Chapter 39 Quantum Mechanics of Atoms

Transcription

1 Chapter 39 Quantum Mechanics of Atoms

2 Units of Chapter 39 Quantum-Mechanical View of Atoms Hydrogen Atom: Schrödinger Equation and Quantum Numbers Hydrogen Atom Wave Functions Complex Atoms; the Exclusion Principle The Periodic Table of Elements X-Ray Spectra and Atomic Number Magnetic Dipole Moments; Total Angular Momentum

3 Units of Chapter 39 Fluorescence and Phosphorescence Lasers Holography

4 39.1 Quantum-Mechanical View of 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 describes an electron probability distribution: Atoms

5 39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers Potential energy for the hydrogen atom:

6 39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers The time-independent Schrödinger equation in three dimensions is then: Equation 39-1 goes here. where Equation 39-2 goes here.

7 39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers There are four different quantum numbers needed to specify the state of an electron in an atom. 1. The principal quantum number n gives the total energy. 2. The orbital quantum number gives the angular momentum; l can take on integer values from 0 to n -1. l Equation 39-3 goes here.

8 39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers 3. The magnetic quantum number, m l, gives the direction of the electron s angular momentum, and can take on integer values from to +. l l

9 39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers Figure 39-3 goes here. This plot indicates the quantization of angular momentum direction for l = 2. The other two components of the angular momentum are undefined.

10 39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers In a magnetic field, the energy levels split depending on m. l

11 39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers 4. The spin quantum number, m s, 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.

12 39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers This table summarizes the four quantum numbers.

13 39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers Conceptual Example 39-1: Possible states for n = 3. How many different states are possible for an electron whose principal quantum number is n = 3?

14 39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers Example 39-2: E and L for n = 3. Determine (a) the energy and (b) the orbital angular momentum for an electron in each of the hydrogen atom states of Example 39 1.

15 39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers Allowed transitions between energy levels occur between states whose value of ldiffer by one: Other, forbidden, transitions also occur but with much lower probability.

16 39.3 Hydrogen Atom Wave Functions The wave function of the ground state of hydrogen has the form: The probability of finding the electron in a volume dv around a given point is then ψ 2 dv.

17 39.3 Hydrogen Atom Wave Functions The ground state is spherically symmetric; the probability of finding the electron at a distance between r and r + dr from the nucleus is:

18 39.3 Hydrogen Atom Wave Functions Example 39-3: Most probable electron radius in hydrogen. Determine the most probable distance r from the nucleus at which to find the electron in the ground state of hydrogen.

19 39.3 Hydrogen Atom Wave Functions Example 39-4: Calculating probability. Determine the probability of finding the electron in the ground state of hydrogen within two Bohr radii of the nucleus.

20 39.3 Hydrogen Atom Wave Functions This figure shows the three probability distributions for n = 2 and l= 1 (the distributions for m l = +1 and m l = -1 are the same), as well as the radial distribution for all n = 2 states.

21 39.4 Complex 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. A neutral atom has Z electrons, as well as Z protons in its nucleus. Z is called the atomic number. l

22 39.4 Complex 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 can have the same set.

23 39.4 Complex Atoms; the Exclusion Principle This chart shows the occupied and some unoccupied states in He, Li, and Na.

24 39.5 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(2 + 1). l

25 39.5 The Periodic Table of the Elements l Each value of is given its own letter symbol.

26 39.5 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 electrons in the subshell as a superscript. For example, here is the groundstate configuration of sodium: 1s 2 2s 2 2p 6 3s 1

27 39.5 The Periodic Table of the Elements This table shows the configuration of the outer electrons only.

28 39.5 The Periodic Table of the Elements Conceptual Example 39-5: Electron configurations. Which of the following electron configurations are possible, and which are not: (a) 1s 2 2s 2 2p 6 3s 3 ; (b) 1s 2 2s 2 2p 6 3s 2 3p 5 4s 2 ; (c) 1s 2 2s 2 2p 6 2d 1?

29 39.5 The Periodic Table of the Elements Atoms with the same number of electrons in their outer shells have similar chemical behaviors. 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.

30 39.6 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. The energy of a level is proportional to Z 2, so the wavelengths corresponding to transitions to the n = 1 state in high-z atoms are in the X-ray range.

31 39.6 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.

32 39.6 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 Z 2.

33 39.6 X-Ray Spectra and Atomic Number Example 39-6: X-ray wavelength. Estimate the wavelength for an n = 2 to n = 1 transition in molybdenum (Z = 42). What is the energy of such a photon?

34 39.6 X-Ray Spectra and Atomic Number Example 39-7: Determining atomic number. High-energy photons are used to bombard an unknown material. The strongest peak is found for X-rays emitted with an energy of 66 kev. Guess what the material is.

35 39.6 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 called bremsstrahlung ( braking radiation ).

36 39.6 X-Ray Spectra and Atomic Number Example 39-8: Cutoff wavelength. What is the shortest-wavelength X-ray photon emitted in an X-ray tube subjected to 50 kv?

37 39.7 Magnetic Dipole Moments; Total Angular Momentum If we consider the electron to be moving in a circle of radius r at speed v around the nucleus, the magnetic dipole moment is given by: where the angular momentum L = mvr. This semiclassical calculation is also valid quantum mechanically as long as the angular momentum is quantized.

38 39.7 Magnetic Dipole Moments; Total Angular Momentum The z component of the dipole moment, where z is defined to be the direction of an external magnetic field, is given by:. The Bohr magneton is defined as:

39 39.7 Magnetic Dipole Moments; Total Angular Momentum Now we can write the z component of the magnetic dipole moment as: An atom placed in a magnetic field will have its energy levels shifted depending on the value of m l; this is the Zeeman effect.

40 39.7 Magnetic Dipole Moments; Total Angular Momentum In the Stern-Gerlach experiment, atoms were sent through a nonuniform magnetic field. This field deflects atoms differently depending on their magnetic dipole moments. Classically, one would expect a continuum of deflection angles.

41 39.7 Magnetic Dipole Moments; Total Angular Momentum Instead, the angles were quantized, corresponding to the quantized values of the magnetic moment. Figure goes here.

42 39.7 Magnetic Dipole Moments; Total Angular Momentum The total angular momentum (the vector sum of the orbital and spin angular momenta) is also quantized: The state of an electron can then be written in the form nl j, such as 2P 3/2 (n = 2, l= 1, j = 3/2) and 1S 1/2 (ground state of hydrogen).

43 39.8 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. Figure goes here.

44 39.8 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 longer to return to the lower state. Meanwhile, the material glows.

45 39.9 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. (a) shows absorption of a photon. (b) 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.

46 39.9 Lasers To obtain coherent light from stimulated emission, two conditions must be met: 1. Most of the atoms must be in the excited state; this is called an inverted population. Figure goes here.

47 39.9 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. Figure goes here.

48 39.9 Lasers The laser beam is narrow, only spreading due to diffraction, which is determined by the size of the end 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.

49 39.9 Lasers A metastable state can also be created through interactions between two sets of atoms, such as in a helium neon laser.

50 39.9 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.

51 39.10 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 threedimensional. 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.

52 Summary of Chapter 39 The electron state in an atom is specified by four numbers; n, l, m l, and m s. 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. m l, the magnetic quantum number, can have values from to +. l l m s, the spin quantum number, can be +½ or -½.

53 Summary of Chapter 39 Energy levels depend on n and, 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. The periodic table reflects shell structure. The X-ray spectrum can give information about inner levels and Z of high-z atoms. l