Quantum Mechanics and Modern Physics

Science & Engineering Magnet High Mr. Puckett

First Atomic Theory

• • The notion that all matter consists of fundamental particles called atoms was first put forward by the Greek philosophers Leucippus and his disciple Democritus, in the 5th century BC. These men taught that everything is composed of infinitely tiny indivisible particles called atoms. The word atom, from the Greek, means "indivisible."

First Atomic Theory continued

• •

The notion of atoms was rejected by other philosophers--most significantly Aristotle , who believed all matter was infinitely divisible. Others believed there were only four elements: earth, air, fire, and water. These "nonatomic" beliefs dominated Western thought for centuries. Only in the early modern era did the concept of atoms regain acceptance. Today, however, atoms are known to be divisible into subatomic particles, such as electrons, protons, neutrons, and quarks [1]

John Dalton’s Atomic Theory

•

John Dalton's Atomic theory in the late 1700's explained the nature of chemical reactions and the similarity of certain elements. It included:

–

A. All elements are composed of tiny indivisible particles called atoms. ( Incorrect in long run)

– –

B. Atoms of the same element are identical. C. Atoms of different elements can combine with one another to form compounds.

–

D. Chemical reactions occur when atoms are separated, joined or rearranged.

JJ Thompson discovers the Electron

• •

The discoverer of the electron as a separate subatomic particle was J.J. Thompson in 1897. He realized that the accepted model of an indivisible atom did not take electrons and protons into account. He used a cathode ray tube that bent an electron beam in E&M fields. He suggested a revised model that was compared to a "plum pudding atom" because it said that negatively charged electrons (raisins) stuck into a lump of positively charged protons (the dough).

JJ Thompson’s Cathode Ray Tube

•

Thompson generated “electric rays” by using a pair of oppositely charged plates that were set in an evacuated tube. When this was done, a small glowing spot appeared at the opposite end of the tube. Thompson noticed that if a magnetic field was applied to the beam, the spot on the opposite side of the tube moved. This implied that the “ray” was composed of a negatively charged particle which responded to the magnetic field according to the equation F = qvB =ma.

Thompson’s Cathode Ray Tube

• Thompson’s CRT measured the charge to mass ratio and gave evidence of electrons.

Electrons from a CRT

What Thompson had discovered were

electrons

, which were being stripped off the cathode by the strong voltage as shown in the diagram. Although he was not able to see individual electrons, the amount of deflection they experienced while traveling through the tube depended upon the electrons mass and charge.

Millikan’s Oil Drop Experiment

• •

Millikan used an atomizer to create tiny drops of oil that were given an electric charge and allowed to fall between two charged plates. The mass of a given oil drop can be calculated by the rate at which it falls when the electric field is turned off. The electric field is then turned on and the drop is brought to a halt. At this point the electrical force on the drop and the gravitational force on the drop are equal (qE=mg) and the total charge on each drop can be calculated.

Millikan’s Oil Drop Experiment

Determined the Elemental Charge on the Electron

• After calculating the charge on many drops, Millikan noticed that the charge on each drop was always a multiple of a common factor, 1.6x10

-19 C, which he reasoned was the fundamental electric charge. Although he did not know it at the time, he had discovered the charge of the electron.

Ernest Rutherford Discovered the Nucleus with the



- Gold Foil Lab

• •

Ernest Rutherford discovered the nucleus in 1911 and proposed the nuclear atom in which electrons surround a dense nucleus.

He thought of the rest of the atom as empty space. But the electrons are negatively charged and the nucleus (protons) are positively charged.

Rutherford’s Discovery of the Nucleus with the Gold Foil Lab

• •

After the discovery that radioactive elements emitted “rays” of various types, physicists rushed out to shine beams of rays at different substances.

Rutherford shone a beam of alpha particles, actually a beam of helium nuclei, at a thin sheet of gold foil. Most alpha particles behaved as expected, being deflected slightly or not at all. However, occasionally an alpha particle would be knocked almost backwards.

