W = H. But a canonical transformation means that the trajectories in the extended phase space, RT*M, are the same:

  2.14) and(2.15) is a formal one: it is the correspondence between the Hamiltonian systems (2.11) and (2.12) -that govern respectively (i) the bicharacteristics of the propagation equation (2.10) for a wave covector obeying the dispersion relation(2.6), and (ii) the trajectories of a classical system -under the condition that the « geometrical optics » limit of the wave equation (2.1) gives the same trajectories as the classical system. Moreover, let us recall that the wave equation can be uniquely deduced from the dispersion equation (2.4) (see the end of § 2.1). However, the correspondence (2.14) gives us the dispersion relation(2.6). But the dispersion relation is just one root of the polynomial dispersion equation (2.4), when the latter is seen as an equation for  = K0.

Thus, if we are given a classical Hamiltonian H and if we seek for the associated « quantum » wave equation, we first (trivially) deduce the dispersion W by the correspondence (2.14) and (2.15), i.e.

(2.16) W(k, x, t ) = [H( k, x, t )]/  . Then: either (i) H is a polynomial in p (at fixed X ), such as, for instance,

(2.17)

  .8)). The latter is found by the correspondence (2.16), in the particular case that the quantum wave equation can be deduced from a classical Hamiltonian by the correspondence (2.18). In the general case, the dispersion equation is found directly from

the wave equation by the transition from Eq. (2.1) to Eq. (2.4), thus by the substitution D  i K , provided the wave equation can be put in a form where it is the same for each component of the wave function; then, several dispersions (whence several C vectors) may be defined by extraction (see § (2.2)). In any case, the definition of C depends on the wave equation. Hence, the natural condition (i) implies that the very definition of a « momentum state » should depend on the quantum wave equation. Let us assume that this is the one-particle Schrödinger equation (2.19), associated with the non-relativistic Hamiltonian (2.17). The group velocity is then (3.1) C = k /m, so that, in the case of the usual Schrödinger equation, condition (i) is equivalent to the requirement that (i)' the spatial wave covector k is a function of t only

1

. Thus x = k(t), which is equivalent to say that the phase  has the form

(3.2)

  The arbitrary numbers R and  determine the constant « spatial extension » of a such wave function, which may be taken to be R' = R/. As outlined above,

	(3.17)	(t, x) = A R [x a (t )] exp { i [ k(t ).x  f (t )] },	da/dt = k (t )/m,
	as square-integrable classical momentum states, i.e., the group velocity (3.1) is spatially
	uniform, the Schrödinger equation and the dispersion equation (3.12) are satisfied. The wave
	covector k(t ) and the function f (t ) are determined (up to their arbitrary initial value) by the
	potential V(t, x); in particular, the momentum p(t )=k (t ) evolves according to Newton's
	second law (3.15).
	p. 390].

3

Probably, A must also have a singularity if one demands instead that A vanish outside a bounded open domain D and be harmonic in D, but perhaps not on its boundary: e.g. this is true if D is the ball |x|< R', as for AR 

[7,

This is not true in general, e.g. it is wrong for the (scalar) Schrödinger equation in the presence of a magnetic vector potential A: in that case, one finds C = [k -(q/c)A]/m (consistently with the fact that, for a classical particle

In the case where a magnetic field is present, the expression of the group velocity involves A that depends on the gauge condition (note 1). Therefore, one is really enforced to impose the wave equation: if one changes the gauge condition, the wave function, hence also the wave covector k, will then transform according to a simple rule[START_REF] Landau | Mécanique Quantique (3 rd French Edition)[END_REF], and it is easy to verify that the group velocity is then independent of the gauge.