Quantum Mechanics

Atomic Theory

dalton_atomsThe ancient philosopher, Heraclitus, maintained that everything is in a state of flux. Nothing escapes change of some sort (it is impossible to step into the same river). On the other hand, Parmenides argued that everything is what it is, so that it cannot become what is not (change is impossible because a substance would have to transition through nothing to become something else, which is a logical contradiction). Thus, change is incompatible with being so that only the permanent aspects of the Universe could be considered real.
An ingenious escape was proposed in the fifth century B.C. by Democritus. He hypothesized that all matter (plus space and time) is composed of tiny indestructible units, called atoms. This idea seems motivated by the question of how finely one can go on cutting up matter. While Democritus performed no experiments and had only the flimsiest evidence for postulating the existence of atoms, his theory was kept alive by the Roman poet Lucretius which survived the Dark Ages to be discovered in 1417. The atoms in Democritus theory themselves remain unchanged, but move about in space to combine in various ways to form all macroscopic objects. Early atomic theory stated that the characteristics of an object are determined by the shape of its atoms. So, for example, sweet things are made of smooth atoms, bitter things are made of sharp atoms.

atom_propIn this manner permanence and flux are reconciled and the field of atomic physics was born. Although Democritus' ideas were to solve a philosophical dilemma, the fact that there is some underlying, elemental substance to the Universe is a primary driver in modern physics, the search for the ultimate subatomic particle. It was John Dalton, in the early 1800's, who determined that each chemical element is composed of a unique type of atom, and that the atoms differed by their masses. He devised a system of chemical symbols and, having ascertained the relative weights of atoms, arranged them into a table. In addition, he formulated the theory that a chemical combination of different elements occurs in simple numerical ratios by weight, which led to the development of the laws of definite and multiple proportions.

He then determined that compounds are made of molecules, and that molecules are composed of atoms in definite proportions. Thus, atoms determine the composition of matter, and compounds can be broken down into their individual elements.

The first estimates for the sizes of atoms and the number of atoms per unit volume where made by Joesph Loschmidt in 1865. Using the ideas of kinetic theory, the idea that the properties of a gas are due to the motion of the atoms that compose it, Loschmidt calculated the mean free path of an atom based on diffusion rates. His result was that there are 6.022x1023 atoms per 12 grams of carbon. And that the typical diameter of an atom is 10-8 centimeters.

states_of_matterMatter exists in four states: solid, liquid, gas and plasma. Plasmas are only found in the coronae and cores of stars. The state of matter is determined by the strength of the bonds between the atoms that makes up matter. Thus, is proportional to the temperature or the amount of energy contained by the matter.

The change from one state of matter to another is called a phase transition. For example, ice (solid water) converts (melts) into liquid water as energy is added. Continue adding energy and the water boils to steam (gaseous water) then, at several million degrees, breaks down into its component atoms.

macro_microAtomic theory is the field of physics that describes the characteristics and properties of atoms that make up matter. The key point to note about atomic theory is the relationship between the macroscopic world (us) and the microscopic world of atoms. For example, the macroscopic world deals with concepts such as temperature and pressure to describe matter. The microscopic world of atomic theory deals with the kinetic motion of atoms to explain macroscopic quantities.
Temperature is explained in atomic theory as the motion of the atoms (faster = hotter). Pressure is explained as the momentum transfer of those moving atoms on the walls of the container (faster atoms = higher temperature = more momentum/hits = higher pressure).


Thomson Atom

The next great step forward in the understanding of atoms was accomplished by John Thomson. Using a cathode ray scope, Thomson determined that all matter, whatever its source, contains particles of the same kind that are much less massive than the atoms of which they form a part. They are now called electrons, although he originally called them corpuscles.

cathodeHis discovery was the result of an attempt to solve a long-standing controversy regarding the nature of cathode rays, which occur when an electric current is driven through a vessel from which most of the air or other gas has been pumped out.

By applying an improved vacuum technique, Thomson was able to put forward a convincing argument that these rays were composed of particles.

Furthermore, these rays seemed to be composed of the same particles, or corpuscles, regardless of what kind of gas carried the electric discharge or what kinds of metals were used as conductors. From these ideas he developed the idea that atoms are made of negative electrons embedded in a gel of positive charge (a "plum pudding" model).thomson

Thomson's conclusion that the corpuscles were present in all kinds of matter was strengthened during the next three years, when he found that corpuscles with the same properties could be produced in other ways; e.g., from hot metals. Thomson may be described as "the man who split the atom" for the first time, although "chipped" might be a better word, in view of the size and number of electrons.


