1. Chapter 28 Quantum Physics

2. Simplification Models Particle Model Allowed us to ignore unnecessary details of an object when studying its behavior Systems and rigid objects Extension of particle model Wave Model Two new models Quantum particle Quantum particle under boundary conditions

3. Blackbody Radiation An object at any temperature is known to emit thermal radiation Characteristics depend on the temperature and surface properties The thermal radiation consists of a continuous distribution of wavelengths from all portions of the em spectrum

4. Blackbody Radiation, cont At room temperature, the wavelengths of the thermal radiation are mainly in the infrared region As the surface temperature increases, the wavelength changes It will glow red and eventually white The basic problem was in understanding the observed distribution in the radiation emitted by a black body Classical physics didn’t adequately describe the observed distribution

5. Blackbody Radiation, final A black body is an ideal system that absorbs all radiation incident on it The electromagnetic radiation emitted by a black body is called blackbody radiation

6. Blackbody Approximation A good approximation of a black body is a small hole leading to the inside of a hollow object The nature of the radiation leaving the cavity through the hole depends only on the temperature of the cavity walls

7. Blackbody Experiment Results The total power of the emitted radiation increases with temperature Stefan’s Law P =  A e T 4 For a blackbody, e = 1 The peak of the wavelength distribution shifts to shorter wavelengths as the temperature increases Wien’s displacement law max T = 2.898 x 10 -3 m. K

8. Stefan’s Law – Details P =  Ae T 4 P is the power  is the Stefan-Boltzmann constant  = 5.670 x 10 -8 W / m 2. K 4 Was studied in Chapter 17

9. Wien’s Displacement Law max T = 2.898 x 10 -3 m. K max is the wavelength at which the curve peaks T is the absolute temperature The wavelength is inversely proportional to the absolute temperature As the temperature increases, the peak is “displaced” to shorter wavelengths

10. Intensity of Blackbody Radiation, Summary The intensity increases with increasing temperature The amount of radiation emitted increases with increasing temperature The area under the curve The peak wavelength decreases with increasing temperature

11. Ultraviolet Catastrophe At short wavelengths, there was a major disagreement between classical theory and experimental results for black body radiation This mismatch became known as the ultraviolet catastrophe You would have infinite energy as the wavelength approaches zero

12. Max Planck 1858 – 1947 He introduced the concept of “quantum of action” In 1918 he was awarded the Nobel Prize for the discovery of the quantized nature of energy

13. Planck’s Theory of Blackbody Radiation In 1900, Planck developed a structural model for blackbody radiation that leads to an equation in agreement with the experimental results He assumed the cavity radiation came from atomic oscillations in the cavity walls Planck made two assumptions about the nature of the oscillators in the cavity walls

14. Planck’s Assumption, 1 The energy of an oscillator can have only certain discrete values E n E n = n h ƒ n is a positive integer called the quantum number h is Planck’s constant ƒ is the frequency of oscillation This says the energy is quantized Each discrete energy value corresponds to a different quantum state

15. Planck’s Assumption, 2 The oscillators emit or absorb energy only in discrete units They do this when making a transition from one quantum state to another The entire energy difference between the initial and final states in the transition is emitted or absorbed as a single quantum of radiation An oscillator emits or absorbs energy only when it changes quantum states

16. Energy-Level Diagram An energy-level diagram shows the quantized energy levels and allowed transitions Energy is on the vertical axis Horizontal lines represent the allowed energy levels The double-headed arrows indicate allowed transitions

17. Correspondence Principle Quantum results must blend smoothly with classical results when the quantum number becomes large Quantum effects are not seen on an everyday basis since the energy change is too small a fraction of the total energy Quantum effects are important and become measurable only on the submicroscopic level of atoms and molecules

18. Photoelectric Effect The photoelectric effect occurs when light incident on certain metallic surfaces causes electrons to be emitted from those surfaces The emitted electrons are called photoelectrons The effect was first discovered by Hertz

