Chapter 28Quantum Theory

Quantum RegimeMacroscopic world explanations fail at the atomic-

scale worldNewtonian mechanicsMaxwell’s equations describing electromagnetism

The atomic-scale world is referred to as the quantum regime

Quantum refers to a very small increment or parcel of energy

The discovery and development of quantum theory began in the late 1800s and continued during the early 1900s

Waves vs. ParticlesIn the world of Newton

and Maxwell, energy can be carried by particles and waves

Waves produce an interference pattern when passed through a double slit

Classical particles (bullets) will pass through one of the slits and no interference pattern will be formed

Section 28.1

Particles and Waves, ClassicalWaves exhibit inference; particles do notParticles often deliver their energy in discrete

amountsThe energy carried or delivered by a wave is not

discreteThe energy carried by a wave is described by its

intensityThe amount of energy absorbed depends on the

intensity and the absorption time

Section 28.1

Interference with Electrons

The separation between waves and particles is not found in the quantum regime

Electrons are used in a double slit experimentThe blue lines show the probability of the electrons

striking particular locationsSection 28.1

Interference with Electrons, cont.The probability curve of the electrons has the same form

as the variation of light intensity in the double-slit interference experiment

The experiment shows that electrons undergo constructive interference at certain locations on the screen

At other locations, the electrons undergo destructive interferenceThe probability for an electron to reach those location is

very small or zeroThe experiment also shows aspects of particle-like

behavior since the electrons arrive one at a time at the screen

Section 28.1

Particles and Waves, QuantumAll objects, including light and electrons, can exhibit

interferenceAll objects, including light and electrons, carry

energy in discrete amountsThese discrete “parcels” are called quanta

Section 28.1

Work FunctionIn the 1880s, studies of

what happens when light is shone onto a metal gave some results that could not be explained with the wave theory of light

The work function, Wc is the minimum energy required to remove a single electron from a piece of metal

Section 28.2

Work Function, cont.A metal contains electrons that are free to move

around within the metalThe electrons are still bound to the metal and need

energy to be removed from the atomThis energy is the work function

The value of the work function is different for different metals

If V is the electric potential at which electrons begin to jump across the vacuum gap, the work function is Wc = eV

Section 28.2

Work Functions of Metals

Section 28.2

Photoelectric EffectAnother way to extract

electrons from a metal is by shining light onto it

Light striking a metal is absorbed by the electrons

If an electron absorbs an amount of light energy greater than Wc, it is ejected off the metal

This is called the photoelectric effect

Section 28.2

Photoelectric Effect, cont.No electrons are

emitted unless the light’s frequency is greater than a critical value ƒc

When the frequency is above ƒc, the kinetic energy of the emitted electrons varies linearly with the frequency

Photoelectric Effect, ExplanationTrying to explain the photoelectric effect with the classical

wave theory of light presented two difficultiesExperiments showed that the critical frequency is

independent of the intensity of the light Classically, the energy is proportional to the intensity It should always be possible to eject electrons by increasing the

intensity to a sufficiently high value Below the critical frequency there, are no ejected electrons no

matter how great the light intensityThe kinetic energy of an ejected electron is independent of

the light intensity Classical theory predicts increasing the intensity will cause the

ejected electrons to have a higher kinetic energy Experiments actually show the electron kinetic energy depends on

the light’s frequency

Section 28.2

PhotonsEinstein proposed that light carries energy in discrete

quanta, now called photonsEach photon carries a parcel of energy Ephoton = hƒ

h is a constant of nature called Planck’s constanth = 6.626 x 10-34 J ∙ s

A beam of light should be thought of as a collection of photonsEach photon has an energy dependent on its frequency

If the intensity of monochromatic light is increased, the number of photons is increased, but the energy carried by each photon does not change

Section 28.2

Photoelectric Effect, Explanation 2Photon explanation accounts for the difficulties in the

classical explanationThe absorption of light by an electron is just like a

collision between two particles, a photon and an electron The photon carries an energy that is absorbed by the

electron If this energy is less the work function, the electron is not

able to escape from the metal The energy of a single photon depends on frequency but not

on the light intensity

Section 28.2

Explanation 2, cont.The kinetic energy of the ejected electrons depends on

light frequency but not intensityThe critical frequency corresponds to photons whose

energy is equal to the work function

h ƒc = Wc This photon is just ejected and would have no kinetic

energyIf the photon has a higher energy, the difference goes into

kinetic energy of the ejected electron

KEelectron = h ƒ - h ƒc = h ƒ - Wc

This linear relationship is what was found experimentallySection 28.2

Momentum of a PhotonA light wave with energy E also carries a certain

momentum

“Particles” of light called photons carry a discrete amount of both energy and momentum

