ELEMENTS OF QUANTUM MECHANICS
Wave-particle duality properties of the particles of matter.
§ 1 of the De Broglie waves

In 1924 Louis de Broglie (French physicist) concluded that the duality of light should be extended to particles of matter - electrons. De Broglie's hypothesis was the fact that the electron which the corpuscular properties (charge, mass) are studied for a long time, has also wave properties, ie under certain conditions behaves like a wave.

Quantitative relations between the corpuscular and wave properties of particles, such as for photons.

De Broglie's idea was, that this relationship is universal, valid for all wave processes. Any particle that has momentum p corresponds to the wave length of which is given by de Broglie.

 - de Broglie wave

p =mv- momentum of the particle, h - Planck constant.

De Broglie waves, which are sometimes called electron waves are not electromagnetic.

In 1927, Davisson and Germer (U.S. physicist) confirmed the hypothesis of de Broglie discovered the diffraction of electrons by a crystal of nickel. Diffraction peaks correspond to the formula Wolfe - Bragg 2dsinj = nl, and the Bragg wavelength was exactly equal

.

Further confirmation of the de Broglie hypothesis in experiments LS Tartakovsky and H. Thomson, the observed Diffraction pattern when a beam of fast electrons (Е» 50 keV) through the foil of different metals. Then was discovered diffraction of neutrons, protons, atomic beams and molecular beams. New methods of research materials - neutron diffraction and electron diffraction and electron optics emerged.

            Macroscopic body must also have all the properties of (m = 1 kg, therefore,  l = 6.62·10^-31 m - can not be detected by modern methods - so macrobody considered only as corpuscles).

§ 2 Properties of de Broglie wave

• Suppose that a particle of mass m moving with speed v. Then the phase velocity of the de Broglie

.

Because c > v then the phase velocity of the de Broglie waves faster than light in a vacuum ( v_ph could be more and could be less then с, as opposed to group).

Group velocity

2. therefore, the group velocity of the de Broglie waves equal to the speed of the particle.
For a photon

i.e. the group velocity is equal to the speed of light.

• De Broglie waves have dispersion. Substituting  in  we find that v_ph= f(λ). Because of the dispersion of the de Broglie waves can be represented as a wave packet, as he instantly "diffused" (disappears) during 10^-26 s.

§3 of the Heisenberg uncertainty relation

Microparticles in some cases behave as waves, as in other corpuscles. They do not apply the laws of classical physics, particles and waves. In quantum physics, it is proved that the microparticle can not apply the concept of the path, but we can say that the particle is in a given volume of space with some probability P. Reducing the volume, we will reduce the probability of finding the particle in it. Probabilistic description of the path (or position) of the particle leads to the fact that the momentum and, consequently, the speed of the particle can be determined with a certain accuracy.

            Further, it is impossible to speak of a wavelength at a given point, and it follows that if we are just given the coordinates of X, then there is nothing we can say about the momentum of the particle, because. Only considering long section DC we can determine the momentum of the particle. The more DC, the better Dр and conversely, the smaller DC, greater the uncertainty in finding Dр.

            Heisenberg uncertainty relation sets a limit to the simultaneous determination of the accuracy of the canonically conjugate variables, which include the position and momentum, energy and time.

Heisenberg uncertainty relation: the product of two conjugate values ??uncertainties quantities can not be in the order of magnitude less than the Planck constant h

(sometimes written )

So. for microparticles no states in which its position and momentum at the same time would have the exact values. The smaller the uncertainty of one value, the greater the uncertainty of the other.

The uncertainty relation is the quantum limit the applicability of classical mechanics to microobjects.

The larger m, the smaller the uncertainty in determining the position and velocity. at m = 10^-12 kg, ℓ = 10^-6 and Δx = 1% ℓ, Δv = 6,62·10^-14 m/s, i.e. will have no effect at all speeds, with which dust can move, i.e. armatures for their wave properties do not play a role

            Let the electron moves in a hydrogen atom. Assume Δx»10^-10 m (about the size of an atom, i.e., the electron belongs to a given atom). Then  

Δv = 7,27·10⁶ m/s. According to classical mechanics, the motion of the radius r » 0,5·10^-10 m; v = 2,3·10^-6 m/s. I.e. velocity uncertainty on the order of magnitude more speed, therefore, can not apply the laws of classical mechanics to the microworld.