19. Photoelectric Effect Apparatus When the tube is kept in the dark, the ammeter reads zero When plate E is illuminated by light having an appropriate wavelength, a current is detected by the ammeter The current arises from photoelectrons emitted from the negative plate (E) and collected at the positive plate (C)

20. Photoelectric Effect, Results At large values of  V, the current reaches a maximum value All the electrons emitted at E are collected at C The maximum current increases as the intensity of the incident light increases When  V is negative, the current drops When  V is equal to or more negative than  V s, the current is zero

21. Photoelectric Effect Feature 1 Dependence of photoelectron kinetic energy on light intensity Classical Prediction Electrons should absorb energy continually from the electromagnetic waves As the light intensity incident on the metal is increased, the electrons should be ejected with more kinetic energy Experimental Result The maximum kinetic energy is independent of light intensity The current goes to zero at the same negative voltage for all intensity curves

22. Photoelectric Effect Feature 2 Time interval between incidence of light and ejection of photoelectrons Classical Prediction For very weak light, a measurable time interval should pass between the instant the light is turned on and the time an electron is ejected from the metal This time interval is required for the electron to absorb the incident radiation before it acquires enough energy to escape from the metal Experimental Result Electrons are emitted almost instantaneously, even at very low light intensities Less than 10 -9 s

23. Photoelectric Effect Feature 3 Dependence of ejection of electrons on light frequency Classical Prediction Electrons should be ejected at any frequency as long as the light intensity is high enough Experimental Result No electrons are emitted if the incident light falls below some cutoff frequency, ƒ c The cutoff frequency is characteristic of the material being illuminated No electrons are ejected below the cutoff frequency regardless of intensity

24. Photoelectric Effect Feature 4 Dependence of photoelectron kinetic energy on light frequency Classical Prediction There should be no relationship between the frequency of the light and the electric kinetic energy The kinetic energy should be related to the intensity of the light Experimental Result The maximum kinetic energy of the photoelectrons increases with increasing light frequency

25. Photoelectric Effect Features, Summary The experimental results contradict all four classical predictions Einstein extended Planck’s concept of quantization to electromagnetic waves All electromagnetic radiation can be considered a stream of quanta, now called photons A photon of incident light gives all its energy hƒ to a single electron in the metal

26. Photoelectric Effect, Work Function Electrons ejected from the surface of the metal and not making collisions with other metal atoms before escaping possess the maximum kinetic energy K max K max = hƒ –   is called the work function The work function represents the minimum energy with which an electron is bound in the metal

27. Some Work Function Values

28. Photon Model Explanation of the Photoelectric Effect Dependence of photoelectron kinetic energy on light intensity K max is independent of light intensity K depends on the light frequency and the work function The intensity will change the number of photoelectrons being emitted, but not the energy of an individual electron Time interval between incidence of light and ejection of the photoelectron Each photon can have enough energy to eject an electron immediately

29. Photon Model Explanation of the Photoelectric Effect, cont Dependence of ejection of electrons on light frequency There is a failure to observe photoelectric effect below a certain cutoff frequency, which indicates the photon must have more energy than the work function in order to eject an electron Without enough energy, an electron cannot be ejected, regardless of the light intensity

30. Photon Model Explanation of the Photoelectric Effect, final Dependence of photoelectron kinetic energy on light frequency Since K max = hƒ –  As the frequency increases, the kinetic energy will increase Once the energy of the work function is exceeded There is a linear relationship between the kinetic energy and the frequency

31. Cutoff Frequency The lines show the linear relationship between K and ƒ The slope of each line is h The absolute value of the y-intercept is the work function The x-intercept is the cutoff frequency This is the frequency below which no photoelectrons are emitted

32. Cutoff Frequency and Wavelength The cutoff frequency is related to the work function through ƒ c =  / h The cutoff frequency corresponds to a cutoff wavelength Wavelengths greater than c incident on a material having a work function  do not result in the emission of photoelectrons