Photons have two properties that are different than classical particlesPhotons do not have any massPhotons exhibit interference effects

photon

E hƒ hp

c c λ

Section 28.2

Blackbody RadiationBlackbody radiation is

emitted over a range of wavelengths

To the eye, the color of the cavity is determined by the wavelength at which the radiation intensity is largest

Section 28.2

Blackbody Radiation, ClassicalThe blackbody intensity curve has the same shape for a

wide variety of objectsElectromagnetic waves form standing waves as they

reflect back and forth inside the oven’s cavityThe frequencies of the standing waves follow the pattern

ƒn = n ƒ where n = 1, 2, 3, … There is no limit to the value of n, so the frequency can

be infinitely largeBut as the frequency increases, so does the energyClassical theory predicts that the blackbody intensity

should become infinite as the frequency approaches infinity

Section 28.2

Blackbody Radiation, QuantumThe disagreement between the classical predictions

and experimental observations was called the “ultraviolet catastrophe”

Planck proposed solving the problem by assuming the energy in a blackbody cavity must come in discrete parcels

Each parcel would have energy E = h ƒn His theory fit the experimental results, but gave no

reason why it workedPlanck’s work is generally considered to be the

beginning of quantum theorySection 28.2

Particle-Wave Nature of LightSome phenomena can only be understood in terms

of the particular nature of lightPhotoelectric effectBlackbody radiation

Light also has wave properties at the same timeInterference

Light has both wave-like and particle-like properties

Section 28.2

Wave-like Properties of ParticlesThe notion that the properties of both classical

waves and classical particles are present at the same time is also called wave-particle duality and it essential for understanding the microscale world

The possibility that all particles are capable of wave-like properties was first proposed by Louis de Broglie

De Broglie suggested that if a particle has a momentum p, its wavelength is

hλ

p

Section 28.3

Electron InterferenceTo test de Broglie’s

hypothesis, an experiment was designed to observe interference involving classical particles

The experiment showed conclusively that electrons have wavelike properties

The calculated wavelength was in good agreement with de Broglie’s theory

Section 28.3

Wavelengths of Macroscopic ParticlesFrom de Broglie’s equation and using the classical

expression for kinetic energy

As the mass of the particle (object) increases, its wavelength decreases

In principle, you could observe interference with baseballsHas not yet been observed

h hλ

p m(KE)

2

Section 28.3

Electron Spin Electrons have another

quantum property that involves their magnetic behavior

An electron has a magnetic moment, a property associated with electron spin

Classically, the electron’s magnetic moment can be thought of as spinning ball of charge

Section 28.4

Electron Spin, cont.The spinning ball of

charge acts like a collection of current loops

This produces a magnetic field

It acts like a small bar magnet

Therefore, it is attracted to or repelled from the poles of other magnets

Section 28.4

Electron Spin, Directions

When a beam of electrons passes near one end of a bar magnetic, there are two directions of deflection observed

Two orientations for the electron magnetic moment are possibleClassical theory predicts the moment may point in any

directionSection 28.4

Electron Spin, Direction, cont.Classically, the electrons should deflect over a range of

anglesObserving only two directions of deflection indicates

there are only two possible orientations for the magnetic moment

The electron magnetic moment is quantized with only two possible values

Quantization of the electron’s magnetic moment applies to both direction and magnitude

All electrons under all circumstances act as identical bar magnets

Section 28.4

Quantization of Electron SpinClassical explanation of electron spin

Circulating charge acts as a current loopThe current loops produce a magnetic fieldThis result is called the spring magnetic momentYou can also say the electron has spring angular

momentumThe classical ideas do not explain the two directions after

the beam of electrons pass the magnetQuantum explanation

Only spin up or spin down are possibleOther quantum particles also have spin angular

momentum and a resulting magnetic moment

Section 28.4

Wave FunctionIn the quantum world, the motion of a particle-wave

is described by its wave functionThe wave function can be calculated from

Schrödinger’s equationDeveloped by Erwin Schrödinger, one of the inventors

of quantum theorySchrödinger’s equation plays a role similar to

Newton’s laws of motion since it tells how the wave function varies with time

In many situations, the solutions of the Schrödinger equation are similar to standing waves

Section 28.5

Wave Function Example

An electron is confined to a particular region of spaceA classical particle would travel back and forth inside

the boxThe wave function for the electron is described by

standing wavesTwo possible waves are shown

Section 28.5

Wave Function Example, cont.The wave function solutions correspond to electrons

with different kinetic energiesThe wavelengths of the standing waves are different

