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ُExperimental foundation of quantum mechanics

الكلية كلية العلوم     القسم قسم الكيمياء     المرحلة 4
أستاذ المادة عباس عبد علي دريع الصالحي       21/10/2019 18:03:59
University of Babylon Undergraduate Studies
College of Sciences
Department of Chemistry
Course No. Chsc. 424 Physical chemistry
Fourth year - Semester 1
Credit Hour: 3 hrs.

Lectures of Quantum mechanics
Scholar year 2019-2020
Prof. Dr Abbas A-Ali Draea
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Lecture No. Three: Experimental foundation of Modren Quantum mechanics
De Broglie experiment.
Electron diffraction.
Compton scattering.
Photoelectric effect.
Black body radiation.
Uncertainty principles of Heisenberg.
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INTRODUCTION:
The experiments have been done and the phenomena of double character behaviour of electrons (wave-like nature of particles) were discovered within the time. The results may be used to measure the wavelength of a travelling electron as a function of its momentum. A fundamental relation between momentum and wavelength is predicted by quantum theory. The analysis tested the theory and determine an accurate value for Planck’s constant h.
3-1 De Broglie experiment :( De Broglie Wave)
De Broglie was postulated that the wavelength of objects (The wave-like properties of matter)was given by ? =h/p, where h is Planck’s constant, and p = mv is the linear momentum.
De Broglie s thesis started from the hypothesis, "that to each portion of energy with a proper mass m0 one may associate a periodic phenomenon of the frequency ?0, such that one finds: h?0 = m0c2. The frequency ?0 is to be measured, of course, in the rest frame of the energy packet.
De Broglie reasoned that his hypothetical intrinsic particle periodic phenomenon is in phase with that phase wave. This was his basic matter-wave conception. He noted the phase wave does not transfer energy.
While the concept of waves being associated with matter is correct, De Broglie did not leap directly to the final understanding of quantum mechanics with no missteps. There are conceptual problems with the approach that De Broglie took in his thesis that he was not able to resolve. These difficulties were resolved by Erwin Schr?dinger, who developed the wave mechanics approach, starting from a somewhat different basic hypothesis.
3-2 Experiment of electron diffraction:

In 1927, however, Clinton Davisson and Lester Germer discovered experimental proof of the wave-like properties of matter particularly electrons (This discovery was quite by mistake!). They were studying electron reflection from a nickel target. That is, electrons were “diffracting” from the crystal planes much like light diffracts from a grating, leading to constructive and destructive interference. Electron diffraction is an extremely important tool used to study new materials.


Figure: 3-1: Electron Diffraction from atomic layers in a crystal.

Consider planes of atoms in a crystal as shown in Fig. 3-1, separated by distance d. Electron ”waves” reflect from each of these planes. Since the electron is wave-like, the combination of the reflections from each interface will lead to an interference pattern. This is completely analogous to light interference, arising, for example, from different path lengths in the Fabry-Perot or Michelson interferometers. The de Broglie wavelength for the electron is given by ? = h/p, where p can be calculated by knowing the energy of the electrons when they leave the “electron gun”:
P^2/2m=e.V_a ------------(1)
Where Va is the accelerating potential. The condition for constructive interference is that the path length difference for the two waves shown in Fig. 3-1 be a multiple of a wavelength. This leads to Bragg’s Law:
n? = 2d sin ? --------- (2)
Where n = 1, 2, is integer (n is order of diffraction). In this experiment, only the first-order diffraction n = 1 is observed. Therefore, the intra-atomic distance in a crystal can be calculated by measuring the angle of electron diffraction and their wavelength (i.e. their momentum):
d=?/(2 sin ?)= 1/(2 sin ? h) . h/?(2e.me.Va) -------(3)
where h is Planck’s constant, e is the electronic charge, me is the electron’s mass, and Va is the accelerating voltage.
Summary of the experiment:
Davisson and Germer designed and built a vacuum apparatus for the purpose of measuring the energies of electrons scattered from a metal surface. Electrons from a heated filament (Tungsten metal W) were accelerated by a voltage and allowed to strike the surface of nickel metal.
The electron beam was directed at the nickel target, which could be rotated to observe angular dependence of the scattered electrons. Their electron detector (called Faraday box) was mounted on an arc so that it could be rotated to observe electron at different angles. It was a great surprise to them to find that at certain angles there was a peak in the intensity of the scattered by the electron beam. This peak indicated wave behaviour for the electrons and could be interpreted by Bragg law to give values for the lattice spacing in nickel crystal.


