1. Chapter 28
Quantum Theory

2. Quantum Regime
Macroscopic world explanations fail at the atomic-scale world
Newtonian mechanics
Maxwell’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

3. Waves vs. Particles
In 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

4. Particles and Waves, Classical
Waves exhibit inference; particles do not
Particles often deliver their energy in discrete amounts
The energy carried or delivered by a wave is not discrete
The energy carried by a wave is described by its intensity
The amount of energy absorbed depends on the intensity and the absorption time
Section 28.1

5. Interference with Electrons
The separation between waves and particles is not found in the quantum regime
Electrons are used in a double slit experiment
The blue lines show the probability of the electrons striking particular locations
Section 28.1

6. 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 interference
The probability for an electron to reach those location is very small or zero
The experiment also shows aspects of particle-like behavior since the electrons arrive one at a time at the screen
Section 28.1

7. Particles and Waves, Quantum
All objects, including light and electrons, can exhibit interference
All objects, including light and electrons, carry energy in discrete amounts
These discrete “parcels” are called quanta
Section 28.1

8. Work Function
In 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

9. Work Function, cont.
A metal contains electrons that are free to move around within the metal
The electrons are still bound to the metal and need energy to be removed from the atom
This 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

10. Work Functions of Metals
Section 28.2

11. Photoelectric Effect
Another 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

12. 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

13. Photoelectric Effect, Explanation
Trying to explain the photoelectric effect with the classical wave theory of light presented two difficulties
Experiments 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 intensity
The 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

14. Photons
Einstein proposed that light carries energy in discrete quanta, now called photons
Each photon carries a parcel of energy Ephoton = hƒ
h is a constant of nature called Planck’s constant
h = x J ∙ s
A beam of light should be thought of as a collection of photons
Each 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

15. Photoelectric Effect, Explanation 2
Photon explanation accounts for the difficulties in the classical explanation
The 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

16. Explanation 2, cont.
The kinetic energy of the ejected electrons depends on light frequency but not intensity
The 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 energy
If 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 experimentally
Section 28.2

17. Momentum of a Photon
A 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 particles
Photons do not have any mass
Photons exhibit interference effects
Section 28.2

18. Blackbody Radiation
Blackbody 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

19. Blackbody Radiation, Classical
The blackbody intensity curve has the same shape for a wide variety of objects
Electromagnetic waves form standing waves as they reflect back and forth inside the oven’s cavity
The 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 large
But as the frequency increases, so does the energy
Classical theory predicts that the blackbody intensity should become infinite as the frequency approaches infinity
Section 28.2

20. Blackbody Radiation, Quantum
The 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 worked
Planck’s work is generally considered to be the beginning of quantum theory
Section 28.2

21. Particle-Wave Nature of Light
Some phenomena can only be understood in terms of the particular nature of light
Photoelectric effect
Blackbody radiation
Light also has wave properties at the same time
Interference
Light has both wave-like and particle-like properties
Section 28.2

22. Wave-like Properties of Particles
The 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
Section 28.3

23. Electron Interference
To 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

24. Wavelengths of Macroscopic Particles
From 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 baseballs
Has not yet been observed
Section 28.3

25. 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

26. 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

27. 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 possible
Classical theory predicts the moment may point in any direction
Section 28.4

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

29. Quantization of Electron Spin
Classical explanation of electron spin
Circulating charge acts as a current loop
The current loops produce a magnetic field
This result is called the spring magnetic moment
You can also say the electron has spring angular momentum
The classical ideas do not explain the two directions after the beam of electrons pass the magnet
Quantum explanation
Only spin up or spin down are possible
Other quantum particles also have spin angular momentum and a resulting magnetic moment
Section 28.4

30. Wave Function
In the quantum world, the motion of a particle-wave is described by its wave function
The wave function can be calculated from Schrödinger’s equation
Developed by Erwin Schrödinger, one of the inventors of quantum theory
Schrödinger’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