Given by de Broglie’s equationAfter finding the wave function, one can calculate

the position and velocity of the electronBut does not give a single value

The wave function allows for the calculation of the probability of finding the electron at different locations in space

Section 28.5

Heisenberg Uncertainty PrincipleFor a particle-wave, quantum effects place

fundamental limits on the precision of measuring position or velocity

The standing waves are the electron, so there is an inherent uncertainty in its position

There is some probability for finding the electron at virtually any spot in the box

The uncertainty, Δx, is approximately the size of the boxThis uncertainty is due to the wave nature of the

electron

Section 28.5

Uncertainty, Example

Electrons are incident on a narrow slitThe electron wave is diffracted as it passes through the slitThe interference pattern gives a measure of how the wave function

of the electron is distributed throughout space after it passes through the slit

The width of the slit affects the interference pattern The narrower the slit, the broader the distribution pattern

Section 28.5

Uncertainties in Position and MomentumThe position of the electron passing through a slit is

known with an uncertainty Δx equal to the width of the slit

Since the outgoing electrons have a spread in their momentum along x, there is some uncertainty Δpx in the x component of the momentum

The uncertainties Δx and Δp are absolute limits set by quantum theory

Section 28.5

Heisenberg Uncertainty PrincipleThe Heisenberg Uncertainty Principle gives the

lower limit on the product of Δx and Δp

The relationship holds for any quantum situation and for any wave-particle

hx p

π

4

Section 28.5

Explaining the Uncertainty PrincipleThe Heisenberg uncertainty principle dictated that in the

quantum regime, the uncertainties in x and p are connected

Under the very best of circumstance, the product of Δx and Δp is a constant, proportional to h

If you measure a particle-wave’s position with great accuracy, you must accept a large uncertainty its momentum

If you know the momentum very accurately, you must accept a large position uncertainty

You cannot make both uncertainties small at the same time

Section 28.5

Heisenberg Time-Energy UncertaintyYou can also derive a relation between the

uncertainties in the energy ΔE of a particle and the time interval Δt over which this energy is measured or generated

The Heisenberg energy-time uncertainty principle is

The uncertainty in energy measured over a time period is negligibly small for a macroscopic object

hE t

π

4

Section 28.5

Heisenberg Uncertainty Principle, finalQuantum theory and the uncertainty principle mean that

there is always a trade-off between the uncertaintiesIt is not possible, even in principle, to have perfect

knowledge of both x and pThis suggest that there is always some inherent uncertainty

in our knowledge of the physical universeQuantum theory says that the world is inherently

unpredictableFor any macroscale object, the uncertainties in the real

measurement will always be much larger than the inherent uncertainties due to the Heisenberg uncertainty relation

Section 28.5

Third Law of ThermodynamicsAccording to the Third Law of Thermodynamics, it is

not possible to reach the absolute zero of temperature

In a classical kinetic theory picture, the speed of all particles would be zero at absolute zeroThere is nothing in classical physics to prevent that

In quantum theory, the Heisenberg uncertainty principle indicates that the uncertainty in the speed of a particle cannot be zero

The uncertainty principle provides a justification of the third law of thermodynamics

Section 28.5

Tunneling

According to classical physics, an electron trapped in a box cannot escape

A quantum effect called tunneling allows an electron to escape under certain circumstances

Quantum theory allows the electron’s wave function to penetrate a short distance into the wall

Section 28.6

Tunneling, cont.The wave function extends a short distance into the

classically forbidden regionAccording to Newton’s mechanics, the electron must

stay completely inside the box and cannot go into the wall

If two boxes are very close together so that the walls between them are very thin, the wave function can extend from one box into the next box

The electron has some probability for passing through the wall

Section 28.6

Scanning Tunneling MicroscopeA scanning tunneling

microscope (STM) operates by using tunneling

A very sharp tip is positioned near a conducting surface

If the separation is large, the space between the tip and the surface acts as a barrier for electron flow

Scanning Tunneling Microscope, cont.The barrier is similar to a wall since it prevents

electrons from leaving the metalIf the tip is brought very close to the surface, an

electron may tunnel between themThis produces a tunneling current

By measuring this current as the tip is scanned over the surface, it is possible to construct an image of how atoms are arranged on the surface

The tunneling current is highest when the tip is closest to an atom

Section 28.6

STM Image

Section 28.6

STM, finalTunneling plays a dual role in the operation of the

STMThe detector current is produced by tunneling

Without tunneling there would be no image

Tunneling is needed to obtain high resolution The tip is very sharp, but still has some rounding The electrons can tunnel across many different paths