Rutherford Atom

Ernest Rutherford is considered the father of nuclear physics. Indeed, it could be said that Rutherford invented the very language to describe the theoretical concepts of the atom and the phenomenon of radioactivity. Particles named and characterized by him include the alpha particle, beta particle and proton. Rutherford overturned Thomson's atom model in 1911 with his well-known gold foil experiment in which he demonstrated that the atom has a tiny, massive nucleus.

radioactivityBy the turn of the 20th century, physicists knew that certain elements emitted fast moving particles of two flavors, alpha particles and beta particles. These elements were typically very heavy (i.e. their atom nuclei were massive) such as uranium and radium. Today we know that heavy nuclei are unstable and `decay', meaning that they spontaneously split into smaller nuclei and emit stray particles. This is called radioactivity.

nucleus_exp1

The alpha particle was heavy and positively charged, we now know that it is the helium nuclei (2 protons and 2 neutrons). The beta particle was light and negatively charged, the electron. Rutherford designed an experiment to use the alpha particles emitted by a radioactive element as probes to the unseen world of atomic structure.

What was observed was the following:

  • most alpha particles were observed to pass straight through the gold foil
  • a few were scattered at large angles
  • some even bounced back toward the source
  • only a positively charged and relatively heavy target particle, such as the proposed nucleus, could account for such strong repulsion

rutherford_atomResults can best explained by a model for the atom as a tiny, dense, positively charged core called a nucleus, in which nearly all the mass is concentrated, around which the light, negative constituents, called electrons, circulate at some distance, much like planets revolving around the Sun
The Rutherford atomic model has been alternatively called the nuclear atom, or the planetary model of the atom.


Planck's Curve

Parallel to efforts to understand the inner workings of the atom was research in the late-1800's to understand how matter emits energy. The energy an object emits as a function of the wavelength of light is called its continuous spectrum. The field of science that studies spectrum is called spectroscopy. One of the primary results from the field of spectroscopy was the discovery that the spectrum of an object changes with temperature.

The Austrian physicist Josef Stefan found in 1879 that the total radiation energy per unit time emitted by a heated surface per unit area increases as the fourth power of its absolute temperature T (Kelvin scale). This means that the Sun's surface, which is at T = 6,000 K, radiates per unit area (6,000/300)4 = 204 = 160,000 times more electromagnetic energy than does the same area of the Earth's surface, which is taken to be T = 300 K. In 1889 another Austrian physicist, Ludwig Boltzmann, used the second law of thermodynamics to derive this temperature dependence for an ideal substance that emits and absorbs all frequencies. Such an object that absorbs light of all colours looks black, and so was called a blackbody.

The wavelength or frequency distribution of blackbody radiation was studied in the 1890s by Wilhelm Wien of Germany. It was his idea to use as a good approximation for the ideal blackbody an oven with a small hole. Any radiation that enters the small hole is scattered and reflected from the inner walls of the oven so often that nearly all incoming radiation is absorbed and the chance of some of it finding its way out of the hole again can be made exceedingly small. The radiation coming out of this hole is then very close to the equilibrium blackbody electromagnetic radiation corresponding to the oven temperature. Wien found that the radiative energy 'dW' per wavelength interval 'd' has a maximum at a certain wavelength 'm' and that the maximum shifts to shorter wavelengths as the temperature 'T' is increased, as illustrated in the figure below.

Wien's law of the shift of the radiative power maximum to higher frequencies as the temperature is raised expresses in a quantitative form commonplace observations. Warm objects emit infrared radiation, which is felt by the skin; near T = 950 K a dull red glow can be observed; and the color brightens to orange and yellow as the temperature is raised. The tungsten filament of a light bulb is T = 2,500 K hot and emits bright light, yet the peak of its spectrum is still in the infrared according to Wien's law. The peak shifts to the visible yellow when the temperature is T = 6,000 K, like that of the Sun's surface.

