Chapter 28 Quantum Mechanics of Atoms

2. Chapter 28Quantum Mechanics of Atoms

3. Contents of Chapter 28 • Quantum Mechanics—A New Theory • The Wave Function and Its Interpretation;the Double-Slit Experiment • The Heisenberg Uncertainty Principle • Philosophic Implications; Probability versus Determinism • Quantum-Mechanical View of Atoms

4. Contents of Chapter 28 • Quantum Mechanics of the Hydrogen Atom;Quantum Numbers • Multielectron Atoms; the Exclusion Principle • The Periodic Table of Elements • X-Ray Spectra and Atomic Number • Fluorescence and Phosphorescence • Lasers • Holography

5. 28-1 Quantum Mechanics—A New Theory Quantum mechanics incorporates wave-particle duality, and successfully explains energy states in complex atoms and molecules, the relative brightness of spectral lines, and many other phenomena. It is widely accepted as being the fundamental theory underlying all physical processes.

6. 28-1 Quantum Mechanics—A New Theory Quantum mechanics is essential to understanding atoms and molecules, but can also have effects on larger scales.

7. 28-2 The Wave Function and Its Interpretation; the Double-Slit Experiment An electromagnetic wave has oscillating electric and magnetic fields. What is oscillating in a matter wave? This role is played by the wave function, Ψ. The square of the wave function at any point is proportional to the number of electrons expected to be found there. For a single electron, the wave function is the probability of finding the electron at that point.

8. 28-2 The Wave Function and Its Interpretation; the Double-Slit Experiment For example: the interference pattern is observed after many electrons have gone through the slits. If we send the electrons through one at a time, we cannot predict the path any single electron will take, but we can predict the overall distribution.

9. 28-3 The Heisenberg Uncertainty Principle Quantum mechanics tells us there are limits to measurement—not because of the limits of our instruments, but inherently. This is due to the wave-particle duality, and to interaction between the observing equipment andthe object being observed.

10. 28-3 The Heisenberg Uncertainty Principle Imagine trying to see an electron with a powerful microscope. At least one photon must scatter off the electron and enter the microscope, but in doing soit will transfer some of its momentum to the electron.

11. 28-3 The Heisenberg Uncertainty Principle The uncertainty in the momentum of the electron is taken to be the momentum of the photon—it could transfer anywhere from none to all of its momentum. In addition, the position can only be measured to about one wavelength of the photon.

12. 28-3 The Heisenberg Uncertainty Principle (28-1) Combining, we find the combination of uncertainties: This is called the Heisenberg uncertainty principle. It tells us that the position and momentum cannot simultaneously be measured with precision.

13. 28-3 The Heisenberg Uncertainty Principle (28-2) This relation can also be written as a relation between the uncertainty in time and the uncertainty in energy: This says that if an energy state only lasts for a limited time, its energy will be uncertain. It also says that conservation of energy can be violated if the time isshort enough.

14. 28-4 Philosophic Implications;Probability versus Determinism The world of Newtonian mechanics is a deterministic one. If you know the forces on an object and its initial velocity, you can predict where it will go. Quantum mechanics is very different—you can predict what masses of electrons will do, but have no idea what any individual one will.

15. 28-5 Quantum-Mechanical View of Atoms Since we cannot say exactly where an electron is, the Bohr picture of the atom, with electrons in neat orbits, cannot be correct. Quantum theory describesan electron probabilitydistribution; this figureshows the distributionforthe ground state ofhydrogen.

16. 28-6 Quantum Mechanics of the Hydrogen Atom; Quantum Numbers There are four different quantum numbers needed to specify the state of an electron in an atom. • Principal quantum number n gives the total energy:

17. 28-6 Quantum Mechanics of the Hydrogen Atom; Quantum Numbers (28-3) 2. Orbital quantum number lgives the angular momentum; l can take on integer values from0 to n − 1. 3. The magnetic quantum number, ml, gives the direction of the electron’s angular momentum, and can take on integer values from −lto +l.

18. 28-6 Quantum Mechanics of the Hydrogen Atom; Quantum Numbers This plot indicates the quantization of angular momentum direction forl = 2. The other two components of the angular momentum are undefined.

19. 28-6 Quantum Mechanics of the Hydrogen Atom; Quantum Numbers 4. The spin quantum number, ms, which for an electron can take on the values +½ and −½. The need for this quantum number was found by experiment; spin is an intrinsically quantum mechanical quantity, although it mathematically behaves as a form of angular momentum.

20. 28-6 Quantum Mechanics of the Hydrogen Atom; Quantum Numbers This table summarizes the four quantum numbers.

21. 28-6 Quantum Mechanics of the Hydrogen Atom; Quantum Numbers The angular momentum quantum numbers do not affect the energy level much, but they do change the spatial distribution of the electron cloud.

22. 28-6 Quantum Mechanics of the Hydrogen Atom; Quantum Numbers “Allowed”transitions between energy levels occur between states whose value of l differ by one: Other, “forbidden,”transitions also occur but with much lower probability.

23. 28-7 Multielectron Atoms;the Exclusion Principle Complex atoms contain more than one electron, so the interaction between electrons must be accounted for in the energy levels. This means that the energy depends on both n and l. A neutral atom has Z electrons, as well as Z protons in its nucleus. Z is called the atomic number.