See fig. 28.17 C The majority of electrons that tunnel follow the shortest path The STM can form images of individual atoms although the

tip is larger than the atoms

Section 28.6

Color VisionWave theory cannot explain color visionLight is detected in the retina at the back of the eyeThe retina contains rods and cones

Both are light-sensitive cellsWhen the cells absorb light, they generate an

electrical signal that travels to the brainRods are more sensitive to low light intensities and

are used predominately at nightCones are responsible for color vision

Section 28.7

RodsAbout 10% of the light that enters your eye reaches

the retinaThe other 90% is reflected or absorbed by the cornea

and other parts of the eyeThe absorption of even a single photon by a rod cell

causes the cell to generate a small electrical signalThe signal from an individual cell is not sent directly

to the brainThe eye combine the signals from many rod cells

before passing the combination signal along the optic nerve

Section 28.7

ConesThe retina contains three

types of cone cellsThey respond to light of

different colorsThe brain deduces the

color of light by combining the signals from all three types of cones

Each type of cone cell is most sensitive to a particular frequency, independent of the light intensity

Section 28.7

Cones, cont.The explanation of color vision depends on two

aspects of quantum theoryLight arrives at the eye as photons whose energy

depends on the frequency of the light When an individual photon is absorbed by a cone, the

energy of the photon Is taken up by a pigment molecule within the cell

The energy of the pigment molecule is quantized Photon absorption is possible because the difference in

energy levels in the various pigments match the energy of the photon

Cones, final

In the simplified energy level diagram (A), a pigment molecule can absorb a photon only if the photon energy precisely matches the pigment energy level

More realistically (C), a range of energies is absorbedQuantum theory and the existence of quantized energies for

both photons and pigment molecules lead to color vision

Section 28.7

The Nature of QuantaThe principles of conservation of energy,

momentum, and charge are believed to hold true under all circumstancesMust allow for the existence of quanta

The energy and momentum of a photon come in discrete quantized units

Electric charge also comes in quantized unitsThe true nature of electrons and photons are

particle-waves

Section 28.8

Puzzles About QuantaThe relation between gravity and quantum theory is a

major unsolved problemNo one knows how Planck’s constant enters the theory of

gravitation or what a quantum theory of gravity looks likeWhy are there two kinds of charge?

Why do the positive and negative charges come in the same quantized units?

What new things happen in the regime where the micro- and macroworlds meet?How do quantum theory and the uncertainty principle apply

to living things?

Section 28.8

Review!Quantum Mechanics

Work Function and Photoelectric Effect

PhotonsEphoton = hƒ

h is Planck’s constanth = 6.626 x 10-34 J ∙ s

photon

E hƒ hp

c c λ

De Broglie WavelengthWave Particle Duality of

Classical Objects

h hλ

p m(KE)

2

Electron ‘Spin’

Stern Gerlach Experiment

Chapter 29Atomic Theory

Atomic TheoryMatter is composed of atomsAtoms are composed of electrons, protons, and

neutronsAtoms were discovered after Galileo, Newton, and

Maxwell and most other physicists discussed so far had completed their work

Quantum theory explains the way atoms are put together

The central goal of atomic theory is to understand why different elements have different propertiesCan explain the organization of the periodic table

Structure of the AtomBy about 1890, most physicists and chemists believed

matter was composed of atomsIt was widely believed that atoms were indivisibleEvidence for this picture of the atoms were the gas laws

and the law of definite proportionsThe law of definite proportions says that when a compound

is completely broken down into its constituent elements, the masses of the constituent always have the same proportions

It is now known that all the elements were composed of three different types of particlesElectrons, protons, and neutrons

Section 29.1

Questions to be Answered by Atomic TheoryWhat are the basic properties of these atomic

building blocks?Mass, charge, size, etc. of each particle

How do just these three building blocks combine to make so many different kinds of atoms?