31. Wave Function Example
An electron is confined to a particular region of space
A classical particle would travel back and forth inside the box
The wave function for the electron is described by standing waves
Two possible waves are shown
Section 28.5

32. Wave Function Example, cont.
The wave function solutions correspond to electrons with different kinetic energies
The wavelengths of the standing waves are different
Given by de Broglie’s equation
After finding the wave function, one can calculate the position and velocity of the electron
But 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

33. Heisenberg Uncertainty Principle
For 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 box
This uncertainty is due to the wave nature of the electron
Section 28.5

34. Uncertainty, Example Electrons are incident on a narrow slit
The electron wave is diffracted as it passes through the slit
The 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

35. Uncertainties in Position and Momentum
The 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

36. Heisenberg Uncertainty Principle
The 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
Section 28.5

37. Explaining the Uncertainty Principle
The 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

38. Heisenberg Time-Energy Uncertainty
You 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
Section 28.5

39. Heisenberg Uncertainty Principle, final
Quantum theory and the uncertainty principle mean that there is always a trade-off between the uncertainties
It is not possible, even in principle, to have perfect knowledge of both x and p
This suggest that there is always some inherent uncertainty in our knowledge of the physical universe
Quantum theory says that the world is inherently unpredictable
For 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

40. Third Law of Thermodynamics
According 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 zero
There 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

41. 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

42. Tunneling, cont.
The wave function extends a short distance into the classically forbidden region
According 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

43. Scanning Tunneling Microscope
A 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

44. Scanning Tunneling Microscope, cont.
The barrier is similar to a wall since it prevents electrons from leaving the metal
If the tip is brought very close to the surface, an electron may tunnel between them
This 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

45. STM Image
Section 28.6

46. STM, final Tunneling plays a dual role in the operation of the STM
The 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 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

47. Color Vision Wave theory cannot explain color vision
Light is detected in the retina at the back of the eye
The retina contains rods and cones
Both are light-sensitive cells
When the cells absorb light, they generate an electrical signal that travels to the brain
Rods are more sensitive to low light intensities and are used predominately at night
Cones are responsible for color vision
Section 28.7

48. Rods About 10% of the light that enters your eye reaches the retina
The other 90% is reflected or absorbed by the cornea and other parts of the eye
The absorption of even a single photon by a rod cell causes the cell to generate a small electrical signal
The signal from an individual cell is not sent directly to the brain
The eye combine the signals from many rod cells before passing the combination signal along the optic nerve
Section 28.7

49. Cones The retina contains three types of cone cells
They respond to light of different colors
The 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

50. Cones, cont.
The explanation of color vision depends on two aspects of quantum theory
Light 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

51. 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 absorbed
Quantum theory and the existence of quantized energies for both photons and pigment molecules lead to color vision
Section 28.7

52. The Nature of Quanta
The principles of conservation of energy, momentum, and charge are believed to hold true under all circumstances
Must allow for the existence of quanta
The energy and momentum of a photon come in discrete quantized units
Electric charge also comes in quantized units
The true nature of electrons and photons are particle-waves
Section 28.8

53. Puzzles About Quanta
The relation between gravity and quantum theory is a major unsolved problem
No one knows how Planck’s constant enters the theory of gravitation or what a quantum theory of gravity looks like
Why 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

54. Review!
Quantum Mechanics

55. Work Function and Photoelectric Effect

56. Photons
Ephoton = hƒ
h is Planck’s constant
h = x J ∙ s

57. De Broglie Wavelength
Wave Particle Duality of Classical Objects

58. Electron ‘Spin’

59. Stern Gerlach Experiment

60. Chapter 29
Atomic Theory

61. Atomic Theory Matter is composed of atoms
Atoms are composed of electrons, protons, and neutrons
Atoms 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 properties
Can explain the organization of the periodic table

