waves and particles

Problem's answer

Note: No Quantum Mechanic (At the beginning of the solution)

We have the classical solution for this problem in two parts.

First part.

For in the ether medium n, we will obtain,

On page 2,

It is "assumed" in the problem, that the atoms have the same velocity.

Substituting in (a),

On page 4, we have,

D2 from the output of the Interf3p program, for .

Thus,

Substituting in (b),

On the second part for,

We will find,

But,

Then,

Substituting in (b)

But on page 6,

D2 from the output of Interf3p program for

We obtain that,

There are two answers because the way that is outlined this problem, called as the first and second part.

First part, for , in the ether medium n, with the application of Michel.fr4 program and the interference program Interf3p, obtaining for the uncertainty principle the following momentum and positions.

From the output of Interf3p program:

Momentum Positions

Where, (Freire's enunciated)

Following the relations from the wave-particles duality and uncertainty principle, we have,

Where

We represent geometrically those momentum and positions as,

B represents the minimum size of the hole, Y represents the minimum size of the spot and D1 is the distance between the hole and the spot.

Is the uncertainty in the z velocity of the silver atoms,

Dividing by V0,

In the y-direction the atoms have a velocity assumed for simplicity to be the same for all atoms. In Michel.fr4 program output we have the time (th-tp), what distance would be d?

Looking at the expression (th-tp) in the output of Michel.fr4 program, we have that,

So we can say,

Also we have the following relation,

And,

First part.

B (diameter of the hole)=

Y (diameter of the spot)=

D1 (distance between the

hole and the spot)=

Second part.

For in the ether medium [1+(n-1)/10], the application of Michel.fr4 and Interf3p program, obtaining for the uncertainty principle the following momentum and positions.

Momentum Positions

Where, (Freire's enunciated)

Following the relations from the wave-particles duality and uncertainty principle,

We can geometrically represent them,

The velocity of the silver atoms will be,

The relation of,

But,

And,

Or,

Second part.

B (diameter of the hole)=

Y (diameter of the spot)=

D1 (distance between the

hole and the spot) =

We found in both parts, the minimum size hole for the minimum size spot and the distance between them.

Distinction between waves and particles.

It is essential that one understand the distinction between waves and particles. A classical particle is something that has: position, momentum, kinetic energy, mass and electric charge. A classical wave has the attributes of: wavelength, frequency, velocity, amplitude of the disturbance, intensity, energy and momentum.

The most distinctive difference between the two is that a particle can be localized, whereas a wave is spread out and occupies a relatively large position of space. There is one important difference between photons and massive objects in the way their waves and particles are related. Because for a photon, only one rule is required to get both wavelength and frequency from a photon's particles of energy and momentum. A massive object, on the other hand, requires separate rules for its wavelength and frequency

A brief description of the wave matter.

The subject of matter waves cannot be treated meaningfully without employing far greater mathematical sophistication than has been necessary heretofore. The reader will encounter functional notation, operators, partial differential equations, probability, and so-called complex algebra.

Since a traveling wave moving in both directions this functional relation can be generalized too,

Obtaining by the second-order partial differential equation for y with respect to x and t, we have,

Since the equation is linear, any linear combination of solutions will also be a solution. This implies the principle of superposition and gives the usual method of forming a standing wave by a linear combination of traveling waves moving in opposite directions.

The Schrödinger equation.

Since the concept of matter waves is not a result of previous physical theories, it is impossible to derive the corresponding wave equation for a particle. We are using the wave model, however, and the equation is developed in a manner analogous to that used for other waves.

The dependent variable of a matter wave is called the wave function and is denoted by (called capital psi).

From the equations,

and

The propagation number of a wave is proportional to the momentum of the particle and the angular frequency is proportional to the energy; therefore we can write;

Using the Moivre's theorem, expressing in exponential; differentiated with respect to x and t,

The partial derivatives with respect to x and t, are connected by means of the relation between the energy and the momentum.

The total energy and momentum are related by the expression for energy conservation of a particle,

To obtain the wave equation is multiplied by and the differential operators and are substituted for p and E. We obtain,

This is the Schrödinger equation for one-dimensional matter waves. This equation differs from the classical wave equation by having a term with no derivative . In addition the Schrödinger equation contains i, which makes it a complex equation.

In particular, the complex nature of indicates that cannot be observed like the displacement of a string and also means that we should not look for a medium which transmits our matter waves. There is no "ether" for matter waves, as opposed to the existence of, say, water for water waves.

I want to point out, that the correspondence with the new established classical ideas , is classically no longer a complex, but a real number.

Through the correspondence principle keeps the theory of an analogy to the classical wave equation.

There is some conflict between previous classical ideas and those of the theory of matter waves. For example, the concept of a classical trajectory of a particle must be discarded and replaced by a probability distribution spread over a large region of space. But, with the application of Freire's enunciated that major flaw in the development of the wave theory of matter, no need the probability to find out the particle position in a certain region.

Because the wave is spread out and occupies a relatively large position of space, I would say equivalent to an expansion of (nq*Y) in the space. That is why it is need to find out the particle through the probability.

For example, in this problem on the Michel.fr4 program for , we have

We observe that the coordinate (Y) is divided by the number (nq), to obtain the real particle's location and the application of Freire's enunciated .

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