Experiments determined the properties and behavior of the particles

The behavior cannot be explained by Newton’s mechanics

The ideas of quantum mechanics are needed to understand the structure of the atom

Section 29.1

Plum Pudding ModelElectrons were the first

building-block particle to be discovered

The model suggested that the positive charge of the atom is distributed as a “pudding” with electrons suspended throughout the “pudding”

Section 29.1

Plum Pudding Model, cont.A neutral atom has zero total electric charge

An atom must contain a precise amount of positive “pudding”

How was that accomplished?Physicists studied how atoms collide with other

atomic-scale particlesExperiments carried out by Rutherford, Geiger and

Marsden

Section 29.1

Planetary ModelRutherford expected

the relatively massive alpha particles would pass freely through the plum-pudding atom

A small number of alpha particles were actually deflected through very large anglesSome bounced

backwardSection 29.1

Planetary Model, cont.The reaction of the alpha particle could not be explained

by the plum-pudding modelRutherford realized that all the positive charge in an atom

must be concentrated in a very small volumeThe mass and density of the positive charge was about the

same as the alpha particleMost alpha particles missed this dense region and

passed through the atomOccasionally an alpha particle collided with the dense

region, giving it a large deflectionHe concluded that atoms contain a nucleus that is

positively charged and has a mass much greater than that of the electron

Section 29.1

Planetary Model, finalRutherford suggested that the atom is a sort of

miniature solar systemThe electrons orbit the nucleus just as the planets

orbit the sunThe electrons must move in orbits to avoid falling into

the nucleus as a result of the electric forceThe atomic nucleus contains protons

The charge on a proton is +eSince the total charge on an atom is zero, the

number of protons must equal the number of electrons

Section 29.1

Atomic Number and NeutronsThe atomic number of the element is the number of

protons its containsSymbolized by Z

Nuclei, except for hydrogen, also contain neutronsThe neutron is a neutral particle

Zero net electric chargeThe neutron was discovered in the 1930sProtons are positively charged and repel each otherThe protons are attracted to the neutrons by an

additional force that overcomes the Coulomb repulsion and holds the nucleus together

Section 29.1

Energy of Orbiting ElectronThe planetary model of

the hydrogen atom is shown

Contains one proton and one electron

The electric force supplies a centripetal force

The speed of the electron is

kev

mr

2

Section 29.1

Energy of Orbiting Electron, cont.This speed corresponds to a kinetic energy of the

electron of 1.2 x 10-18 J = 7.5 eVThis is close to the measured ionization energy of

the hydrogen atom of 13.6 eVThe ionization energy is the energy required to remove

an electron from an atom in the gas phaseThe electron also has potential energy

The change in potential energy when the atom is ionized is 14 eV

Section 29.1

Major Problem with the Planetary ModelStability of the electron orbit

Since the electrons are undergoing accelerated motion, they should emit electromagnetic radiation

As the electron loses energy, it should spiral into inward to the nucleus

The atom would be inherently unstable

It should only last a fraction of a second

There was no way to fix the planetary model to make the atom stable

Quantum Theory SolutionQuantum theory avoids the problem of unstable

electronsQuantum theory says the electrons are not simple

particles that obey Newton’s laws and spiral into the nucleus

The electron is a wave-particle described by a wave function with discrete energy levels

Electrons gain or lose energy only when they undergo a transition between energy levels

Section 29.1

Atomic SpectraThe best evidence that an electron can exist only in

discrete energy levels comes from the radiation an atom emits or absorbs when an electron undergoes a transition from one energy level to another

This was related to the question of what gives an object its color

Physicists of that time knew about the relationship between blackbody radiation and temperature

Section 29.2

Sun’s Spectra

The sun’s spectrum shows sharp dips superimposed on the smooth blackbody curve

The dips are called lines because of their appearanceThe dips show up as dark linesThe locations of the dips indicate the wavelengths at which the

light intensity is lower than the expected blackbody value

Section 29.2

Formation of Spectra

When light from a pure blackbody source passes through a gas, atoms in the gas absorb light at certain wavelengths

The values of the wavelengths have been confirmed in the laboratory

Section 29.2

Absorption and EmissionThe dark spectral lines are called absorption linesThe atoms can also produce an emission spectrumThe absorption and emission lines occur at the same

wavelengthsThe pattern of spectral lines is different for each elementQuestions

Why do the lines occur at specific wavelengths?Why do absorption and emission lines occur at the same

wavelength?What determines the pattern of wavelengths?Why are the wavelengths different for different elements?

Section 29.2

Photon EnergyThe energy of a photon is Ephoton = h ƒSince energy is conserved, the energy of the photon is the

difference in the energy of the atom before and after emission or absorption

Since atomic emission occurs only at certain discrete wavelengths, the energy of the orbiting electron can only have certain discrete values

According to Newton’s mechanics, the radius of the electron’s orbit can have a continuous range of values

Based on Newton’s mechanics, there is no way for the planetary orbit picture to give discrete electron energies So there is no way to explain the existence of discrete spectral

linesThe problem is resolved in quantum mechanics’ explanation of

the electron’s state in terms of a wave function instead of an orbitSection 29.2

Atomic Energy LevelsThe energy of an atom is

quantizedThe energy of an absorbed

or emitted photon is equal to the difference in energy between two discrete atomic energy levels

