ELEMENTS OF QUANTUM MECHANICS
Wave-particle duality properties of the particles of matter.
§ 1 of the De Broglie waves
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.
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.
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
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 ( vph could be more and could be less then с, as opposed to group).
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 vph= 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
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 )
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
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
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
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)
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).
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...yn, then it can be in a state y, described by a linear combination of these functions:
Сn (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
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.
- 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:
Е - the total energy of the particle, a constant in a stationary field.
Substituting (2) → (1):
- Schrödinger equation for stationary states.
There are infinitely many solutions. By applying the boundary conditions are taken solutions having a physical meaning.
wave functions must be regular, i.e.,
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 En 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
from the boundary conditions follows
y(0) = 0,
В = 0
From the boundary condition
The energy of the particles of En in "potential well" with infinitely high walls accepts only certain discrete values, i.e. quantized. Quantized energies of En 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 En (or are in the quantum state n)
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→ ∞ Еn 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
§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.
Schrodinger equation for region 1 and 3:
for region 2:
The solution of these differential equations:
Temporary wave function for the area 1:
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:
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
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, whichEmin = 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).