Both these relationships were synthesized by physicist Max Planck into what is called Planck's curve. All objects emit energy in the shape of Planck's curve, where the amount and the peak energy vary only as the temperature of the body.radiation_curve

The problem with Planck's curve is that it does not agree with Rutherford's model of the atom. Atoms absorb and emit light through a process called scattering. Photons fly near the atoms and are deflected. Sometimes their motion pushes the atom (where push means electromagnetic forces), and the photon loses energy (i.e. becomes redder). Sometimes the atom pushes the photon, and the photon gains energy (i.e. becomes bluer). However, given the nature of atoms, the photons should receive more energy than the atoms, so there should be more and more blue photons, but clearly the Planck curve drops off at short wavelengths. This is called the UV catastrophe, which was resolved by quantum physics.


Spectrospocy

a3comboThe fact that the temperature and energy generation of an object can be determined by its Planck curve made the study of spectrum the key component to stellar astronomy. The amount of energy emitted from stars is determined by measuring their brightness or the amount of light they emit. This is called photometry. However, two major developments expanded our understanding of the chemical make-up of stars. They were:
The invention of the spectroscope, a device that separates white light into component colors called a spectrum. And the discovery that elements emit a unique spectrum, i.e. produce a unique chemical fingerprint in the spectrum.

The two discoveries combined to produce a new field called spectroscopy, and allowed astronomers to measure the chemical composition of stars for the first time.

solar_spectThe discovery of spectra lines was made by Fraunhofer who, in the early 1800's, magnified the Sun's spectrum and discovered dark lines which could be identified with particular elements (based on spectra in the laboratories).

heliumEnglish astronomer Lockyer, in the late-1800's, discovered an unknown element in the Sun, i.e. a set of spectral lines which did not correspond to elements in the lab. He named this element helium (Latin for Sun element).

By the 20th century, spectral lines for all the elements in the periodic table have been classified and astronomers could examine the chemical composition of stars and planets. What was missing was an explanation for why these lines should exist. Why did particular atoms absorb photons at particular wavelengths?


Kirchhoff's Laws

The first clue to the origin of spectral lines was found by Kirchhoff who, in the mid-1800's, developed the three laws of spectroscopic analysis. Gustav Kirchhoff (b. March 12, 1824, Knigsberg, Prussia [now Kaliningrad, Russia]--d. Oct. 17, 1887, Berlin, Ger.), German physicist who, with the chemist Robert Bunsen, firmly established the theory of spectrum analysis (a technique for chemical analysis by analyzing the light emitted by a heated material), which Kirchhoff applied to determine the composition of the Sun.kirchhoff

In 1847 Kirchhoff became Privatdozent (unsalaried lecturer) at the University of Berlin and three years later accepted the post of extraordinary professor of physics at the University of Breslau. In 1854 he was appointed professor of physics at the University of Heidelberg, where he joined forces with Bunsen and founded spectrum analysis. They demonstrated that every element gives off a characteristic colored light when heated to incandescence. This light, when separated by a prism, has a pattern of individual wavelengths specific for each element. Applying this new research tool, they discovered two new elements, cesium (1860) and rubidium (1861).

Kirchhoff went further to apply spectrum analysis to study the composition of the Sun. He found that when light passes through a gas, the gas absorbs those wavelengths that it would emit if heated. He used this principle to explain the numerous dark lines (Fraunhofer lines) in the Sun's spectrum. That discovery marked the beginning of a new era in astronomy.

In 1875 Kirchhoff was appointed to the chair of mathematical physics at the University of Berlin. Most notable of his published works are Vorlesungen ber mathematische Physik (4 vol., 1876-94; "Lectures on Mathematical Physics") and Gesammelte Abhandlungen (1882; supplement, 1891; "Collected Essays").

There remains a problem with the spectra and the Rutherford model of the atom. Rutherford atoms can absorb or emit photons at any wavelength because the orbit of the electron can take on any distance from the nucleus. However, the spectral lines indicate that there are preferred orbits for the electrons, that they have certain discrete distances from the nuclei. This `quantization' of electron orbits leads to the next great breakthrough in modern physics, quantum mechanics.


Planck's Constant

planck_electronThe UV catastrophe and the dilemma of spectral lines were already serious problems for attempts to understand how light and matter interact. Planck also noticed another fatal flaw in our physics by demonstrating that the electron in orbit around the nucleus accelerates. Acceleration means a changing electric field (the electron has charge), when means photons should be emitted. But, then the electron would lose energy and fall into the nucleus. Therefore, atoms shouldn't exist!

