1. Chapter 27 Quantum Physics

2. Need for Quantum Physics Problems remained from classical mechanics that relativity didn’t explain Problems remained from classical mechanics that relativity didn’t explain Blackbody Radiation Blackbody Radiation The electromagnetic radiation emitted by a heated object The electromagnetic radiation emitted by a heated object Photoelectric Effect Photoelectric Effect Emission of electrons by an illuminated metal Emission of electrons by an illuminated metal Spectral Lines Spectral Lines Emission of sharp spectral lines by gas atoms in an electric discharge tube Emission of sharp spectral lines by gas atoms in an electric discharge tube

3. Development of Quantum Physics 1900 to 1930 1900 to 1930 Development of ideas of quantum mechanics Development of ideas of quantum mechanics Also called wave mechanics Also called wave mechanics Highly successful in explaining the behavior of atoms, molecules, and nuclei Highly successful in explaining the behavior of atoms, molecules, and nuclei Quantum Mechanics reduces to classical mechanics when applied to macroscopic systems Quantum Mechanics reduces to classical mechanics when applied to macroscopic systems Involved a large number of physicists Involved a large number of physicists Planck introduced basic ideas Planck introduced basic ideas Mathematical developments and interpretations involved such people as Einstein, Bohr, Schrödinger, de Broglie, Heisenberg, Born and Dirac Mathematical developments and interpretations involved such people as Einstein, Bohr, Schrödinger, de Broglie, Heisenberg, Born and Dirac

4. Blackbody Radiation An object at any temperature is known to emit electromagnetic radiation An object at any temperature is known to emit electromagnetic radiation Sometimes called thermal radiation Sometimes called thermal radiation Stefan’s Law describes the total power radiated Stefan’s Law describes the total power radiated The spectrum of the radiation depends on the temperature and properties of the object The spectrum of the radiation depends on the temperature and properties of the object

5. Blackbody Radiation Graph Experimental data for distribution of energy in blackbody radiation Experimental data for distribution of energy in blackbody radiation As the temperature increases, the total amount of energy increases As the temperature increases, the total amount of energy increases Shown by the area under the curve Shown by the area under the curve As the temperature increases, the peak of the distribution shifts to shorter wavelengths As the temperature increases, the peak of the distribution shifts to shorter wavelengths

6. Wien’s Displacement Law The wavelength of the peak of the blackbody distribution was found to follow Wein’s Displacement Law The wavelength of the peak of the blackbody distribution was found to follow Wein’s Displacement Law λ max T = 0.2898 x 10 -2 m K λ max T = 0.2898 x 10 -2 m K λ max is the wavelength at the curve’s peak λ max is the wavelength at the curve’s peak T is the absolute temperature of the object emitting the radiation T is the absolute temperature of the object emitting the radiation

7. The Ultraviolet Catastrophe Classical theory did not match the experimental data Classical theory did not match the experimental data At long wavelengths, the match is good At long wavelengths, the match is good At short wavelengths, classical theory predicted infinite energy At short wavelengths, classical theory predicted infinite energy At short wavelengths, experiment showed no energy At short wavelengths, experiment showed no energy This contradiction is called the ultraviolet catastrophe This contradiction is called the ultraviolet catastrophe

8. Planck’s Resolution Planck hypothesized that the blackbody radiation was produced by resonators Planck hypothesized that the blackbody radiation was produced by resonators Resonators were submicroscopic charged oscillators Resonators were submicroscopic charged oscillators The resonators could only have discrete energies The resonators could only have discrete energies E n = n h ƒ E n = n h ƒ n is called the quantum number n is called the quantum number ƒ is the frequency of vibration ƒ is the frequency of vibration h is Planck’s constant, 6.626 x 10 -34 J s h is Planck’s constant, 6.626 x 10 -34 J s Key point is quantized energy states Key point is quantized energy states