33. Applications of the Photoelectric Effect Detector in the light meter of a camera Phototube Used in burglar alarms and soundtrack of motion picture films Largely replaced by semiconductor devices Photomultiplier tubes Used in nuclear detectors and astronomy

34. Arthur Holly Compton 1892 - 1962 Director at the lab of the University of Chicago Discovered the Compton Effect Shared the Nobel Prize in 1927

35. The Compton Effect, Introduction Compton and coworkers dealt with Einstein’s idea of photon momentum Einstein proposed a photon with energy E carries a momentum of E/c = hƒ / c Compton and others accumulated evidence of the inadequacy of the classical wave theory The classical wave theory of light failed to explain the scattering of x-rays from electrons

36. Compton Effect, Classical Predictions According to the classical theory, electromagnetic waves of frequency ƒ o incident on electrons should Accelerate in the direction of propagation of the x- rays by radiation pressure Oscillate at the apparent frequency of the radiation since the oscillating electric field should set the electrons in motion Overall, the scattered wave frequency at a given angle should be a distribution of Doppler-shifted values

37. Compton Effect, Observations Compton’s experiments showed that, at any given angle, only one frequency of radiation is observed

38. Compton Effect, Explanation The results could be explained by treating the photons as point-like particles having energy hƒ and momentum hƒ / c Assume the energy and momentum of the isolated system of the colliding photon- electron are conserved Adopted a particle model for a well-known wave This scattering phenomena is known as the Compton Effect

39. Compton Shift Equation The graphs show the scattered x-ray for various angles The shifted peak, ', is caused by the scattering of free electrons This is called the Compton shift equation

40. Compton Wavelength The unshifted wavelength, o, is caused by x-rays scattered from the electrons that are tightly bound to the target atoms The shifted peak, ', is caused by x-rays scattered from free electrons in the target The Compton wavelength is

41. Photons and Waves Revisited Some experiments are best explained by the photon model Some are best explained by the wave model We must accept both models and admit that the true nature of light is not describable in terms of any single classical model Light has a dual nature in that it exhibits both wave and particle characteristics The particle model and the wave model of light complement each other

42. Louis de Broglie 1892 – 1987 Originally studied history Was awarded the Nobel Prize in 1929 for his prediction of the wave nature of electrons

43. Wave Properties of Particles Louis de Broglie postulated that because photons have both wave and particle characteristics, perhaps all forms of matter have both properties The de Broglie wavelength of a particle is

44. Frequency of a Particle In an analogy with photons, de Broglie postulated that particles would also have a frequency associated with them These equations present the dual nature of matter particle nature, m and v wave nature, and ƒ

45. Davisson-Germer Experiment If particles have a wave nature, then under the correct conditions, they should exhibit diffraction effects Davission and Germer measured the wavelength of electrons This provided experimental confirmation of the matter waves proposed by de Broglie

46. Electron Microscope The electron microscope depends on the wave characteristics of electrons The electron microscope has a high resolving power because it has a very short wavelength Typically, the wavelengths of the electrons are about 100 times shorter than that of visible light

47. Quantum Particle The quantum particle is a new simplification model that is a result of the recognition of the dual nature of light and of material particles In this model, entities have both particle and wave characteristics We much choose one appropriate behavior in order to understand a particular phenomenon

48. Ideal Particle vs. Ideal Wave An ideal particle has zero size Therefore, it is localized in space An ideal wave has a single frequency and is infinitely long Therefore, it is unlocalized in space A localized entity can be built from infinitely long waves

49. Particle as a Wave Packet Multiple waves are superimposed so that one of its crests is at x = 0 The result is that all the waves add constructively at x = 0 There is destructive interference at every point except x = 0 The small region of constructive interference is called a wave packet The wave packet can be identified as a particle

50. Wave Envelope The blue line represents the envelope function This envelope can travel through space with a different speed than the individual waves