62. Structure of the Atom
By about 1890, most physicists and chemists believed matter was composed of atoms
It was widely believed that atoms were indivisible
Evidence for this picture of the atoms were the gas laws and the law of definite proportions
The 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 particles
Electrons, protons, and neutrons
Section 29.1

63. Questions to be Answered by Atomic Theory
What 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

64. Plum Pudding Model
Electrons 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

65. 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 particles
Experiments carried out by Rutherford, Geiger and Marsden
Section 29.1

66. Planetary Model
Rutherford 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 angles
Some bounced backward
Section 29.1

67. Planetary Model, cont.
The reaction of the alpha particle could not be explained by the plum-pudding model
Rutherford realized that all the positive charge in an atom must be concentrated in a very small volume
The mass and density of the positive charge was about the same as the alpha particle
Most alpha particles missed this dense region and passed through the atom
Occasionally an alpha particle collided with the dense region, giving it a large deflection
He concluded that atoms contain a nucleus that is positively charged and has a mass much greater than that of the electron
Section 29.1

68. Planetary Model, final
Rutherford suggested that the atom is a sort of miniature solar system
The electrons orbit the nucleus just as the planets orbit the sun
The electrons must move in orbits to avoid falling into the nucleus as a result of the electric force
The atomic nucleus contains protons
The charge on a proton is +e
Since the total charge on an atom is zero, the number of protons must equal the number of electrons
Section 29.1

70. Atomic Number and Neutrons
The atomic number of the element is the number of protons its contains
Symbolized by Z
Nuclei, except for hydrogen, also contain neutrons
The neutron is a neutral particle
Zero net electric charge
The neutron was discovered in the 1930s
Protons are positively charged and repel each other
The protons are attracted to the neutrons by an additional force that overcomes the Coulomb repulsion and holds the nucleus together
Section 29.1

71. Energy of Orbiting Electron
The 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
Section 29.1

72. Energy of Orbiting Electron, cont.
This speed corresponds to a kinetic energy of the electron of 1.2 x J = 7.5 eV
This is close to the measured ionization energy of the hydrogen atom of 13.6 eV
The ionization energy is the energy required to remove an electron from an atom in the gas phase
The electron also has potential energy
The change in potential energy when the atom is ionized is 14 eV
Section 29.1

73. Major Problem with the Planetary Model
Stability 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

74. Quantum Theory Solution
Quantum theory avoids the problem of unstable electrons
Quantum 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

75. Atomic Spectra
The 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

76. Sun’s Spectra
The sun’s spectrum shows sharp dips superimposed on the smooth blackbody curve
The dips are called lines because of their appearance
The dips show up as dark lines
The locations of the dips indicate the wavelengths at which the light intensity is lower than the expected blackbody value
Section 29.2

77. 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

78. Absorption and Emission
The dark spectral lines are called absorption lines
The atoms can also produce an emission spectrum
The absorption and emission lines occur at the same wavelengths
The pattern of spectral lines is different for each element
Questions
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

79. Photon Energy The energy of a photon is Ephoton = h ƒ
Since 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 lines
The problem is resolved in quantum mechanics’ explanation of the electron’s state in terms of a wave function instead of an orbit
Section 29.2

80. Atomic Energy Levels The energy of an atom is quantized
The 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

81. Bohr Model of the Atom
Experiments showed that Rutherford’s planetary model of the atom did not work
Niels Bohr invented another model called the Bohr model
Although not perfect, this model included ideas of quantum theory
Based on Rutherford’s planetary model
Included discrete energy levels
Section 29.3

82. Ideas In Bohr’s Model Circular electron orbits Use hydrogen
For simplification
Use hydrogen
Simplest atom
Postulated only certain electron orbits are allowed
To explain discrete spectral lines
Only specific values of r are allowed
Then only specific energies are allowed based on the values of r
Energy level diagrams can be used to show absorption and emission of photons
Explained the experimental evidence
Section 29.3