9. Photoelectric Effect When light is incident on certain metallic surfaces, electrons are emitted from the surface When light is incident on certain metallic surfaces, electrons are emitted from the surface This is called the photoelectric effect This is called the photoelectric effect The emitted electrons are called photoelectrons The emitted electrons are called photoelectrons The effect was first discovered by Hertz The effect was first discovered by Hertz The successful explanation of the effect was given by Einstein in 1905 The successful explanation of the effect was given by Einstein in 1905 Received Nobel Prize in 1921 for paper on electromagnetic radiation, of which the photoelectric effect was a part Received Nobel Prize in 1921 for paper on electromagnetic radiation, of which the photoelectric effect was a part

10. Photoelectric Effect Schematic When light strikes E, photoelectrons are emitted When light strikes E, photoelectrons are emitted Electrons collected at C and passing through the ammeter are a current in the circuit Electrons collected at C and passing through the ammeter are a current in the circuit C is maintained at a positive potential by the power supply C is maintained at a positive potential by the power supply

11. Photoelectric Current/Voltage Graph The current increases with intensity, but reaches a saturation level for large ΔV’s The current increases with intensity, but reaches a saturation level for large ΔV’s No current flows for voltages less than or equal to –ΔV s, the stopping potential No current flows for voltages less than or equal to –ΔV s, the stopping potential The stopping potential is independent of the radiation intensity The stopping potential is independent of the radiation intensity

12. Features Not Explained by Classical Physics/Wave Theory No electrons are emitted if the incident light frequency is below some cutoff frequency that is characteristic of the material being illuminated No electrons are emitted if the incident light frequency is below some cutoff frequency that is characteristic of the material being illuminated The maximum kinetic energy of the photoelectrons is independent of the light intensity The maximum kinetic energy of the photoelectrons is independent of the light intensity

13. More Features Not Explained The maximum kinetic energy of the photoelectrons increases with increasing light frequency The maximum kinetic energy of the photoelectrons increases with increasing light frequency Electrons are emitted from the surface almost instantaneously, even at low intensities Electrons are emitted from the surface almost instantaneously, even at low intensities

14. Einstein’s Explanation A tiny packet of light energy, called a photon, would be emitted when a quantized oscillator jumped from one energy level to the next lower one A tiny packet of light energy, called a photon, would be emitted when a quantized oscillator jumped from one energy level to the next lower one Extended Planck’s idea of quantization to electromagnetic radiation Extended Planck’s idea of quantization to electromagnetic radiation The photon’s energy would be E = hƒ The photon’s energy would be E = hƒ Each photon can give all its energy to an electron in the metal Each photon can give all its energy to an electron in the metal The maximum kinetic energy of the liberated photoelectron is KE = hƒ – Φ The maximum kinetic energy of the liberated photoelectron is KE = hƒ – Φ Φ is called the work function of the metal Φ is called the work function of the metal

15. Explanation of Classical “Problems” The effect is not observed below a certain cutoff frequency since the photon energy must be greater than or equal to the work function The effect is not observed below a certain cutoff frequency since the photon energy must be greater than or equal to the work function Without this, electrons are not emitted, regardless of the intensity of the light Without this, electrons are not emitted, regardless of the intensity of the light The maximum KE depends only on the frequency and the work function, not on the intensity The maximum KE depends only on the frequency and the work function, not on the intensity

16. More Explanations The maximum KE increases with increasing frequency The maximum KE increases with increasing frequency The effect is instantaneous since there is a one-to-one interaction between the photon and the electron The effect is instantaneous since there is a one-to-one interaction between the photon and the electron

17. Verification of Einstein’s Theory Experimental observations of a linear relationship between KE and frequency confirm Einstein’s theory Experimental observations of a linear relationship between KE and frequency confirm Einstein’s theory The x-intercept is the cutoff frequency The x-intercept is the cutoff frequency

18. Photocells Photocells are an application of the photoelectric effect Photocells are an application of the photoelectric effect When light of sufficiently high frequency falls on the cell, a current is produced When light of sufficiently high frequency falls on the cell, a current is produced Examples Examples Streetlights, garage door openers, elevators Streetlights, garage door openers, elevators