The Dense Nucleus makes itself Known in a Big Way

• • •

Rutherford said, “This unexpected result was equivalent to firing an artillery shell at a sheet of tissue paper and having the artillery shell bounce back!” These results implied that the alpha particles would occasionally strike a small, incredibly dense object – Rutherford had discovered the nucleus! Note: this question was asked in both 82 and 97.

Rutherford’s Gold Foil Lab

Rutherford Experiment CloseUp

The Bohr Atomic Model (Solar System)

• • •

Neils Bohr developed the next stage of the atomic theory in 1913 with the Bohr model. It proposes that the electrons are in concentric circular orbits around the nucleus. The model was patterned after our solar system with the sun in the center (nucleus) and the planets (electrons) orbiting around it. The energy that the orbiting provides prevents the electrons from falling into the nucleus

Energy Levels of Electrons

• • •

The ENERGY LEVEL of an electron is the region around the nucleus where it is most likely to be found. The different energy levels are analogous to the rungs of a ladder. The higher you go up the ladder (away from the nucleus) the higher the energy .

Electrons can also climb the ladder and jump from one energy level to the next ( energy must be provided or taken away in the proper amount).

Energy Orbitals of the Electrons

• Electron energy orbitals are the regions where there is the greatest chance to find them as clouds

Electron Orbitals

• •

Electrons cannot stay between levels and will naturally migrate to their appropriate level. However, unlike the rungs of a ladder, the energy levels are not equally spaced. A QUANTUM of energy is the amount of energy required to move an electron from its present energy to the next higher level. Thus the energies of electrons are said to be quantized. The term , quantum leap, is used to describe an abrupt change

The Birth of Quantum Mechanics

• • It all began when Max Planck (1900) was trying to explain the glow of a hot glowing “blackbody” like an electric stove eye. A black object absorbs all wavelengths of light, yet glows red with high temperature. Higher temps yield yellow and white light. The spectrum fit an empirical formula when he assumed that the energy was not continuous, but small discrete amounts. These amounts were called

Quanta

Origin of the Word Quantum

• •

The light emitted by a glowing piece of iron, for instance, was actually "grainy," composed of minuscule light "grains" too small to be seen. Planck called a light "grain" a quantum, from the Latin word meaning "how much?"

Temperature and Wavelength of Light

• • Wein’s Law was the basis for the wavelength calculations based upon temperature for Planck’s energy constant.

Formula:  T = 2.9 x 10 -3 m .

K

Quanta comes in Specific Amounts

• • •

Planck proposed that electrons, for some unknown reason, can give off light only in certain specific amounts of light energy- in quanta. Only whole quanta can be given off, never a fraction of a quantum. The energy of these quanta varies directly with the frequency of the light. Energetic light of higher frequency, such as violet or ultraviolet light, consists of higher-energy quanta than does light of lower frequency, such as red or infrared light.

Planck’s Constant Describes the Energy of a Quantum

• •

The energy of Planck’s constant is the energy needed to promote electrons to the next higher energy orbital; based upon frequency. E = hf The formula became E=nhf is the whole number multiple of “h” (Planck’s constant = 6.6 x10 -34 where “n” J .

s) and “f” is the frequency of light photons.

Planck’s Constant of KE vs. Frequency

Planck’s Constant Energy Level

•

Planck's constant is expressed in terms of energy multiplied by time--a unit called action--and may be given in erg-seconds or joule-seconds. An erg is defined as the amount of energy needed to raise a milligram (roughly the weight of a grain of sand) a distance of 1 centimeter (about 1/3 inch). This is not a great deal of energy.

Einstein Uses Planck’s Constant for the Photoelectric Effect

• •

In 1905 the German-born physicist Albert Einstein used Planck's quantum theory to explain the photoelectric effect, in which charged particles such as electrons are emitted from certain materials when light (electromagnetic radiation) strikes the materials mostly metals.

This is the topic of Einstein’s Nobel Prize not Relativity.

Planck’s Threshold Electron Ejection

Einstein Explained Planck’s Constant with the PE of the Photoelectric Effect

• • •

Albert Einstein said that the electrons around an atom were trapped in a potential energy “well”. If an electron was to escape the well it would have to be struck by a single photon of light which would have enough energy to “kick” the electron out of the well. Chemists call this the ionization constant – the amount of energy needed to remove electrons.