The wavelength (or frequency) of the line gives the spacing between the atom’s energy levels

Explained the experimental evidence of discrete spectral lines

Section 29.2

Bohr Model of the AtomExperiments showed that Rutherford’s planetary

model of the atom did not workNiels Bohr invented another model called the Bohr

modelAlthough not perfect, this model included ideas of

quantum theoryBased on Rutherford’s planetary modelIncluded discrete energy levels

Section 29.3

Ideas In Bohr’s ModelCircular electron orbits

For simplificationUse hydrogen

Simplest atomPostulated only certain electron orbits are allowed

To explain discrete spectral linesOnly specific values of r are allowedThen only specific energies are allowed based on the

values of rEnergy level diagrams can be used to show absorption

and emission of photonsExplained the experimental evidence

Section 29.3

Energy Levels

Each allowed orbit is a quantum state of the electronE1 is the ground state

The state of lowest possible energy for the atomOther states are excited statesPhotons are emitted when electrons fall from higher to lower statesWhen photons are absorbed, the electron undergoes a transition to a

higher state

Section 29.3

Angular Momentum and rTo determine the allowed values of r, Bohr proposed

that the orbital angular momentum of the electron could only have certain values

n = 1, 2, 3, … is an integer and h is Planck’s constantCombining this with the orbital motion of the

electron, the radii of allowed orbits can be found

Section 29.3

2h

nL

22

22

4 mke

hnr

Values of rThe only variable is n

The other terms in the equation for r are constantsThe orbital radius of an electron in a hydrogen atom

can have only these valuesShows the orbital radii are quantized

The smallest value of r corresponds to n = 1This is called the Bohr radius of the hydrogen atom

and is the smallest orbit allowed in the Bohr modelFor n = 1, r = 0.053 nm

Section 29.3

Energy ValuesThe energies corresponding to the allowed values of r

can also be calculated

The only variable is n, which is an integer and can have values n = 1, 2, 3, …

Therefore, the energy levels in the hydrogen atom are also quantized

For the hydrogen atom, this becomes

tot elec

π k e mE KE PE

h n

2 2 4

2 2

2 1

tot

. eVE

n

2

13 6

Section 29.3

Energy Level Diagram for HydrogenThe negative energies

come from the convention that PEelec = 0 when the electron is infinitely far from the proton

The energy required to take the electron from the ground state and remove it from the atom is the ionization energy

The arrows show some possible transitions leading to emissions of photons

Section 29.3

Quantum Theory and the Kinetic Theory of GasesQuantum theory explains the claim that the collisions

between atoms in a gas are elasticAt room temperature, the kinetic energy of the

colliding atoms is smaller than the spacing between the ground and the excited states

A collision does not involve enough energy to cause a transition to a higher level

The atoms stay in their ground state None of their kinetic energy is converted into potential

energy of the atomic electrons

Section 29.3

X-Rays from AtomsThe highest photon energy available in a hydrogen

atom is in the ultraviolet part of the electromagnetic spectrum

Other atoms can emit much more energetic photonsMay applications use X-ray photons obtained from

an electron transition from E2 to E1 in heavier atomsThis are called K X-raysSee table 29.1 for the energy of K X-rays produced by

some elements

Section 29.3

Continuous SpectrumIf an absorbed photon has

more energy than is needed to ionize an atom, the extra energy goes into the kinetic energy of the ejected electron

This final energy can have a range of values and so the absorbed energy can have a range of values

This produces a continuous spectrum

Section 29.3

Quantized Angular MomentumBohr’s suggestion that the angular momentum of the

electron is quantized was completely newOther assumptions could be traced to Einstein’s theory

of the photon and conservation of energy in atomic transitions

The assumption of quantized angular momentum can be understood in terms of de Broglie’s theoryWhich came about 10 years after Bohr made the

assumptionDe Broglie stated that electrons have a wave

character, with a wavelength of λ = h / p

Section 29.3

Bohr and de Broglie

The allowed electron orbits in the Bohr model correspond to standing waves that fit into the orbital circumference

Since the circumference has to be an integer number of wavelengths, 2 π r = n λ

This leads to Bohr’s condition for angular momentum

Section 29.3

Problems with Bohr’s ModelThe Bohr model was successful for atoms with one

electronH, He+, etc.