19. X-Rays Electromagnetic radiation with short wavelengths Electromagnetic radiation with short wavelengths Wavelengths less than for ultraviolet Wavelengths less than for ultraviolet Wavelengths are typically about 0.1 nm Wavelengths are typically about 0.1 nm X-rays have the ability to penetrate most materials with relative ease X-rays have the ability to penetrate most materials with relative ease Discovered and named by Roentgen in 1895 Discovered and named by Roentgen in 1895

20. The Compton Effect Compton directed a beam of x-rays toward a block of graphite Compton directed a beam of x-rays toward a block of graphite He found that the scattered x-rays had a slightly longer wavelength that the incident x- rays He found that the scattered x-rays had a slightly longer wavelength that the incident x- rays This means they also had less energy This means they also had less energy The amount of energy reduction depended on the angle at which the x-rays were scattered The amount of energy reduction depended on the angle at which the x-rays were scattered The change in wavelength is called the Compton shift The change in wavelength is called the Compton shift

21. Compton Scattering Compton assumed the photons acted like other particles in collisions Compton assumed the photons acted like other particles in collisions Energy and momentum were conserved Energy and momentum were conserved The shift in wavelength is The shift in wavelength is

22. Compton Scattering, final The quantity h/m e c is called the Compton wavelength The quantity h/m e c is called the Compton wavelength Compton wavelength = 0.00243 nm Compton wavelength = 0.00243 nm Very small compared to visible light Very small compared to visible light The Compton shift depends on the scattering angle and not on the wavelength The Compton shift depends on the scattering angle and not on the wavelength Experiments confirm the results of Compton scattering and strongly support the photon concept Experiments confirm the results of Compton scattering and strongly support the photon concept

23. QUICK QUIZ 27.1 An x-ray photon is scattered by an electron. The frequency of the scattered photon relative to that of the incident photon (a) increases, (b) decreases, or (c) remains the same.

24. QUICK QUIZ 27.1 ANSWER (b). Some energy is transferred to the electron in the scattering process. Therefore, the scattered photon must have less energy (and hence, lower frequency) than the incident photon.

25. QUICK QUIZ 27.2 A photon of energy E 0 strikes a free electron, with the scattered photon of energy E moving in the direction opposite that of the incident photon. In this Compton effect interaction, the resulting kinetic energy of the electron is (a) E 0, (b) E, (c) E 0  E, (d) E 0 + E, (e) none of the above.

26. QUICK QUIZ 27.2 ANSWER (c). Conservation of energy requires the kinetic energy given to the electron be equal to the difference between the energy of the incident photon and that of the scattered photon.

27. QUICK QUIZ 27.3 A photon of energy E 0 strikes a free electron with the scattered photon of energy E moving in the direction opposite that of the incident photon. In this Compton effect interaction, the resulting momentum of the electron is (a) E 0 /c (b) E 0 /c(d) (E 0  E)/c (e) (E  E o )/c

28. QUICK QUIZ 27.3 ANSWER (c). Conservation of momentum requires the momentum of the incident photon equal the vector sum of the momenta of the electron and the scattered photon. Since the scattered photon moves in the direction opposite that of the electron, the magnitude of the electron’s momentum must exceed that of the incident photon.

29. Photons and Electromagnetic Waves Light has a dual nature. It exhibits both wave and particle characteristics Light has a dual nature. It exhibits both wave and particle characteristics Applies to all electromagnetic radiation Applies to all electromagnetic radiation The photoelectric effect and Compton scattering offer evidence for the particle nature of light The photoelectric effect and Compton scattering offer evidence for the particle nature of light When light and matter interact, light behaves as if it were composed of particles When light and matter interact, light behaves as if it were composed of particles Interference and diffraction offer evidence of the wave nature of light Interference and diffraction offer evidence of the wave nature of light