51. Speeds Associated with Wave Packet The phase speed of a wave in a wave packet is given by This is the rate of advance of a crest on a single wave The group speed is given by This is the speed of the wave packet itself

52. Speeds, cont The group speed can also be expressed in terms of energy and momentum This indicates that the group speed of the wave packet is identical to the speed of the particle that it is modeled to represent

53. Electron Diffraction, Set-Up

54. Electron Diffraction, Experiment Parallel beams of mono-energetic electrons are incident on a double slit The slit widths are small compared to the electron wavelength An electron detector is positioned far from the slits at a distance much greater than the slit separation

55. Electron Diffraction, cont If the detector collects electrons for a long enough time, a typical wave interference pattern is produced This is distinct evidence that electrons are interfering, a wave-like behavior The interference pattern becomes clearer as the number of electrons reaching the screen increases

56. Electron Diffraction, Equations A minimum occurs when This shows the dual nature of the electron The electrons are detected as particles at a localized spot at some instant of time The probability of arrival at that spot is determined by finding the intensity of two interfering waves

57. Electron Diffraction, Closed Slits If one slit is closed, the maximum is centered around the opening Closing the other slit produces another maximum centered around that opening The total effect is the blue line It is completely different from the interference pattern (brown curve)

58. Electron Diffraction Explained An electron interacts with both slits simultaneously If an attempt is made to determine experimentally which slit the electron goes through, the act of measuring destroys the interference pattern It is impossible to determine which slit the electron goes through In effect, the electron goes through both slits The wave components of the electron are present at both slits at the same time

59. Werner Heisenberg 1901 – 1976 Developed matrix mechanics Many contributions include Uncertainty Principle Rec’d Nobel Prize in 1932 Prediction of two forms of molecular hydrogen Theoretical models of the nucleus

60. The Uncertainty Principle, Introduction In classical mechanics, it is possible, in principle, to make measurements with arbitrarily small uncertainty Quantum theory predicts that it is fundamentally impossible to make simultaneous measurements of a particle’s position and momentum with infinite accuracy

61. Heisenberg Uncertainty Principle, Statement The Heisenberg Uncertainty Principle states if a measurement of the position of a particle is made with uncertainty  x and a simultaneous measurement of its x component of momentum is made with uncertainty  p, the product of the two uncertainties can never be smaller than

62. Heisenberg Uncertainty Principle, Explained It is physically impossible to measure simultaneously the exact position and exact momentum of a particle The inescapable uncertainties do not arise from imperfections in practical measuring instruments The uncertainties arise from the quantum structure of matter

63. Heisenberg Uncertainty Principle, Another Form Another form of the Uncertainty Principle can be expressed in terms of energy and time This suggests that energy conservation can appear to be violated by an amount  E as long as it is only for a short time interval  t

64. Probability – A Particle Interpretation From the particle point of view, the probability per unit volume of finding a photon in a given region of space at an instant of time is proportional to the number N of photons per unit volume at that time and to the intensity

65. Probability – A Wave Interpretation From the point of view of a wave, the intensity of electromagnetic radiation is proportional to the square of the electric field amplitude, E Combining the points of view gives

66. Probability – Interpretation Summary For electromagnetic radiation, the probability per unit volume of finding a particle associated with this radiation is proportional to the square of the amplitude of the associated em wave The particle is the photon The amplitude of the wave associated with the particle is called the probability amplitude or the wave function The symbol is 

67. Wave Function The complete wave function  for a system depends on the positions of all the particles in the system and on time The function can be written as  (r 1, r 2, … r j …., t) =  (r j )e -i  t r j is the position of the jth particle in the system  = 2  ƒ is the angular frequency

68. Wave Function, con’t The wave function is often complex-valued The absolute square |  | 2 =  is always real and positive  * is the complete conjugate of  It is proportional to the probability per unit volume of finding a particle at a given point at some instant The wave function contains within it all the information that can be known about the particle