83. Energy Levels Each allowed orbit is a quantum state of the electron
E1 is the ground state
The state of lowest possible energy for the atom
Other states are excited states
Photons are emitted when electrons fall from higher to lower states
When photons are absorbed, the electron undergoes a transition to a higher state
Section 29.3

84. Angular Momentum and r
To 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 constant
Combining this with the orbital motion of the electron, the radii of allowed orbits can be found
Section 29.3

85. Values of r The only variable is n
The other terms in the equation for r are constants
The orbital radius of an electron in a hydrogen atom can have only these values
Shows the orbital radii are quantized
The smallest value of r corresponds to n = 1
This is called the Bohr radius of the hydrogen atom and is the smallest orbit allowed in the Bohr model
For n = 1, r = nm
Section 29.3

86. Energy Values
The 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
Section 29.3

87. Energy Level Diagram for Hydrogen
The 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

88. Quantum Theory and the Kinetic Theory of Gases
Quantum theory explains the claim that the collisions between atoms in a gas are elastic
At 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

89. X-Rays from Atoms
The highest photon energy available in a hydrogen atom is in the ultraviolet part of the electromagnetic spectrum
Other atoms can emit much more energetic photons
May applications use X-ray photons obtained from an electron transition from E2 to E1 in heavier atoms
This are called K X-rays
See table 29.1 for the energy of K X-rays produced by some elements
Section 29.3

90. Continuous Spectrum
If 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

91. Quantized Angular Momentum
Bohr’s suggestion that the angular momentum of the electron is quantized was completely new
Other 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 theory
Which came about 10 years after Bohr made the assumption
De Broglie stated that electrons have a wave character, with a wavelength of λ = h / p
Section 29.3

92. 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

93. Problems with Bohr’s Model
The Bohr model was successful for atoms with one electron
H, 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 theory
It was a “transition theory” that help pave the way from Newton’s mechanics to modern quantum mechanics
Section 29.3

94. Modern Quantum Mechanics
Modern 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 equations
The solution gives the wave function, including its dependence on position and time
Four quantum numbers are required for a full description of the electron in an atom
Bohr’s model used only one
Section 29.4

95. Quantum Numbers, Summary
Section 29.4

96. Principle Quantum Number
n is the principle quantum number
It can have values n = 1, 2, 3, …
It is roughly similar to Bohr’s quantum number
As n increases, the average distance from the electron to the nucleus increases
State with a particular value of n are referred to as a “shell”
Section 29.4

97. Orbital Quantum Number
ℓ is the orbital quantum number
Allowed values are ℓ = 0, 1, 2, … n - 1
The 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

98. Orbital Magnetic Quantum Number
m 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

99. Spin Quantum Number s 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

100. Electron Shells and Probabilities
A particular quantized electron state is specified by all four of the quantum number n, ℓ, m and s
The solution of Schrödinger’s equation also gives the wave function of each quantum state
From 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

101. Electron Clouds
Section 29.4

102. Electron Cloud Example
Ground state of hydrogen
n = 1
The only allowed state for ℓ is ℓ = 0
This is an “s state”
The only allowed state for m is m = 0
The 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 A
Section 29.4

103. Hydrogen Electrons, final
The 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 s
This is not true for atoms with more than one electron
Section 29.4

104. Anti Hydrogen!

105. Multielectron Atoms
The electron energy levels of multielectron atoms follow the same pattern as hydrogen
Use the same quantum numbers
The electron distributions are also similar
There are two main differences between hydrogen and multielectron atoms
The values of the electron energies are different for different atoms
The spatial extent of the electron probability clouds varies from element to element
Section 29.5

106. Pauli Exclusion Principle
Each quantum state can be occupied by only one electron
Each electron must occupy its own quantum state, different from the states of all other electrons
This is called the Pauli exclusion principle
Each electron is described by a unique set of quantum numbers
Section 29.5