30. Wave Properties of Particles In 1924, Louis de Broglie postulated that because photons have wave and particle characteristics, perhaps all forms of matter have both properties In 1924, Louis de Broglie postulated that because photons have wave and particle characteristics, perhaps all forms of matter have both properties Furthermore, the frequency and wavelength of matter waves can be determined Furthermore, the frequency and wavelength of matter waves can be determined

31. de Broglie Wavelength and Frequency The de Broglie wavelength of a particle is The de Broglie wavelength of a particle is The frequency of matter waves is The frequency of matter waves is

32. The Davisson-Germer Experiment They scattered low-energy electrons from a nickel target They scattered low-energy electrons from a nickel target They followed this with extensive diffraction measurements from various materials They followed this with extensive diffraction measurements from various materials The wavelength of the electrons calculated from the diffraction data agreed with the expected de Broglie wavelength The wavelength of the electrons calculated from the diffraction data agreed with the expected de Broglie wavelength This confirmed the wave nature of electrons This confirmed the wave nature of electrons Other experimenters have confirmed the wave nature of other particles Other experimenters have confirmed the wave nature of other particles

33. QUICK QUIZ 27.4 A non-relativistic electron and a non- relativistic proton are moving and have the same de Broglie wavelength. Which of the following are also the same for the two particles: (a) speed, (b) kinetic energy, (c) momentum, (d) frequency?

34. QUICK QUIZ 27.4 ANSWER (c). Two particles with the same de Broglie wavelength will have the same momentum p = mv. If the electron and proton have the same momentum, they cannot have the same speed because of the difference in their masses. For the same reason, remembering that KE = p 2 /2m, they cannot have the same kinetic energy. Because the kinetic energy is the only type of energy an isolated particle can have, and we have argued that the particles have different energies, Equation 27.15 tells us that the particles do not have the same frequency.

35. QUICK QUIZ 27.5 We have seen two wavelengths assigned to the electron, the Compton wavelength and the de Broglie wavelength. Which is an actual physical wavelength associated with the electron: (a) the Compton wavelength, (b) the de Broglie wavelength, (c) both wavelengths, (d) neither wavelength?

36. QUICK QUIZ 27.5 ANSWER (b). The Compton wavelength, λ C = h/m e c, is a combination of constants and has no relation to the motion of the electron. The de Broglie wavelength, λ = h/m e v, is associated with the motion of the electron through its momentum.

37. The Electron Microscope The electron microscope depends on the wave characteristics of electrons The electron microscope depends on the wave characteristics of electrons Microscopes can only resolve details that are slightly smaller than the wavelength of the radiation used to illuminate the object Microscopes can only resolve details that are slightly smaller than the wavelength of the radiation used to illuminate the object The electrons can be accelerated to high energies and have small wavelengths The electrons can be accelerated to high energies and have small wavelengths

38. The Uncertainty Principle When measurements are made, the experimenter is always faced with experimental uncertainties in the measurements When measurements are made, the experimenter is always faced with experimental uncertainties in the measurements Classical mechanics offers no fundamental barrier to ultimate refinements in measurements Classical mechanics offers no fundamental barrier to ultimate refinements in measurements Classical mechanics would allow for measurements with arbitrarily small uncertainties Classical mechanics would allow for measurements with arbitrarily small uncertainties

39. Importance of Hydrogen Atom Hydrogen is the simplest atom Hydrogen is the simplest atom The quantum numbers used to characterize the allowed states of hydrogen can also be used to describe (approximately) the allowed states of more complex atoms The quantum numbers used to characterize the allowed states of hydrogen can also be used to describe (approximately) the allowed states of more complex atoms This enables us to understand the periodic table This enables us to understand the periodic table

40. More Reasons the Hydrogen Atom is so Important The hydrogen atom is an ideal system for performing precise comparisons of theory and experiment The hydrogen atom is an ideal system for performing precise comparisons of theory and experiment Also for improving our understanding of atomic structure Also for improving our understanding of atomic structure Much of what we know about the hydrogen atom can be extended to other single- electron ions Much of what we know about the hydrogen atom can be extended to other single- electron ions For example, He + and Li 2+ For example, He + and Li 2+

