_{1}

^{*}

It is shown that a single-particle wave function Ψ, obtained (Landau, 1930) as a solution of the Schr?dinger equation (for a charged particle in a homogeneous magnetic field), and an operator relation of (or equation ) lead to the dynamic description of one-dimensional many-particle quantum filamentary states. Thus, one can overcome the problem, connected with the finding of many-body wave function as solution of the Schr?dinger equation with a very tangled Hamiltonian for multi-body system. An effect of nonlocality appears. The dependence of the linear density of particles on the magnetic field and on the number of particles in the one- dimension filamentary multiparticle quantum structure is calculated.

There is a set of interesting questions which have no clear answers. Why is our world not the continuous medium of particles? Why are there infinitely great variety of structures? The micro structures are the corner- stones of all macro structures. At the same time, the micro structures are described by quantum theory. There- fore, it would be interesting to find the answer to a question: what is the cause for arising of the quantum struc- tures? Why are ones of them stable, whereas others exist during only very short time interval? Here, we try to find the beginning of the answer to these questions.

Our attempt will be done for a particular case. The subject of our investigation is the motion of charged particles in a spatially homogeneous magnetic field. In the non-relativistic quantum mechanics, the Schrödinger equation for a particle in the homogeneous magnetic field was decided long ago (Landau, 1930), and the wave function and the energy spectrum were obtained [

momentum, velocity, coordinate, mass, and time

Obtained trajectories (dependence of coordinates on time) have a probabilistic nature (because of the pro- babilistic nature of the wave function). However, namely theirs geometry is most informative and important for applications. In Section 4, it is shown that every trajectory is one-dimensional trajectory which represents a sequence of segments at the ends of which there are quantum turning points [

proposed equation,

function

In the classical turning point, the velocity of electron is equal to zero

point the sign of velocity is changed on the opposite one [

So in our scheme, in the

The conditions for the motion of the center (the point

It would be incorrect to apply the concept of a quantum ensemble for a group of interacting theoretical electrons, where each electron would have its own individual wave function. In order to avoid this mis- understanding, in Section 2 (according to [

given by the wave function

solution of which is a trajectory consisting of the n isolated segments (see

In the experiment, we have a real particle, however in the quantum theory we can deal only with the statistical quantum ensemble. The quantum mechanics, formulated on the principles of the quantum ensembles, is des- cribed in [

To obtain the trajectory of a quantum ensemble, we use [

Here,

The wave function

For carrying out the mathematical calculations, we can choose the operator of coordinate

This operator

or

In [

For calculation of a trajectory for movement of the linear quantum oscillator, it is necessary to insert its wave function

For arbitrary value of n, the wave function of linear oscillator is [

where

For

where

According to [

Let us determine the complex variables

variables [

An equality of real parts of Equation (9) gives equation^{1}

Thus, the movement of a quantum oscillator with a quantum number of ^{2} with its finite size on the

The result (10) can also be expressed in the variables which are used by authors [

For example, for these variables

At the trajectory calculation, we will follow an algorithm described in Section 3, use the substitution (11) which facilitates the mathematical calculations. In this case, according to (11), we should name

For n = 1 the function (5) is

After integration of (12) between the points

The real part of secondary equation in (13) is^{3}

From (12)-(14), we should see that ξ = ±1 are coordinates of two turning points (where the velocity

the coordinate

For

or

After integration of (16) between the points

From the left-hand equality in (17), we obtain

The real part of (18) is^{4}

From (16)-(19), we should see that the points

There is approximation

able is changing within only one of two intervals:

For

or

After integration of (21) between the points

where

From (22), we have

The real part of (24) is^{5}

From (21)-(23), we should see that the points

There are approximations

oscillates (see (25)), if the

According to (5), for

or

After integration of (27) between the points

where

From (28), we obtain

The real part of (30) is

From (27)-(29), we should see that the points

There are approximations

oscillates (see (31)), if the

In Section 4, it is shown that a trajectory of one-dimensional quantum oscillator, described by wave function

Therefore, for any constant quantum number of n, the isolated segments can be filled with electrons like it is shown in the

Thus, a linear quantum oscillator with quantum number of n is described by the wave function

one from another and filled with electrons. Therefore, for simplicity, we can suppose that the energy

For simplicity, we have neglected the spin term in the formula for energy

fore in

linear density of electrons in filamentary structure grows with growing n because of reduce of

In Section 4, it is shown the number of segments, their average length^{6}

and coordinates of their endpoints are strongly correlated with a quantum number of n (see

To determine the value of

So, to find the points of divergence^{7} (the turning points)

^{6}For example,

^{7}That is, the values of

In (5), the functions of

Now we should find linear density

sverse magnetic field

get

where

The value of

demonstrated for different n. Thus, in one-dimensional filamentary structure, the linear density

with growing

The motion inside the one-dimensional filamentary structures has been calculated in Section 4. To describe the

motion of filamentary structure as a whole, we will follow [

(see coordinate

To obtain more information concerning the quantum objects, we offer to use well-known relation between the

operators of the momentum, velocity, coordinate, mass and time

wave function has to be treated only within the theory of quantum statistical ensembles (see [

If a wave function has an oscillating form (as it is for the wave function of a harmonic oscillator at^{8} [

In [

The movement of one-dimensional filamentary quantum structure (in the crossed electric and magnetic fields) is discussed in Section 5.

The energy of oscillator is shared between electrons which form one-dimensional filamentary quantum struc- ture. Therefore, it seems interesting to investigate the possible correlations between the oscillations of energy per one electron (

The dependence of average distance between the particles (of one-dimensional filamentary quantum structure) on the external transverse magnetic field

The dependence, presented in

^{8}where the velocity takes the value

In Section 5.1, it is discussed that an energy per one electron of filamentary structure is reduced with growing quantum number of n (

In Equation (35), there is no dependence of

frequency

particles to manipulate with them.

Offered approximation could be applied to investigate of two-dimensional structures in magnetic field. How- ever, these structures could be formed instantly and locally in any place where there is two-dimensional movement of charged fermions in transverse homogeneous magnetic field. Change in the direction of the magnetic field leads to the collapse of one-dimensional quantum filamentary structures and to their instant formation in a new plane transverse to the new direction of the magnetic field. To the casual observer it looks like chaos (or “crazy” dance of particles) whereas, in fact, it is continuous transformation from one quantum structure to another.

Therefore, investigation of local magnetic fields in different structures could be useful for understanding the nature of appearance of structures.

The possible applications of offered approach could be found in [

I am grateful to V. M. Chudakov for useful discussions, K. G. Gulamov for help and V. Sh. Navotny for providing the Internet communications. My special thanks to E. N. Tikhonov for his patience and support.

From Equation (4), one can see that the y (or

equation

there is an extremum of the

An Equation (5) shows identity of points of extremum for the

Equation (33) opens the way to calculate these points of extremum by an Equation (34), and then to obtain the dependence of

The computational problem is that the numerical coefficients of ^{20} for

whereas it would be interesting to know the linear density of particles

could be useful information, related to the two-dimension electron density on the graphene surface, or connected with density of a two-dimensional plasma instability in the process of nucleosynthesis, or related with other collective phenomena.