This question was asked on the AP test in 1997.

The KE of Electrons with Escape Velocity from the Atom

•

Photons with a frequency of f o have just enough energy to accomplish this. Photons with higher frequencies not only have enough energy for the electron to escape, but have extra energy to give the electron additional kinetic energy, KE max in the diagram.

Work Function: Exciting the Electrons Up

• • The Energy required to take the electron from the one level and promote it to a higher level is found with the Work Function: W=  E = hf o ; where h is Planck’s constant and f o is the threshold frequency to promote the electron.

The KE of an ejected electron is the quantum energy – the work function. KE= hf – hf o The difference is the amount of energy for the kinetic energy ½ mv 2 . .

Photon Energy Problem

Photon Speed and Energy

Photoelectric Effect Diagram

• In this lab the light shines on the metal and has enough energy that it knocks electrons off the metal into a detector that causes a current through the circuit.

Einstein proposes Quanta Energy Levels of Electrons

• •

Einstein also proposed that electrons, besides emitting electromagnetic radiation in quanta, also absorb it in quanta. Einstein's work demonstrated that electromagnetic radiation has the characteristics of both a wave--because the fields of which it is composed rise and fall in strength--and a particle--because the energy is contained in separate "packets." These packets were later called PHOTONS.

Compton’s Scattering Effect

• •

This experiment was similar to the Rutherford’s experiment except that the beam was composed of particles of light, called photons. In this case a photon stuck an atom, knocked an electron off the target, and was then deflected. The only way a photon can “knock” an atom out of an electron is if the photon had momentum. This suggested that photons were particles. However, the scattered photon did not seem to change speed during the collision, but rather changed their frequency.

Compton’s Scattering Effect

• • This suggested that photons were actually waves that travel at the speed of light, changing frequency as energy is lost. Compton’s conclusion? Photons can act as both waves and as particles depending on the situation. This question was asked in 1982.

Michelson – Morley Determined the Speed of Light

• • Michelson and Morley first proposed the experiment to find the speed of the Earth through the “ether” that filled the universe. A single beam of light was split into two paths and then rejoined at an observation scope. If the Earth was traveling to the right through the “Ether Wind” the light traveling at right angles to the wind would be “blown off course” and require more time to reach the telescope.

Michelson – Morley Experiment Continued

• • • By allowing the two beams to interfere with each other, sight differences in the speed of the two beams could be calculated. By measuring the difference in the speed of the two beams, the speed of the Earth through the “Ether” could be worked out. It turned out that no matter how the experiment was set up, the speed of light in both directions remained constant. Thus… no Ether.

Note: Although this question was asked in both 1982 and 1987, relativity is no longer on the AP

Michelson-Morley Experiment

• The Michelson- Morley Interferometer was used to measure the speed of light and measure the “ Ether”. It never found the ether, but did establish 2.99 x 10 8 m/s as the speed of light.

Nuclear Reactions

• • •

It was originally thought that the fundamental particle of matter was the atom – and that atoms could neither be created nor destroyed. The discovery that atoms were made up of protons, neutrons, and electrons suggested the possibility that one type of atom could be transformed into another type of atom by adding or subtracting these fundamental particles.

The reactions are either FUSION or FISSION.

Nuclear Fusion in the SUN !

• •

This happens every day as hydrogen isotopes are transformed into helium in the sun. In this type of equation, atoms are written in the form where X is the atomic symbol, Z is the atomic number of the atom (basically the number of protons an atom contains) and A is the mass number of the atom (the total number of protons and neutrons in the atom). A Z X Example: 2 1 H is deuterium (heavy hydrogen)

Nuclear Fusion Formula:

• • Fusion is the nuclear reaction that combines the hydrogen isotopes into helium and releases huge amounts of energy as in the sun and stars.

The equation is: 2 1 H + 3 1 H  4 2 He + 1 0 n +E

Nuclear Fission Reaction

• Fission is the process of breaking down the nucleus by physical bombardment with neutrons of other decaying radioactive atoms. • This is the type of reaction that is used in modern nuclear reactors and was the first “Atomic Bomb” mechanism.