At last the wavelength of the electron beam is 0.165nm but the same wavelength of electron beam according to De Broglie is 0.166nm, this very much closer between them and another provident to failure of Bohr theory.
3-3 Compton scattering:
Compton scattering phenomena is discovered by Arthur Holly Compton, is the inelastic scattering of a photon by a charged particle, usually an electron. It results in a decrease in energy (increase in wavelength) of the photon (which may be X-ray or gamma-ray photon), called the Compton Effect. Part of the energy of the photon is transferred to the recoiling electron. Inverse Compton scattering exists, in which a charged particle transfers part of its energy to a photon.
Compton scattering is an example of inelastic scattering of light by a free charged particle, where the wavelength of the scattered light is different from that of the incident radiation. In Compton s original experiment, the energy of the X-ray photon (?17 keV) was very much larger than the binding energy of the atomic electron, so the electrons could be treated as being free. The amount by which the light s wavelength changes is called the Compton shift. Although nuclear Compton scattering exists, Compton scattering usually refers to the interaction involving only the electrons of an atom.
The effect is important because it demonstrates that light cannot be explained purely as a wave phenomenon. Thomson scattering, the classical theory of an electromagnetic wave scattered by charged particles, cannot explain low-intensity shifts in wavelength. The light must behave as if it consists of particles if we are to explain low-intensity Compton scattering.



Figure:3-2. Diagram of Compton scattering phenomena.

Applying the conservation laws of energy, momentum and mass according to the axis.
The energy of photon before interference - Energy of photon after interference = Energy of electron after interference- Energy of electron before interference.
hc/?- hc/?^- = mc^2- m_o c^2--------4
The difference in Energy of photon = Difference in Energy of Electron





3-4 Photoelectric effect:
The photoelectric effect or photoemission is the production of electrons or other free carriers when light is shone onto a material. Electrons emitted in this manner can be called photoelectrons. The phenomenon is commonly studied in electronic physics, as well as in fields of chemistry, such as quantum chemistry or electrochemistry.
According to classical electromagnetic theory, this effect can be attributed to the transfer of energy from the light to an electron. From this perspective, an alteration in either the intensity or wavelength of light would induce changes in the rate of emission of electrons from the metal.
Electrons are dislodged only by the impingement of photons when those photons reach or exceed a threshold frequency (energy). Below that threshold, no electrons are emitted from the metal regardless of the light intensity or the length of time of exposure to the light. To make sense of the fact that light can eject electrons even if its intensity is low.
The photoelectric effect requires photons with energies approaching zero (in the case of negative electron affinity) to over 1 MeV for core electrons in elements with a high atomic number. Emission of conduction electrons from typical metals usually requires a few electron-volts, corresponding to short-wavelength visible or ultraviolet light. Study of the photoelectric effect led to important steps in understanding the quantum nature of light and electrons and influenced the formation of the concept of wave-particle duality.
Emission mechanism:
The photons of a light beam have a characteristic energy proportional to the frequency of the light.
In the photoemission process, if an electron within some material absorbs the energy of one photon and acquires more energy than the work function (the electron binding energy) of the material, it is ejected. If the photon energy is too low, the electron is unable to escape the material.
An increase in the intensity of low-frequency light will only increase the number of low-energy photons sent over a given interval of time, this change in intensity will not create any single photon with enough energy to dislodge an electron.
The energy of the emitted electrons does not depend on the intensity of the incoming light, but only on the energy (equivalent frequency) of the individual photons. It is an interaction between the incident photon and the outermost electrons.
All of the energy from one photon must be absorbed and used to liberate one electron from atomic binding, or else the energy is re-emitted.
If the photon energy is absorbed, some of the energy liberates the electron from the atom, and the rest contributes to the electron s kinetic energy as a free particle.


Figure: 3-4. Photoelectric effect (low energy phenomena).