41. Early Models of the Atom J.J. Thomson’s model of the atom J.J. Thomson’s model of the atom A volume of positive charge A volume of positive charge Electrons embedded throughout the volume Electrons embedded throughout the volume A change from Newton’s model of the atom as a tiny, hard, indestructible sphere A change from Newton’s model of the atom as a tiny, hard, indestructible sphere

42. Early Models of the Atom, 2 Rutherford Rutherford Planetary model Planetary model Based on results of thin foil experiments Based on results of thin foil experiments Positive charge is concentrated in the center of the atom, called the nucleus Positive charge is concentrated in the center of the atom, called the nucleus Electrons orbit the nucleus like planets orbit the sun Electrons orbit the nucleus like planets orbit the sun

43. Difficulties with the Rutherford Model Atoms emit certain discrete characteristic frequencies of electromagnetic radiation Atoms emit certain discrete characteristic frequencies of electromagnetic radiation The Rutherford model is unable to explain this phenomena The Rutherford model is unable to explain this phenomena Rutherford’s electrons are undergoing a centripetal acceleration and so should radiate electromagnetic waves of the same frequency Rutherford’s electrons are undergoing a centripetal acceleration and so should radiate electromagnetic waves of the same frequency The radius should steadily decrease as this radiation is given off The radius should steadily decrease as this radiation is given off The electron should eventually spiral into the nucleus The electron should eventually spiral into the nucleus It doesn’t It doesn’t

44. Emission Spectra A gas at low pressure has a voltage applied to it A gas at low pressure has a voltage applied to it A gas emits light characteristic of the gas A gas emits light characteristic of the gas When the emitted light is analyzed with a spectrometer, a series of discrete bright lines is observed When the emitted light is analyzed with a spectrometer, a series of discrete bright lines is observed Each line has a different wavelength and color Each line has a different wavelength and color This series of lines is called an emission spectrum This series of lines is called an emission spectrum

45. Examples of Spectra

46. Emission Spectrum of Hydrogen – Equation The wavelengths of hydrogen’s spectral lines can be found from The wavelengths of hydrogen’s spectral lines can be found from R H is the Rydberg constant R H is the Rydberg constant R H = 1.0973732 x 10 7 m -1 R H = 1.0973732 x 10 7 m -1 n is an integer, n = 1, 2, 3, … n is an integer, n = 1, 2, 3, … The spectral lines correspond to different values of n The spectral lines correspond to different values of n

47. Spectral Lines of Hydrogen The Balmer Series has lines whose wavelengths are given by the preceding equation The Balmer Series has lines whose wavelengths are given by the preceding equation Examples of spectral lines Examples of spectral lines n = 3, λ = 656.3 nm n = 3, λ = 656.3 nm n = 4, λ = 486.1 nm n = 4, λ = 486.1 nm

48. Absorption Spectra An element can also absorb light at specific wavelengths An element can also absorb light at specific wavelengths An absorption spectrum can be obtained by passing a continuous radiation spectrum through a vapor of the gas An absorption spectrum can be obtained by passing a continuous radiation spectrum through a vapor of the gas The absorption spectrum consists of a series of dark lines superimposed on the otherwise continuous spectrum The absorption spectrum consists of a series of dark lines superimposed on the otherwise continuous spectrum The dark lines of the absorption spectrum coincide with the bright lines of the emission spectrum The dark lines of the absorption spectrum coincide with the bright lines of the emission spectrum

49. Applications of Absorption Spectrum The continuous spectrum emitted by the Sun passes through the cooler gases of the Sun’s atmosphere The continuous spectrum emitted by the Sun passes through the cooler gases of the Sun’s atmosphere The various absorption lines can be used to identify elements in the solar atmosphere The various absorption lines can be used to identify elements in the solar atmosphere Led to the discovery of helium Led to the discovery of helium