The model does not correctly explain the properties of atoms or ions that contain two or more electrons

Physicists concluded that the Bohr model is not the correct quantum theoryIt was a “transition theory” that help pave the way from

Newton’s mechanics to modern quantum mechanics

Section 29.3

Modern Quantum MechanicsModern quantum mechanics depends on the ideas

of wave functions and probability densities instead of mechanical ideas of position and motion

To solve a problem in quantum mechanics, you use Schrödinger’s equationsThe solution gives the wave function, including its

dependence on position and timeFour quantum numbers are required for a full

description of the electron in an atomBohr’s model used only one

Section 29.4

Quantum Numbers, Summary

Section 29.4

Principle Quantum Numbern is the principle quantum number

It can have values n = 1, 2, 3, …It is roughly similar to Bohr’s quantum numberAs n increases, the average distance from the electron

to the nucleus increasesState with a particular value of n are referred to as a

“shell”

Section 29.4

Orbital Quantum Numberℓ is the orbital quantum

numberAllowed values are ℓ = 0,

1, 2, … n - 1The angular momentum

of the electron is proportional to ℓ States with ℓ = 0 have no

angular momentum

See the table for shorthand letters for varies ℓ values

Section 29.4

Orbital Magnetic Quantum Numberm is the orbital magnetic quantum number

It has allowed values of m = - ℓ, -ℓ + 1, … , -1, 0, 1 … , ℓ

You can think of m as giving the direction of the angular momentum of the electron in a particular state

Section 29.4

Spin Quantum Numbers is the spin quantum number

s = + ½ or – ½ These are often referred to as “spin up” and “spin down”

This gives the direction of the electron’s spin angular momentum

Section 29.4

Electron Shells and ProbabilitiesA particular quantized electron state is specified by

all four of the quantum number n, ℓ, m and sThe solution of Schrödinger’s equation also gives

the wave function of each quantum stateFrom the wave function, you can calculate the

probability for finding the electron at different location around the nucleus

Plots of probability distributions for an electron are often called “electron clouds”

Section 29.4

Electron Clouds

Section 29.4

Electron Cloud ExampleGround state of hydrogen

n = 1The only allowed state for ℓ is ℓ = 0

This is an “s state”

The only allowed state for m is m = 0The allowed states for s are s = ± ½

The probability of finding an electron at a particular location does not depend on s, so both of these states have the same probability

The electron probability distribution forms a spherical “cloud” around the nucleus See fig. 29.17 A

Section 29.4

Hydrogen Electrons, finalThe electron probability distributions for all states

are independent of the value of the spin quantum number

For the hydrogen atom, the electron energy depends only on the value of n and is independent of ℓ, m and sThis is not true for atoms with more than one electron

Section 29.4

Anti Hydrogen!

http://www.nature.com/news/2010/101117/full/468355a.html

Multielectron AtomsThe electron energy levels of multielectron atoms

follow the same pattern as hydrogenUse the same quantum numbers

The electron distributions are also similarThere are two main differences between hydrogen

and multielectron atomsThe values of the electron energies are different for

different atomsThe spatial extent of the electron probability clouds

varies from element to element

Section 29.5

Pauli Exclusion PrincipleEach quantum state can be occupied by only one

electronEach electron must occupy its own quantum state,

different from the states of all other electronsThis is called the Pauli exclusion principleEach electron is described by a unique set of

quantum numbers

Section 29.5

Electric Distribution

The direction of the arrow represents the electron’s spin

In C, the He electrons have different spins and obey the Pauli exclusion principle Section 29.5

Electron ConfigurationThere is a useful shorthand notation for showing electron

configurationsExamples:

1s1 1 – n =1 s – ℓ = 0 Superscript 1 – 1 electron No information about electron spin

1s22s22p2 2 electrons in n = 1 with ℓ = 0 2 electrons in n = 2 with ℓ = 0 2 electrons in n = 2 with ℓ = 1

Section 29.5

Filling Energy LevelsThe energy of each level depends mainly on the

value of nIn multielectron atoms, the order of energy levels is

more complicatedFor shells higher than n = 2, the energies of

subshells from different shells being to overlapIn general, the energy levels fill with electrons in the

following order:

1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f

Section 29.5

Order of Energy Levels

Section 29.5

Quiz!How many electrons are in an atom with electrons

filled up to 4s?A) 20B) 18 C) 16 D) 12E) 42

Chemical Properties of ElementsQuantum theory explains why the periodic table has

its structureThe periodic table was developed by Dmitry

Mendeleyev in the late 1860’sMendeleyev and other chemists had noticed that

many elements could be grouped according to their chemical properties

Mendeleyev organized his table by grouping related elements in the same column

His table had a number of “holes” because many elements had not yet been discovered