Nuclear Fission Reaction

• As the first nucleus decays it gives off 3 neutrons that strike other atoms and cause them to decay in a cascading reaction.

Nuclear Power Reactor

• A nuclear reactor is a complex heat exchanger with a steam driven generator.

The Infamous: E = mc 2

• • Under normal conditions, the total number of fundamental particles in an atomic reaction remains constant. The exception usually occurs in particle accelerators, black holes, and other unpleasant environments where there is enough excess energy to create new matter according to the Einstein’s equation E=mc 2 . An electron and a proton can also combine to form a neutron, which usually occurs only under very high pressures.

Mass and Energy are Equivalent

• •

Atoms form because it requires less energy for two protons and two neutrons to exist as a He atom than as separate particles. Since Einstein showed that mass and energy are equivalent (E=mc 2 ) we can directly measure the energy content of atoms by measuring their mass.

The Loss of Mass in Fusion gives ENERGY !!

• •

Since He consists of two protons and two neutrons, we can estimate the mass of a He atom as 2(m n =1.008665 au) + 2(m p =1.007825 au) = 4.032980 au However, the measured mass of the He atom is “only” 4.002603 au, a difference of 0.0030377 au.

Energy Release from Fusion

• • •

This mass deficit may seem small, but if we use Einstein’s formula to convert this mass into energy we get: E=mc 2 E = (0.00303)(6.66x10

-27 4.5 x10 -13 Joules.

kg)(3x10 8 m/s) 2 = This does not seem like a lot of energy, but remember this is for just one atom.

The process of making a mole of He atoms releases 2.7x10

11 J

Radioactive Decay: The Three Products

•

The weak nuclear force in nature is responsible for Radioactive Decay – the spontaneous splitting of radioactive isotopes gradually into more stable elements and energy release.

•

There are three decay products: the Alpha particle, Beta particle and the Gamma Ray.

Decay Particles from Fission

• • •

The Alpha particle (



) is the nucleus of the Helium atom. When it is given off the atomic number reduces by 2 and mass number by 4 and changes to a new element.

The Beta particle (



) is a high speed electron from a neutron and leaving a proton that increases the atomic number by 1. The Gamma Ray (



) is an energy ray without mass.

Alpha Particle Decay

• • Example: When Uranium-238 decays by an alpha decay the result is a helium nucleus and a thorium-234 atom. Notice that as in any chemical equation, the summation of the mass before and after the reaction adds up.

238 92 U  4 2 He + 234 90 Th

Beta Particle Emission

• • • The beta particle is actually a high speed electron. It originates from the decay of a neutron (0) in the nucleus into a proton(+1) and an electron(-1). This causes a change of +1 to the atomic number and a ZERO change to the Mass number. Remember the electron is 1/1830 the mass of a proton.

Example: If an carbon-14 decays into a nitrogen-14 the formula looks like: 14 6 C  14 7 N + e + a neutrino

Gamma Emission

• • The gamma emission is a photon of very high energy. The decay of a nucleus by emissions of a gamma  ray is much like emission of photons by excited atoms. Except this time it is an excited nucleus with a lot more energy. Since it is only energy, there is NO CHANGE in mass or charge. Gamma’s are deadly ionizing radiation. Neutron bombs work from this mechanism.

A Z N*  A Z N + 

Radioactive Half Life

• • The half-life of a radioactive material is the amount of time it takes for ½ of the mass of a radioactive isotope sample to decay spontaneously into new material. Two versions of the formula are:  N =  N  t and N = N o e  t where  is the decay constant and N is the number of radioactive nuclei. The shortcut formula is T ½ = 0.693/  Half-lives can range from a fraction of a second to billions of years. Carbon-14 has a ½ life of 5370 years while U-238 has a 2.3 billion year half life.

Erwin Schrodinger’s Wave Equations for Electron Orbitals

• • Erwin Schrodinger took the atomic model another step in 1926, when he used the new quantum theory to write and solve an equation describing the location and energy level of an electron. The most modern description of the position of the electrons is the Quantum Mechanical Model. It is not a description of an exact pathway of the electron but is concerned with the likelihood of finding an electron in a certain position. The mathematical probability is artistically portrayed as a blurry cloud of negative charge.