50. The Bohr Theory of Hydrogen In 1913 Bohr provided an explanation of atomic spectra that includes some features of the currently accepted theory In 1913 Bohr provided an explanation of atomic spectra that includes some features of the currently accepted theory His model includes both classical and non-classical ideas His model includes both classical and non-classical ideas His model included an attempt to explain why the atom was stable His model included an attempt to explain why the atom was stable

51. Bohr’s Assumptions for Hydrogen The electron moves in circular orbits around the proton under the influence of the Coulomb force of attraction The electron moves in circular orbits around the proton under the influence of the Coulomb force of attraction The Coulomb force produces the centripetal acceleration The Coulomb force produces the centripetal acceleration

52. Bohr’s Assumptions, cont Only certain electron orbits are stable Only certain electron orbits are stable These are the orbits in which the atom does not emit energy in the form of electromagnetic radiation These are the orbits in which the atom does not emit energy in the form of electromagnetic radiation Therefore, the energy of the atom remains constant and classical mechanics can be used to describe the electron’s motion Therefore, the energy of the atom remains constant and classical mechanics can be used to describe the electron’s motion Radiation is emitted by the atom when the electron “jumps” from a more energetic initial state to a lower state Radiation is emitted by the atom when the electron “jumps” from a more energetic initial state to a lower state The “jump” cannot be treated classically The “jump” cannot be treated classically

53. Bohr’s Assumptions, final The electron’s “jump,” continued The electron’s “jump,” continued The frequency emitted in the “jump” is related to the change in the atom’s energy The frequency emitted in the “jump” is related to the change in the atom’s energy It is generally not the same as the frequency of the electron’s orbital motion It is generally not the same as the frequency of the electron’s orbital motion The size of the allowed electron orbits is determined by a condition imposed on the electron’s orbital angular momentum The size of the allowed electron orbits is determined by a condition imposed on the electron’s orbital angular momentum

54. Mathematics of Bohr’s Assumptions and Results Electron’s orbital angular momentum Electron’s orbital angular momentum m e v r = n ħ where n = 1, 2, 3, … m e v r = n ħ where n = 1, 2, 3, … The total energy of the atom The total energy of the atom The energy can also be expressed as The energy can also be expressed as

55. Bohr Radius The radii of the Bohr orbits are quantized The radii of the Bohr orbits are quantized This shows that the electron can only exist in certain allowed orbits determined by the integer n This shows that the electron can only exist in certain allowed orbits determined by the integer n When n = 1, the orbit has the smallest radius, called the Bohr radius, a o When n = 1, the orbit has the smallest radius, called the Bohr radius, a o a o = 0.0529 nm a o = 0.0529 nm

56. Radii and Energy of Orbits A general expression for the radius of any orbit in a hydrogen atom is A general expression for the radius of any orbit in a hydrogen atom is r n = n 2 a o r n = n 2 a o The energy of any orbit is The energy of any orbit is E n = - 13.6 eV/ n 2 E n = - 13.6 eV/ n 2

57. Specific Energy Levels The lowest energy state is called the ground state The lowest energy state is called the ground state This corresponds to n = 1 This corresponds to n = 1 Energy is –13.6 eV Energy is –13.6 eV The next energy level has an energy of –3.40 eV The next energy level has an energy of –3.40 eV The energies can be compiled in an energy level diagram The energies can be compiled in an energy level diagram The ionization energy is the energy needed to completely remove the electron from the atom The ionization energy is the energy needed to completely remove the electron from the atom The ionization energy for hydrogen is 13.6 eV The ionization energy for hydrogen is 13.6 eV

58. Energy Level Diagram The value of R H from Bohr’s analysis is in excellent agreement with the experimental value The value of R H from Bohr’s analysis is in excellent agreement with the experimental value A more generalized equation can be used to find the wavelengths of any spectral lines A more generalized equation can be used to find the wavelengths of any spectral lines