24. 28-7 Multielectron Atoms;the Exclusion Principle In order to understand the electron distributions in atoms, another principle is needed. This is the Pauli exclusion principle: No two electrons in an atom can occupy the same quantum state. The quantum state is specified by the four quantum numbers; no two electrons in the same atom can have the same set.

25. 28-7 Multielectron Atoms;the Exclusion Principle This chart shows the occupied—and some unoccupied—states in He, Li, and Na.

26. 28-8 The Periodic Table of the Elements We can now understand the organization of the periodic table. Electrons with the same n are in the same shell. Electrons with the same n and l are in the same subshell. The exclusion principle limits the maximum number of electrons in each subshell to 2(2l + 1).

27. 28-8 The Periodic Table of the Elements Each value of l is given its own letter symbol.

28. 28-8 The Periodic Table of the Elements Electron configurations are written by giving the value for n, the letter code for l, and the number of electronsin the subshell as a superscript. For example, here is the ground-state configurationof sodium: 1s22s22p63s1

29. 28-8 The Periodic Table of the Elements This table shows the configuration of theouter electrons only.

30. 28-8 The Periodic Table of the Elements Atoms with the same number of electrons in their outer shells have similar chemical behavior. They appear in the same column of the periodic table. The outer columns—those with full, almost full, or almost empty outer shells—are the most distinctive. The inner columns, with partly filled shells, have more similar chemical properties.

31. 28-9 X-Ray Spectra and Atomic Number The effective charge that an electron “sees”is the charge on the nucleus shielded by inner electrons. Only the electrons in the first level see the entire nuclear charge.

32. 28-9 X-Ray Spectra and Atomic Number The energy of a level is proportional to Z2, so the wavelengths corresponding to transitions to the n= 1 state in high-Z atoms are in the X-ray range.

33. 28-9 X-Ray Spectra and Atomic Number Inner electrons can be ejected by high-energy electrons. The resulting X-ray spectrum is characteristic of the element. This example is for molybdenum.

34. 28-9 X-Ray Spectra and Atomic Number Measurement of these spectra allows determination of inner energy levels, as well as Z, as the wavelength of the shortest X-rays is inversely proportional to Z2.

35. 28-9 X-Ray Spectra and Atomic Number The continuous part of the X-ray spectrum comes from electrons that are decelerated by interactions within the material, and therefore emit photons. This radiation is calledbremsstrahlung(“braking radiation”).

36. 28-10 Fluorescence and Phosphorescence If an electron is excited to a higher energy state, it may emit two or more photons of longer wavelength as it returns to the lower level.

37. 28-10 Fluorescence and Phosphorescence Fluorescence occurs when the absorbed photon is ultraviolet and the emitted photons are in the visible range. Phosphorescence occurs when the electron is excited to a metastable state; it can take seconds or more to return to the lower state. Meanwhile, the material glows.

38. 28-11 Lasers A laser produces a narrow, intense beam of coherent light. This coherence means that, at a given cross section, all parts of the beam have the same phase. The top figure shows absorption of a photon. The bottom picture shows stimulated emission—if the atom is already in the excited state, the presence of another photon of the same frequency can stimulate the atom to make the transition to the lower state sooner. These photons are in phase.

39. 28-11 Lasers To obtain coherent light from stimulated emission, two conditions must be met: • Most of the atoms must be in the excited state; this is called an inverted population.

40. 28-11 Lasers 2. The higher state must be a metastable state, so that once the population is inverted, it stays that way. This means that transitions occur through stimulated emission rather than spontaneously.

41. 28-11 Lasers The laser beam is narrow, only spreading due to diffraction, which is determined by the size of theend mirror. An inverted population can be created by exciting electrons to a state from which they decay to a metastable state. This is called optical pumping.

42. 28-11 Lasers A metastable state can also be created through interactions between two sets of atoms, such as in a helium-neon laser.

43. 28-11 Lasers Lasers are used for a wide variety of applications: surgery, machining, surveying, reading bar codes, CDs, and DVDs, and so on. This diagram shows how a CD is read.

44. 28-12 Holography Holograms are created using the coherent light of a laser. The beam is split, allowing the film to record both the intensity and the relative phase of the light. The resulting image, when illuminated by a laser, is three-dimensional. White-light holograms are made with a laser but can be viewed in ordinary light. The emulsion is thick, and contains interference patterns that make the image somewhat three-dimensional.

45. Summary of Chapter 28 • Quantum mechanics is the basic theory at the atomic level; it is statistical rather than deterministic • Heisenberg uncertainty principle: • Electron state in atom is specified by four numbers, n, l, ml, and ms

46. Summary of Chapter 28 • n, the principal quantum number, can have any integer value, and gives the energy of the level • l, the orbital quantum number, can have values from 0 to n− 1 • ml, the magnetic quantum number, can have values from −lto +l • ms, the spin quantum number, can be +½ or −½

47. Summary of Chapter 28 • Energy levels depend on n and l, except in hydrogen. The other quantum numbers also result in small energy differences. • Pauli exclusion principle: no two electrons in the same atom can be in the same quantum state • Electrons are grouped into shells and subshells • Periodic table reflects shell structure • X-ray spectrum can give information about inner levels and Z of high-Z atoms