To resolve this problem, Planck made a wild assumption that energy, at the sub-atomic level, can only be transfered in small units, called quanta. Due to his insight, we call this unit Planck's constant (h). The word quantum derives from quantity and refers to a small packet of action or process, the smallest unit of either that can be associated with a single event in the microscopic world.
Quantum, in physics, discrete natural unit, or packet, of energy, charge, angular momentum, or other physical property. Light, for example, appearing in some respects as a continuous electromagnetic wave, on the submicroscopic level is emitted and absorbed in discrete amounts, or quanta; and for light of a given wavelength, the magnitude of all the quanta emitted or absorbed is the same in both energy and momentum. These particle-like packets of light are called photons, a term also applicable to quanta of other forms of electromagnetic energy such as X rays and gamma rays.

All phenomena in submicroscopic systems (the realm of quantum mechanics) exhibit quantization: observable quantities are restricted to a natural set of discrete values. When the values are multiples of a constant least amount, that amount is referred to as a quantum of the observable. Thus Planck's constant h is the quantum of action, and h/ (i.e., h/2 ) is the quantum of angular momentum, or spin.

quantaChanges of energy, such as the transition of an electron from one orbit to another around the nucleus of an atom, is done in discrete quanta. Quanta are not divisible. The term quantum leap refers to the abrupt movement from one discrete energy level to another, with no smooth transition. There is no "in-between."
The quantization, or "jumpiness'' of action as depicted in quantum physics differs sharply from classical physics which represented motion as smooth, continuous change. Quantization limits the energy to be transfered to photons and resolves the UV catastrophe problem.


Wave Particle Duality

The results from spectroscopy (emission and absorption spectra) can only be explained if light has a particle nature as shown by Bohr's atom and the photon description of light.
This dualism to the nature of light is best demonstrated by the photoelectric effect, where a weak UV light produces a current flow (releases electrons) but a strong red light does not release electrons no matter how intense the red light.

An unusual phenomenon was discovered in the early 1900's. If a beam of light is pointed at the negative end of a pair of charged plates, a current flow is measured. A current is simply a flow of electrons in a metal, such as a wire. Thus, the beam of light must be liberating electrons from one metal plate, which are attracted to the other plate by electrostatic forces. This results in a current flow.

In classical physics, one would expect the current flow to be proportional to the strength of the beam of light (more light = more electrons liberated = more current). However, the observed phenomenon was that the current flow was basically constant with light strength, yet varied strong with the wavelength of light such that there was a sharp cutoff and no current flow for long wavelengths.

Einstein successfully explained the photoelectric effect within the context of the new physics of the time, quantum physics. In his scientific paper, he showed that light was made of packets of energy quantum called photons. Each photon carries a specific energy related to its wavelength, such that photons of short wavelength (blue light) carry more energy than long wavelength (red light) photons. To release an electron from a metal plate required a minimal energy which could only be transfered by a photon of energy equal or greater than that minimal threshold energy (i.e. the wavelength of the light had to be sufficiently short). Each photon of blue light released an electron. But all red photons were too weak. The result is no matter how much red light was shown on the metal plate, there was no current.

The photoelectric earned Einstein the Nobel Prize, and introduced the term "photon'' of light into our terminology.

Einstein explained that light exists in a particle-like state as packets of energy (quanta) called photons. The photoelectric effect occurs because the packets of energy carried by each individual red photons are too weak to knock the electrons off the atoms no matter how many red photons you beamed onto the cathode. But the individual UV photons were each strong enough to release the electron and cause a current flow.

It is one of the strange, but fundamental, concepts in modern physics that light has both a wave and particle state (but not at the same time), called wave-particle dualism.

photoelectric_effectWave/particle duality is the possession by physical entities (such as light and electrons) of both wavelike and particle-like characteristics. On the basis of experimental evidence, the German physicist Albert Einstein first showed (1905) that light, which had been considered a form of electromagnetic waves, must also be thought of as particle-like, or localized in packets of discrete energy. The French physicist Louis de Broglie proposed (1924) that electrons and other discrete bits of matter, which until then had been conceived only as material particles, also have wave properties such as wavelength and frequency. Later (1927) the wave nature of electrons was experimentally established. An understanding of the complementary relation between the wave aspects and the particle aspects of the same phenomenon was announced in 1928.

Dualism is not such a strange concept, consider the following picture, are the swirl moving or not or both?

swirl


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