107. 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

108. Electron Configuration
There is a useful shorthand notation for showing electron configurations
Examples:
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

109. Filling Energy Levels
The energy of each level depends mainly on the value of n
In multielectron atoms, the order of energy levels is more complicated
For shells higher than n = 2, the energies of subshells from different shells being to overlap
In 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

110. Order of Energy Levels
Section 29.5

111. Quiz!
How many electrons are in an atom with electrons filled up to 4s?
A) 20
B) 18
C) 16
D) 12
E) 42

112. Chemical Properties of Elements
Quantum theory explains why the periodic table has its structure
The periodic table was developed by Dmitry Mendeleyev in the late 1860’s
Mendeleyev 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

113. Chemical Properties, cont.
Mendeleyev could not explain why the regularities in the periodic table occurred
The 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

114. Electron Configuration of Some Elements
Section 29.6

115. Electrons and Shells
The electron that forms bonds with other atoms is a valence electron
When a shell has all possible states filled it forms a closed shell
Elements 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 shells
These elements are largely inert
They almost never participate in chemical reactions
Section 29.6

116. Structure of the Periodic Table
Mendeleyev grouped elements into columns according to their common bonding properties and chemical reactions
These properties rely on the valence electrons and can be traced to the electron configurations
The rows correspond to different values of the principle quantum number, n
Since the n = 1 shell can hold only two electrons, the row contains only two elements
The number of elements in each row can be found by using the rules for allowed quantum numbers
Section 29.6

117. Atomic Clocks Atomic clocks are used as global and US time standards
The clocks are based on the accurate measurements of certain spectral line frequencies
Cs atoms are popular
One second is now defined as the time it takes a cesium clock to complete 9,192,631,770 ticks
Section 29.7

118. Incandescent Light Bulbs
The incandescent bulb contains a thin wire filament that carries a large electric current
Type 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

119. Fluorescent Bulbs
This type of bulb uses gas of atoms in a glass container
An electric current is passed through the gas
This produces ions and high-energy electrons
The electrons, ions, and neutral atoms undergo many collisions, causing many of the atoms to be in an excited state
These atoms decay back to their ground state and emit light
Section 29.7

120. Neon and Fluorescent Bulbs
A neon bulb contains a gas of Ne atoms
Fluorescent bulbs often contain mercury atoms
Mercury emits strongly in the ultraviolet
The glass is coated with a fluorescent material
The photons emitted by the Hg atoms are absorbed by the fluorescent coating
The coating atoms are excited to higher energy levels
When the coating atoms undergo transitions to lower energy states, they emit new photons
The coating is designed to emit light throughout the visible spectrum, producing “white” light
Section 29.7

121. Lasers
Lasers depend on the coherent emission of light by many atoms, all at the same frequency
In spontaneous emission, each atom emits photons independently of the other atoms
It is impossible to predict when it will emit a photon
The 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

122. Lasers, cont.
Laser is an acronym for light amplification by stimulated emission of radiation
The light from a laser is thus a coherent source
Mirrors 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

123. Lasers, final Laser can be made with a variety of different atoms
One design uses a mixture of Ne and He gas and is called a helium-neon laser
The photons emitted by the He-Ne laser have a wavelength of about 633 nm
Another common type of laser is based on light produced by light-emitting diodes (LEDs)
These photons have a wavelength around 650 nm
These are used in optical barcode scanners
Section 29.7

124. Force Between Atoms
Consider 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 atoms
A typical bond energy is 10 eV
Section 29.7

125. Force Between Atoms, cont.
Assume the atom is pulled apart by separating the atoms a distance Δx
The 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 N
Section 29.7

126. Quantum Mechanics and Newtonian Mechanics
Quantum 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 regime
Quantum 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

127. Where the Regimes Meet
Physicists are actively studying the area where quantum mechanics and Newtonian mechanics meet
One question concerns the quantum behavior of living organisms such as viruses
Section 29.8