3-5 Black body radiation:
Black-body radiation is one type of electromagnetic radiation within or surrounding a body in thermodynamic equilibrium with its environment, or emitted by a black body and theory to be held at constant, uniform temperature. The radiation has a specific spectrum and intensity that depends only on the temperature of the body.
The thermal radiation spontaneously emitted by many ordinary objects can be approximated as blackbody radiation.
A black-body at room temperature appears black, as most of the energy it radiates is infra-red and cannot be perceived by the human eye. Because the human eye cannot perceive colour at very low light intensities, a black body, viewed in the dark at the lowest just faintly visible temperature, subjectively appears grey (but only because the human eye is sensitive only to black and white at very low intensities - in reality, the frequency of the light in the visible range would still be red, although the intensity would be too low to discern as red), even though its objective physical spectrum peaks in the infrared range. When it becomes a little hotter, it appears dull red. As its temperature increases further it eventually becomes blue-white.
The term black body was introduced by Gustav Kirchhoff in 1860. Black-body radiation is also called complete radiation or temperature radiation or thermal radiation.
Experimentally, black-body radiation may be established best as the ultimately stable steady-state equilibrium radiation in a cavity in a rigid body, at a uniform temperature, that is entirely opaque and is only partly reflective. A closed box of graphite walls at a constant temperature with a small hole on one side produces a good approximation to ideal black-body radiation emanating from the opening.
Black-body radiation becomes a visible glow of light if the temperature of the object is high enough. The Draper point is the temperature at which all solids glow a dim red, about 798 K. At 1000 K, a small opening in the wall of a large uniformly heated opaque-walled cavity (let us call it a furnace), viewed from outside, looks red; at 6000 K, it looks white. No matter how the oven is constructed, or of what material, as long as it is built so that almost all light entering is absorbed by its walls, it will contain a good approximation to black-body radiation. The spectrum, and therefore colour, of the light that comes out, will be a function of the cavity temperature alone. A graph of the amount of energy inside the oven per unit volume and per unit frequency interval plotted versus frequency is called the black-body curve. Different curves are obtained by varying the temperature.
This means that at the thermodynamic equilibrium of the amount for every wavelength in every direction of thermal radiation emitted by a body at temperature T, black or not is equal to the corresponding amount that the body absorbs because it is surrounded by light at temperature T.
When the body is black, the absorption is obvious: the amount of light absorbed is all the light that hits the surface. For a black body much bigger than the wavelength, the light energy absorbed at any wavelength ? per unit time is strictly proportional to the black-body curve. This means that the black-body curve is the amount of light energy emitted by a black body, which justifies the name. This is the condition for the applicability of Kirchhoff s law of thermal radiation: the black-body curve is characteristic of thermal light, which depends only on the temperature of the walls of the cavity, provided that the walls of the cavity are completely opaque and are not very reflective, and that the cavity is in thermodynamic equilibrium. E?T ?1/?------5


Figure: 3-6. Black body radiation spectrum.

3-6 Uncertainty principles of Heisenberg:
The uncertainty principle also called the Heisenberg Uncertainty Principle, or Indeterminacy Principle, articulated (1927) by the German physicist Werner Heisenberg, that the position and the velocity of an object cannot both be measured exactly, at the same time, even in theory. The very concepts of exact position and exact velocity together, in fact, have no meaning in nature.
It is easy to measure both the position and the velocity of, say, an automobile because the uncertainties implied by this principle for ordinary objects are too small to be observed. The complete rule stipulates that the product of the uncertainties in position and velocity is equal to or greater than a tiny physical quantity, or constant (about 10-34 joule-second, the value of the quantity h (where h is Planck s constant). Only for the exceedingly small masses of atoms and subatomic particles does the product of the uncertainties become significant.

The uncertainty principle is alternatively expressed in terms of a particle s momentum and position. The momentum of a particle is equal to the product of its mass times its velocity. The momentum of a wave is given by its wavelength. A wave packet like a photon or electron is a composite of many waves. Therefore, it must be made of many momentums. But how can an object have many momentums?
A characteristic feature of quantum is the principle of complementarity, which "implies the impossibility of any sharp separation between the behaviour of atomic objects and the interaction with the measuring instruments which serve to define the conditions under which the phenomena appear." As a result, "evidence obtained under different experimental conditions cannot be comprehended within a single picture, but must be regarded as complementary in the sense that only the totality of the phenomena exhausts the possible information about the objects." Mathematically, a description of the uncertainty principle as the following, where `x is position and `p is momentum:

This is perhaps the most famous equation next to E=mc2 in physics.
It basically says that the combination of the error in position times the error in momentum must always be greater than Planck s constant. So, you can measure the position of an electron to some accuracy, but then its momentum will be inside a very large range of values. Likewise, you can measure the momentum precisely, but then its position is unknown.


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