59. Generalized Equation For the Balmer series, n f = 2 For the Balmer series, n f = 2 For the Lyman series, n f = 1 For the Lyman series, n f = 1 Whenever an transition occurs between a state, n i to another state, n f (where n i > n f ), a photon is emitted Whenever an transition occurs between a state, n i to another state, n f (where n i > n f ), a photon is emitted The photon has a frequency f = (Ei – Ef)/h and wavelength λ The photon has a frequency f = (Ei – Ef)/h and wavelength λ

60. Bohr’s Correspondence Principle Bohr’s Correspondence Principle states that quantum mechanics is in agreement with classical physics when the energy differences between quantized levels are very small Bohr’s Correspondence Principle states that quantum mechanics is in agreement with classical physics when the energy differences between quantized levels are very small Similar to having Newtonian Mechanics be a special case of relativistic mechanics when v << c Similar to having Newtonian Mechanics be a special case of relativistic mechanics when v << c

61. Successes of the Bohr Theory Explained several features of the hydrogen spectrum Explained several features of the hydrogen spectrum Accounts for Balmer and other series Accounts for Balmer and other series Predicts a value for R H that agrees with the experimental value Predicts a value for R H that agrees with the experimental value Gives an expression for the radius of the atom Gives an expression for the radius of the atom Predicts energy levels of hydrogen Predicts energy levels of hydrogen Gives a model of what the atom looks like and how it behaves Gives a model of what the atom looks like and how it behaves Can be extended to “hydrogen-like” atoms Can be extended to “hydrogen-like” atoms Those with one electron Those with one electron Ze 2 needs to be substituted for e 2 in equations Ze 2 needs to be substituted for e 2 in equations Z is the atomic number of the element Z is the atomic number of the element

62. Modifications of the Bohr Theory – Elliptical Orbits Sommerfeld extended the results to include elliptical orbits Sommerfeld extended the results to include elliptical orbits Retained the principle quantum number, n Retained the principle quantum number, n Added the orbital quantum number, ℓ Added the orbital quantum number, ℓ ℓ ranges from 0 to n-1 in integer steps ℓ ranges from 0 to n-1 in integer steps All states with the same principle quantum number are said to form a shell All states with the same principle quantum number are said to form a shell The states with given values of n and ℓ are said to form a subshell The states with given values of n and ℓ are said to form a subshell

63. Modifications of the Bohr Theory – Zeeman Effect Another modification was needed to account for the Zeeman effect Another modification was needed to account for the Zeeman effect The Zeeman effect is the splitting of spectral lines in a strong magnetic field The Zeeman effect is the splitting of spectral lines in a strong magnetic field This indicates that the energy of an electron is slightly modified when the atom is immersed in a magnetic field This indicates that the energy of an electron is slightly modified when the atom is immersed in a magnetic field A new quantum number, m ℓ, called the orbital magnetic quantum number, had to be introduced A new quantum number, m ℓ, called the orbital magnetic quantum number, had to be introduced m ℓ can vary from - ℓ to + ℓ in integer steps m ℓ can vary from - ℓ to + ℓ in integer steps

64. Modifications of the Bohr Theory – Fine Structure High resolution spectrometers show that spectral lines are, in fact, two very closely spaced lines, even in the absence of a magnetic field High resolution spectrometers show that spectral lines are, in fact, two very closely spaced lines, even in the absence of a magnetic field This splitting is called fine structure This splitting is called fine structure Another quantum number, m s, called the spin magnetic quantum number, was introduced to explain the fine structure Another quantum number, m s, called the spin magnetic quantum number, was introduced to explain the fine structure

65. de Broglie Waves One of Bohr’s postulates was the angular momentum of the electron is quantized, but there was no explanation why the restriction occurred One of Bohr’s postulates was the angular momentum of the electron is quantized, but there was no explanation why the restriction occurred de Broglie assumed that the electron orbit would be stable only if it contained an integral number of electron wavelengths de Broglie assumed that the electron orbit would be stable only if it contained an integral number of electron wavelengths