Section 29.6

Chemical Properties, cont.Mendeleyev could not explain why the regularities in

the periodic table occurredThe electron energy levels and the electron

configuration of the atom are responsible for its chemical properties

When an atom participates in a chemical reaction, some of its electrons combine with electrons from other atoms to form chemical bonds

The bonding electrons are those occupying the highest energy levels

Section 29.6

Electron Configuration of Some Elements

Section 29.6

Electrons and ShellsThe electron that forms bonds with other atoms is a

valence electronWhen a shell has all possible states filled it forms a

closed shellElements in the same column in the periodic table

have the same number of valence electrons The last column in the periodic table contains

elements with completely filled shellsThese elements are largely inertThey almost never participate in chemical reactions

Section 29.6

Structure of the Periodic TableMendeleyev grouped elements into columns

according to their common bonding properties and chemical reactionsThese properties rely on the valence electrons and

can be traced to the electron configurationsThe rows correspond to different values of the

principle quantum number, nSince the n = 1 shell can hold only two electrons, the

row contains only two elementsThe number of elements in each row can be found by

using the rules for allowed quantum numbers

Section 29.6

Atomic Clocks

Atomic clocks are used as global and US time standardsThe clocks are based on the accurate measurements of

certain spectral line frequenciesCs atoms are popularOne second is now defined as the time it takes a cesium

clock to complete 9,192,631,770 ticks

Section 29.7

Incandescent Light BulbsThe incandescent bulb contains a thin wire

filament that carries a large electric currentType developed by Edison

The electrical energy dissipated in the filament heats it to a high temperature

The filament then acts as a blackbody and emits radiation

Section 29.7

Fluorescent Bulbs

This type of bulb uses gas of atoms in a glass containerAn electric current is passed through the gasThis produces ions and high-energy electronsThe electrons, ions, and neutral atoms undergo many

collisions, causing many of the atoms to be in an excited stateThese atoms decay back to their ground state and emit light

Section 29.7

Neon and Fluorescent BulbsA neon bulb contains a gas of Ne atomsFluorescent bulbs often contain mercury atoms

Mercury emits strongly in the ultravioletThe glass is coated with a fluorescent materialThe photons emitted by the Hg atoms are absorbed by

the fluorescent coatingThe coating atoms are excited to higher energy levelsWhen the coating atoms undergo transitions to lower

energy states, they emit new photonsThe coating is designed to emit light throughout the

visible spectrum, producing “white” light

Section 29.7

LasersLasers depend on the coherent emission of light by many

atoms, all at the same frequencyIn spontaneous emission, each atom emits photons

independently of the other atomsIt is impossible to predict when it will emit a photonThe photons are radiated randomly in all directions

In a laser, an atom undergoes a transition and emits a photon in the presence of many other photons that have energies equal to the atom’s transition energy

A process known as stimulated emission causes the light emitted by this atom to propagate in the same direction and with the same phase as surrounding light waves

Section 29.7

Lasers, cont.

Laser is an acronym for light amplification by stimulated emission of radiation

The light from a laser is thus a coherent sourceMirrors are located at the ends of the bulb (laser tube)One of the mirrors lets a small amount of the light pass

through and leave the laser

Section 29.7

Lasers, finalLaser can be made with a variety of different atomsOne design uses a mixture of Ne and He gas and is

called a helium-neon laserThe photons emitted by the He-Ne laser have a

wavelength of about 633 nmAnother common type of laser is based on light

produced by light-emitting diodes (LEDs)These photons have a wavelength around 650 nmThese are used in optical barcode scanners

Section 29.7

Force Between AtomsConsider two hypothetical

atoms and assume they are bound together to form a molecule

The binding energy of a molecule is the energy require to break the chemical bond between the two atomsA typical bond energy is

10 eV

Section 29.7

Force Between Atoms, cont.Assume the atom is pulled apart by separating the

atoms a distance ΔxThe magnitude of the force between the atoms is

A Δx of 1 nm should be enough to break the chemical bond

This gives a force of ~1.6 x 10-19 N

PEF

x

Section 29.7

Quantum Mechanics and Newtonian MechanicsQuantum mechanics is needed in the regime of

electrons and atoms since Newton’s mechanics fails in that area

Newton’s laws work very well in the classical regimeQuantum theory can be applied to macroscopic

objects, giving results that are virtually identical to Newton’s mechanics

Classical objects have extremely short wavelengths, making the quantum theory description in terms of particle-waves unnecessary

Section 29.8

Where the Regimes MeetPhysicists are actively studying the area where

quantum mechanics and Newtonian mechanics meetOne question concerns the quantum behavior of

living organisms such as viruses

Section 29.8