            From relation  follow, that the system has a life time Dt, can not be characterized by a particular value of energy. Energy spread   with the decrease in the average lifetime. Hence, the frequency of the emitted photon must also have an uncertainty of  Dn = DE/h, i.e. spectral lines have a certain width n ± DE/h, will be blurred. By measuring the width of the spectral lines can estimate the order of the lifetime of an atom in an excited state.

§ 4 The wave function and its physical meaning

The diffraction patterns observed for microparticles, characterized the unequal distribution of particulate flows in different directions - there are minimum and maximum in the other direction. The presence of peaks in the diffraction pattern indicates that these lines are distributed de Broglie wave with the greatest intensity. But the intensity will be maximum if in this direction extends the maximum number of particles. I.e. diffraction pattern for microparticles is a manifestation of the statistical (probabilistic) laws in the distribution of particles, where the intensity of the de Broglie wave maximum out there and more particles.
De Broglie waves in quantum mechanics are considered as waves of probability, that is, the probability of finding the particle at different points in space changes the wave law (i.e. ~е^-iωt). But for some points of the probability will be negative (i.e., the particle does not fall within this area.) Max Born (German physicist) suggests that the wave does not change the law of probability itself, and the probability amplitude, which is also called a wave function or y-function (psi - function).

The wave function - a function of position and time.

The square modulus of the psi-function gives the probability that a particle will be found within the scope of volume dV - physical meaning not the psi function but the square of its modulus.

Ψ^* - function of the complex conjugate to Ψ

 (z = a +ib, z^* =a- ib, z^*- complex conjugate)

If a particle is in a finite volume V, then the ability to detect it in this volume is equal to 1 (a certain event)                              

Р = 1 ⇒

In quantum mechanics, it is assumed that Ψ and AΨ, where A = const, describe the same state of the particle. Consequently, the

- normalization condition

integral, , means that it is calculated on the unlimited volume (space).

y - function should be

1) final (as P can not be greater than 1)

2) unique (you can not find a particle under constant conditions with a probability of say 0.01 and 0.9, as the probability to be unique).

3) continuous (from the continuity of space. Always a probability of finding the particle at different points in space, but at different points, it will be different)
4) the wave function satisfies the principle of superposition: If the system can be in different states, described by wave functions y₁,y₂...y_n, then it can be in a state y, described by a linear combination of these functions:

С_n (n=1,2...) - any number of.

With the wave function calculates the average value of any physical quantity of a particle

§ 5 The Schrödinger equation

Schrödinger equation, as well as other basic equations of physics (Newton's equations, Maxwell), not outputed, and is postulated. It should be seen as a starting basic assumption, the validity of which is proved by the fact that all the consequences arising out of it exactly in agreement with the experimental data.

                                                                               (1)

- Temporary Schrödinger equation.

 - nabla - Laplace operator

 - the potential function of a particle in a force field,

Ψ(y, z, t) - unknown function
If the force field in which the particle moves, stationary (i.e., does not change over time), the function U is independent of time and the sense of potential energy. In this case the solution of the Schrödinger equation (i.e., Ψ - function) can be represented as a product of two factors - one depends only on the coordinates, the other - on time only:

                                 (2)

Е - the total energy of the particle, a constant in a stationary field.
Substituting (2) → (1):

                                        (3)

- Schrödinger equation for stationary states.
There are infinitely many solutions. By applying the boundary conditions are taken solutions having a physical meaning.
Boundary conditions:
wave functions must be regular, i.e.,
1) finite;
2) unique;
3) continuous.
Solutions that satisfy the Schrödinger equation, called the eigenfunctions, and the corresponding value of the energy - the energy eigenvalues??. The set of eigenvalues ??is called the spectrum of magnitude. If of E_n takes discrete values, the spectrum – discrete, if continuous - continuous or uninterrupted.

§ 6 The motion of a free particle

Particle is called free if it is not the force field, ie, U = 0.

Schrödinger equation for the stationary states in the cases е:

His solution:   Ψ(x)=Ае ^ikx , where А = const, k = const

And energy eigenvalues:

Because k can take any value, then, consequently, E, take any value, i.e. energy spectrum is continuous.
Temporary wave function

                        (-  wave equation)

I.e. is flat monochrome de Broglie waves.

§7 A particle in the "potential well" rectangular.