69. Wave Function, General Comments, Final The probabilistic interpretation of the wave function was first suggested by Max Born Erwin Schrödinger proposed a wave equation that describes the manner in which the wave function changes in space and time This Schrödinger Wave Equation represents a key element in quantum mechanics

70. Wave Function of a Free Particle The wave function of a free particle moving along the x-axis can be written as  (x) = Ae ikx k = 2  is the angular wave number of the wave representing the particle A is the constant amplitude If  represents a single particle, |  | 2 is the relative probability per unit volume that the particle will be found at any given point in the volume |  | 2 is called the probability density

71. Wave Function of a Free Particle, Cont In general, the probability of finding the particle in a volume dV is |  | 2 dV With one-dimensional analysis, this becomes |  | 2 dx The probability of finding the particle in the arbitrary interval a  x  b is and is the area under the curve

72. Wave Function of a Free Particle, Final Because the particle must be somewhere along the x axis, the sum of all the probabilities over all values of x must be 1 Any wave function satisfying this equation is said to be normalized Normalization is simply a statement that the particle exists at some point in space

73. Expectation Values  is not a measurable quantity Measurable quantities of a particle can be derived from  The average position is called the expectation value of x and is defined as

74. Expectation Values, cont The expectation value of any function of x can also be found The expectation values are analogous to averages

75. Particle in a Box A particle is confined to a one-dimensional region of space The “box” is one- dimensional The particle is bouncing elastically back and forth between two impenetrable walls separated by L Classically, the particle’s momentum and kinetic energy remain constant

76. Wave Function for the Particle in a Box Since the walls are impenetrable, there is zero probability of finding the particle outside the box  (x) = 0 for x L The wave function must also be 0 at the walls The function must be continuous  (0) = 0 and  (L) = 0

77. Potential Energy for a Particle in a Box As long as the particle is inside the box, the potential energy does not depend on its location We can choose this energy value to be zero The energy is infinitely large if the particle is outside the box This ensures that the wave function is zero outside the box

78. Wave Function of a Particle in a Box – Mathematical The wave function can be expressed as a real, sinusoidal function Applying the boundary conditions and using the de Broglie wavelength

79. Graphical Representations for a Particle in a Box

80. Wave Function of the Particle in a Box, cont Only certain wavelengths for the particle are allowed |  | 2 is zero at the boundaries |  | 2 is zero at other locations as well, depending on the values of n The number of zero points increases by one each time the quantum number increases by one

81. Momentum of the Particle in a Box Remember the wavelengths are restricted to specific values Therefore, the momentum values are also restricted

82. Energy of a Particle in a Box We chose the potential energy of the particle to be zero inside the box Therefore, the energy of the particle is just its kinetic energy The energy of the particle is quantized

83. Energy Level Diagram – Particle in a Box The lowest allowed energy corresponds to the ground state E n = n 2 E 1 are called excited states E = 0 is not an allowed state The particle can never be at rest The lowest energy the particle can have, E = 1, is called the zero-point energy

84. Boundary Conditions Boundary conditions are applied to determine the allowed states of the system In the model of a particle under boundary conditions, an interaction of a particle with its environment represents one or more boundary conditions and, if the interaction restricts the particle to a finite region of space, results in quantization of the energy of the system In general, boundary conditions are related to the coordinates describing the problem

85. Erwin Schrödinger 1887 – 1961 Best known as one of the creators of quantum mechanics His approach was shown to be equivalent to Heisenberg’s Also worked with statistical mechanics color vision general relativity

86. Schrödinger Equation The Schrödinger equation as it applies to a particle of mass m confined to moving along the x axis and interacting with its environment through a potential energy function U(x) is This is called the time-independent Schrödinger equation

87. Schrödinger Equation, cont Both for a free particle and a particle in a box, the first term in the Schrödinger equation reduces to the kinetic energy of the particle multiplied by the wave function Solutions to the Schrödinger equation in different regions must join smoothly at the boundaries