66. de Broglie Waves in the Hydrogen Atom In this example, three complete wavelengths are contained in the circumference of the orbit In this example, three complete wavelengths are contained in the circumference of the orbit In general, the circumference must equal some integer number of wavelengths In general, the circumference must equal some integer number of wavelengths 2  r = n λ n = 1, 2, … 2  r = n λ n = 1, 2, …

67. de Broglie Waves in the Hydrogen Atom, cont The expression for the de Broglie wavelength can be included in the circumference calculation The expression for the de Broglie wavelength can be included in the circumference calculation m e v r = n  m e v r = n  This is the same quantization of angular momentum that Bohr imposed in his original theory This is the same quantization of angular momentum that Bohr imposed in his original theory This was the first convincing argument that the wave nature of matter was at the heart of the behavior of atomic systems This was the first convincing argument that the wave nature of matter was at the heart of the behavior of atomic systems Schrödinger’s wave equation was subsequently applied to atomic systems Schrödinger’s wave equation was subsequently applied to atomic systems

68. QUICK QUIZ 28.1 In an analysis relating Bohr's theory to the de Broglie wavelength of electrons, when an electron moves from the n = 1 level to the n = 3 level, the circumference of its orbit becomes 9 times greater. This occurs because (a) there are 3 times as many wavelengths in the new orbit, (b) there are 3 times as many wavelengths and each wavelength is 3 times as long, (c) the wavelength of the electron becomes 9 times as long, or (d) the electron is moving 9 times as fast.

69. QUICK QUIZ 28.1 ANSWER (b). The circumference of the orbit is n times the de Broglie wavelength (2πr = nλ), so there are three times as many wavelengths in the n = 3 level as in the n = 1 level. Also, by combining Equations 28.4, 28.6 and the defining equation for the de Broglie wavelength (λ = h/mv), one can show that the wavelength in the n = 3 level is three times as long.

70. Quantum Mechanics and the Hydrogen Atom One of the first great achievements of quantum mechanics was the solution of the wave equation for the hydrogen atom One of the first great achievements of quantum mechanics was the solution of the wave equation for the hydrogen atom The significance of quantum mechanics is that the quantum numbers and the restrictions placed on their values arise directly from the mathematics and not from any assumptions made to make the theory agree with experiments The significance of quantum mechanics is that the quantum numbers and the restrictions placed on their values arise directly from the mathematics and not from any assumptions made to make the theory agree with experiments

71. Quantum Number Summary The values of n can range from 1 to  in integer steps The values of n can range from 1 to  in integer steps The values of ℓ can range from 0 to n-1 in integer steps The values of ℓ can range from 0 to n-1 in integer steps The values of m ℓ can range from -ℓ to ℓ in integer steps The values of m ℓ can range from -ℓ to ℓ in integer steps Also see Table 28.2 Also see Table 28.2

72. QUICK QUIZ 28.2 How many possible orbital states are there for (a) the n = 3 level of hydrogen? (b) the n = 4 level?

73. QUICK QUIZ 28.2 ANSWER The quantum numbers associated with orbital states are n,, and m. For a specified value of n, the allowed values of range from 0 to n – 1. For each value of, there are (2 + 1) possible values of m. (a) If n = 3, then = 0, 1, or 2. The number of possible orbital states is then [2(0) + 1] + [2(1) + 1] + [2(2) + 1] = 1 + 3 + 5 = 9. (b) If n = 4, one additional value of is allowed ( = 3) so the number of possible orbital states is now 9 + [2(3) + 1] = 9 + 7 = 16

74. QUICK QUIZ 28.3 When the principal quantum number is n = 5, how many different values of (a) and (b) m are possible?

75. QUICK QUIZ 28.3 ANSWER (a) For n = 5, there are 5 allowed values of, namely = 0, 1, 2, 3, and 4. (b) Since m ranges from – to + in integer steps, the largest allowed value of ( = 4 in this case) permits the greatest range of values for m. For n = 5, there are 9 possible values for m : -4, -3, -2, –1, 0, +1, +2, +3, and +4.