The Quantum Atom Model

• QUANTUM MECHANICAL MODEL of the atom designates the energy levels of electron and are designated by 4 numbers to describe the energy level, orbital and sub-orbital and the spin property.

Quantum Mechanics of Atomic Orbitals

• ATOMIC ORBITALS are regions in space where there is a high probability of finding an electron. There are a maximum of two electrons per orbital. They will fill the atomic orbitals in a specific filling pattern.

• Quantum Mechanics is an accounting system to map out the electrons of an atom.

Quantum Numbers of Hydrogen

• • • • The Principle Quantum number is an integer value of the energy orbital.

The Orbital quantum number is the orbital type: s, p, d, f The Magnetic quantum number is the direction of the angular momentum.

The Spin quantum number is the direction of electron rotation: + ½ or - ½ that gives rise to the magnetic properties within the structural domain.

Quantum Numbers for Electrons

Atomic Orbital Shapes

• • • • Different atomic orbitals are denoted by letters. S - orbitals are spherical clouds.

P-orbitals give pear or dumbbell-shaped clouds. The shapes of d-orbitals and f-orbitals are more complex than what we will study this year.

Just as the clouds in the sky that you see, these clouds of probability are not sharp edged. They just gradually disappear.

Light and Atomic Spectra

• • • The work that led to the development of the quantum mechanical model came from the study of light. Light is considered to consist of electromagnetic waves that travel in a vacuum at the speed of 3 x 10 8 meters per second. Spectroscopy is the study of the light emitted by the electrons when they undergo quantum leaps.

Wave Nature of Matter

• • The properties of light had been debated and researched for years. The photoelectric effect, Compton effect and others predict particle nature. Young’s double slit and Compton’s experiments showed the wave nature.

In 1923 Louis de Broglie proposed that all matter (not just photons) had wave properties.

De Broglie Wavelength of Matter

• • De Broglie proposed that the wavelength of a material particle would related to its momentum with the equation:  = h/mv

• deBroglie wavelength problem

Heisenberg Uncertainty Principle

• • • In 1927 Walter Heisenberg developed the uncertainty principle that explains why we cannot measure the position and momentum of an object (electrons) precisely at the same time. We can measure either property accurately, but not both due to the nature of matter / wave duality.

Another form of the same idea relates energy and time. If we measure the position of a photon, then  x   and  t =  x/c so  t =  /c

Heisenberg Uncertainty Example

An analogy of the uncertainty in measurement concept is this picture that you cannot measure the location of cars due to speed.

Heisenberg Uncertainty Principle

Heisenberg Uncertainty Problem

Position Uncertainty of Electron

Wavelength of an Electron

Photoelectron Speed and Energy

Atomic Spectroscopy

• • • Atomic Spectroscopy is the analytical measurement of the quantum energy level jumps of different electron energy states.

It is a spectral analysis of the colors that an atom gives off (or takes in) when it changes energy levels.

It involves either Emission spectroscopy or Absorption spectroscopy.

Atomic Emission Spectroscopy

• In this technique, the atoms are heated up to the point that the thermal energy promotes the electron up to an excited energy level and then measures the color (wavelength) of light that is given off when the electron collapses back into the ground state.

Energy Level Transitions of Electrons

Spectrum Examples

Energy of Photon Example Problem

Wavelength Problem in Spectroscopy

Lasers

• • • A laser is a device that can produce a very narrow intense beam of monochromatic coherent light.

Coherent means that across any cross section of the beam, all parts would have the same phase.

It uses stimulated emission to stay in phase – An excited electron is stuck by a photon of the same energy gives off a double photon.

Laser Stimulated Emission

• When a photon of light at the same frequency hits an excited electron: The electron produces coherent E& M

AP Problems on Quantum Mechanics and Modern Physics

• • • •

Historical Physics

1982 #7 1997 #6

Atomic Physics

1996 #5 1999 #4

Atomic Energy Levels Photoelectric Effect

• • 1992 #4

1995 #4

1980 #3 1988 #6