88. Schrödinger Equation, final  (x) must be continuous  (x) must approach zero as x approaches ±  This is needed so that  (x) obeys the normalization condition d  / dx must also be continuous for finite values of the potential energy

89. Solutions of the Schrödinger Equation Solutions of the Schrödinger equation may be very difficult The Schrödinger equation has been extremely successful in explaining the behavior of atomic and nuclear systems Classical physics failed to explain this behavior When quantum mechanics is applied to macroscopic objects, the results agree with classical physics

90. Potential Wells A potential well is a graphical representation of energy The well is the upward-facing region of the curve in a potential energy diagram The particle in a box is sometimes said to be in a square well Due to the shape of the potential energy diagram

91. Schrödinger Equation Applied to a Particle in a Box In the region 0 < x < L, where U = 0, the Schrödinger equation can be expressed in the form The most general solution to the equation is  (x) = A sin kx + B cos kx A and B are constants determined by the boundary and normalization conditions

92. Schrödinger Equation Applied to a Particle in a Box, cont. Solving for the allowed energies gives The allowed wave functions are given by The second expression is the normalized wave function These match the original results for the particle in a box

93. Application – Nanotechnology Nanotechnology refers to the design and application of devices having dimensions ranging from 1 to 100 nm Nanotechnology uses the idea of trapping particles in potential wells One area of nanotechnology of interest to researchers is the quantum dot A quantum dot is a small region that is grown in a silicon crystal that acts as a potential well Storage of binary information using quantum dots is being researched

94. Quantum Corral Corrals and other structures are used to confine surface electron waves This corral is a ring of 48 iron atoms on a copper surface The ring has a diameter of 143 nm

95. Tunneling The potential energy has a constant value U in the region of width L and zero in all other regions This a called a square barrier U is the called the barrier height

96. Tunneling, cont Classically, the particle is reflected by the barrier Regions II and III would be forbidden According to quantum mechanics, all regions are accessible to the particle The probability of the particle being in a classically forbidden region is low, but not zero According to the Uncertainty Principle, the particle can be inside the barrier as long as the time interval is short and consistent with the Principle

97. Tunneling, final The curve in the diagram represents a full solution to the Schrödinger equation Movement of the particle to the far side of the barrier is called tunneling or barrier penetration The probability of tunneling can be described with a transmission coefficient, T, and a reflection coefficient, R

98. Tunneling Coefficients The transmission coefficient represents the probability that the particle penetrates to the other side of the barrier The reflection coefficient represents the probability that the particle is reflected by the barrier T + R = 1 The particle must be either transmitted or reflected T  e -2CL and can be non zero Tunneling is observed and provides evidence of the principles of quantum mechanics

99. Applications of Tunneling Alpha decay In order for the alpha particle to escape from the nucleus, it must penetrate a barrier whose energy is several times greater than the energy of the nucleus- alpha particle system Nuclear fusion Protons can tunnel through the barrier caused by their mutual electrostatic repulsion

100. More Applications of Tunneling – Scanning Tunneling Microscope An electrically conducting probe with a very sharp edge is brought near the surface to be studied The empty space between the tip and the surface represents the “barrier” The tip and the surface are two walls of the “potential well” The vertical motion of the probe follows the contour of the specimen’s surface and therefore an image of the surface is obtained

101. Cosmic Temperature In the 1940’s, a structural model of the universe was developed which predicted the existence of thermal radiation from the Big Bang The radiation would now have a wavelength distribution consistent with a black body The temperature would be a few kelvins

102. Cosmic Temperature, cont In 1965 two workers at Bell Labs found a consistent “noise” in the radiation they were measuring They were detecting the background radiation from the Big Bang It was detected by their system regardless of direction It was consistent with a back body at about 3 K

103. Cosmic Temperature, Final Measurements at many wavelengths were needed The brown curve is the theoretical curve The blue dots represent measurements from COBE and Bell Labs