Quantization of energy.

We find the energy eigenvalues ??and the corresponding eigenfunctions for a particle in a one-dimensional infinitely deep potential well. Suppose that the particle can move only along the axis x. Let the motion of the particles is limited impenetrable walls x = 0 and x = ℓ. The potential energy U is:

Schrödinger equation for the stationary states for one-dimensional problem

Potential well beyond the particle will not be able to get, so the probability of finding a particle outside the well is 0. Consequently, Ψ outside the well is 0. From the continuity conditions that Ψ = 0 and on the borders of well that is Ψ(0) = Ψ(ℓ) = 0

Within well (0 ≤ x≤ ℓ)   U = 0 and the Schrödinger equation.

by typing   get 

General solution

;

from the boundary conditions follows

y(0) = 0,

so

В = 0

Therefore,

From the boundary condition

Follows

         ⇒

Then               

The energy of the particles of E_n in "potential well" with infinitely high walls accepts only certain discrete values, i.e. quantized. Quantized energies of E_n called energy levels, and the number n, indicating energetic levels of a particle, called the principal quantum number. I.e. particles in a "potential well" can only be a certain energy level E_n (or are in the quantum state n)

Eigenfunctions:

А found from the normalization condition

 - probability density. From Fig. shows that the probability density varies with n: n = 1 particle is likely to be in the middle of the well, but not at the edges, with n = 2 - will be either the left or the right half, but not in the middle well and not on edges, etc. I.e. you can not talk about the trajectory of the particle.

The energy spacing between adjacent energy levels:

When n = 1 has the lowest energy non-zero

There is a minimum energy follows from the uncertainty principle, because,

With increases n the distance between the levels decreases when n→ ∞ Е_n almost continuous, i.e. discontinuity is smoothed, i.e. Bohr's correspondence principle holds: for large values ??of the quantum numbers of the laws of quantum mechanics become the laws of classical physics.
General interpretation of the correspondence principle: any new, more general theory is an extension of classical, not reject it completely, and includes classical, indicating the limits of its applicability.

            §8 The tunnel effect.
The passage of a particle through a potential barrier

For a classical particle: at E > U, it passes over a barrier at E < U - reflected from it, for a quantum: at E > U is the probability that the particle is reflected, for E < U is the probability that it will be through the barrier.

Potential energy

:  

Schrodinger equation for region 1 and 3:

for region 2:           

The solution of these differential  equations:

For 1;

For 2;  

For 3:

Because in region 3 is possible of distribution only of  transmitted wave, then, ⇒, В₃=0.

In region 2, the solution depends on the ratio of Е> U or Е< U. interest is the case Е< U.

       q = ib, where

Then the solution of the Schrödinger equation can be written as:

For 1;

For 2;  

For 3:

The qualitative form of the functions shown in Fig. 2. From Fig. 2 shows that the function is not zero inside the barrier, and in 3 of the form de Broglie waves, if the barrier is not very wide.
The phenomenon of "penetration" of a particle through a potential barrier, called the tunnel effect. The tunnel effect is a specific quantum effect. The passage of a particle can be explained using the uncertainty principle: the uncertainty of momentum Dр in the interval Dx = ℓ is . Associated with this spread in the kinetic energy of the pulse  may be sufficient for the total energy of the particles was greater the potential energy barrier.

§ 9 The linear harmonic oscillator

Linear harmonic oscillator - the system, to perform a one-dimensional oscillatory motion by quasielastic force - is a model for the study of the vibrational motion.

In classical physics - it's spring, physical and mathematical pendulum. In quantum physics - quantum oscillator.

Writing the potential energy in the form of

Schrödinger equation can be written as:

Then the energy eigenvalues:

i.e. the energy of a quantum oscillator takes discrete values, i.e. quantized. The minimum value  - zero-point energy - is a result of uncertainty in the same way as in the case of a particle in a "potential well".

The presence of zero-point means that the particles can not fall to the bottom of well, as the in this case would be precisely determined its momentum p = 0, Dp = 0, ⇒, Dx = ∞ - does not match the uncertainty relation. The presence of zero-point energy contradicts classic ideas, whichE_min = 0. - у energy levels are located at equal distances from each other. The quantum analysis that the particle can be detected outside the area . According to the classical treatment only within –x ≤ x ≤ x (Fig..2).