Chapter 29

1. Chapter 29 Atomic Physics

2. Importance of the Hydrogen Atom • The hydrogen atom is the only atomic system that can be solved exactly • Much of what was learned about the hydrogen atom, with its single electron, can be extended to such single-electron ions as He+ and Li2+

3. More Reasons the Hydrogen Atom is Important • The hydrogen atom proved to be an ideal system for performing precision tests of theory against experiment • Also for improving our understanding of atomic structure • The quantum numbers that are used to characterize the allowed states of hydrogen can also be used to investigate more complex atoms • This allows us to understand the periodic table

4. Final Reason for the Importance of the Hydrogen Atom • The basic ideas about atomic structure must be well understood before we attempt to deal with the complexities of molecular structures and the electronic structure of solids

5. Early Models of the Atom – Newton’s Time • The atom was a tiny, hard indestructible sphere • It was a particle model that ignored any internal structure • The model was a good basis for the kinetic theory of gases

6. Early Models of the Atom – JJ Thomson • J. J. Thomson established the charge to mass ratio for electrons • His model of the atom • A volume of positive charge • Electrons embedded throughout the volume

7. Rutherford’s Thin Foil Experiment • Experiments done in 1911 • A beam of positively charged alpha particles hit and are scattered from a thin foil target • Large deflections could not be explained by Thomson’s model

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

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

10. The Bohr Theory of Hydrogen • 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 • He applied Planck’s ideas of quantized energy levels to orbiting electrons

11. Bohr’s Theory, cont • In this model, the electrons are generally confined to stable, nonradiating orbits called stationary states • Used Einstein’s concept of the photon to arrive at an expression for the frequency of radiation emitted when the atom makes a transition

12. Problem’s With Bohr’s Model • Improved spectroscopic techniques showed many of the single spectral lines were actually a group of closely spaced lines • Single spectral lines could be split into three closely spaced lines when the atom was placed in a magnetic field

13. Mathematical Description of Hydrogen • The solution to Schrödinger’s equation as applied to the hydrogen atom gives a complete description of the atom’s properties • Apply the quantum particle under boundary conditions and determine the allowed wave functions and energies of the atom

14. Mathematical Description, cont • For a three-dimensional system, the boundary conditions will generate three quantum numbers • A fourth quantum number is needed for spin • The potential energy function for the hydrogen atom is

15. Mathematical Description, final • The energies of the allowed states becomes • ao is the Bohr radius • The allowed energy levels depend only on the principle quantum number, n

16. Quantum Numbers, General • The imposition of boundary conditions also leads to two new quantum numbers • Orbital quantum number, • Orbital magnetic quantum number, m

17. Principle Quantum Number • The first quantum number is associated with the radial function • It is called the principle quantum number • It is symbolized by n • The potential energy function depends only on the radial coordinate r • The energies of the allowed states in the hydrogen atom are the same En values found from the Bohr theory

18. Orbital and Orbital Magnetic Quantum Numbers • The orbital quantum number is symbolized by l • It is associated with the orbital angular momentum of the electron • It is an integer • The orbital magnetic quantum number is symbolized by ml • It is also associated with the angular orbital momentum of the electron and is an integer

19. Quantum Numbers, Summary of Allowed Values • The values of n can range from 1 to ¥ • The values of l can range from 0 to n-1 • The values of ml can range from –l to l • Example: • If n = 1, then only l = 0 and ml = 0 are permitted • If n = 2, then l = 0 or 1 • If l = 0 then ml = 0 • If l = 1 then ml may be –1, 0, or 1

20. Quantum Numbers, Summary Table

21. Shells • Historically, all states having the same principle quantum number are said to form a shell • Shells are identified by letters K, L, M … • All states having the same values of n and l are said to form a subshell • The letters s, p,d, f, g, h, .. are used to designate the subshells for which l = 0, 1, 2, 3, …

22. Shell and Subshell Notation, Summary Table

23. Quantum Numbers, final • State with quantum numbers that violate the previous rules cannot exist • They would not satisfy the boundary conditions on the wave function of the system

24. Wave Functions for Hydrogen • The simplest wave function for hydrogen is the one that describes the 1s state and is designated y1s (r) • As y1s (r) approaches zero, r approaches ¥ and is normalized as presented • y1s (r) is also spherically symmetric • This symmetry exists for all s states

25. Probability Density • The probability density for the 1s state is • The radial probability density function, P(r), is the probability per unit radial length of finding the electron in a spherical shell of radius r and thickness dr

26. Radial Probability Density • A spherical shell of radius r and thickness dr has a volume of 4 p r2 dr • The radial probability function is P(r) = 4 p r2|y|2

27. P(r) for 1s State of Hydrogen • The radial probability density function for the hydrogen atom in its ground state is • The peak indicates the most probable location • The peak occurs at the Bohr radius

28. P(r) for 1s State of Hydrogen, cont • The average value of r for the ground state of hydrogen is 3/2 ao • The graph shows asymmetry, with much more area to the right of the peak • According to quantum mechanics, the atom has no sharply defined boundary as suggested by the Bohr theory

29. Electron Clouds • The charge of the electron is extended throughout a diffuse region of space, commonly called an electron cloud • This shows the probability density as a function of position in the xy plane • The darkest area, r = ao, corresponds to the most probable region

30. Electron Clouds, cont • The electron cloud model is quite different from the Bohr model • The electron cloud structure remains the same, on the average, over time • The atom does not radiate when it is in one particular quantum state • This removes the problem of the Rutherford model • Radiation occurs when a transition is made, causing the structure to change in time

31. Wave Function of the 2s state • The next-simplest wave function for hydrogen is for the 2s state • n = 2; l = 0 • The wave function is • y2s depends only on r and is spherically symmetric

32. Comparison of 1s and 2s States • The plot of the radial probability density for the 2s state has two peaks • The highest value of P corresponds to the most probable value • In this case, r » 5ao

33. Physical Interpretation of l • The magnitude of the angular momentum of an electron moving in a circle of radius r is L = me v r • The direction of is perpendicular to the plane of the circle • In the Bohr model, the angular momentum of the electron is restricted to integer multiples of h

34. Physical Interpretation of l, cont • According to quantum mechanics, an atom in a state whose principle quantum number is n can take on the following discrete values of the magnitude of the orbital angular momentum: • That L can equal zero causes great difficulty when attempting to apply classical mechanics to this system • In the quantum mechanical interpretation, the electron cloud for the L = 0 state is spherically symmetrical with no fundamental axis of rotation

35. Physical Interpretation of ml • The atom possesses an orbital angular momentum • Because angular momentum is a vector, its direction must be specified • An orbiting electron can be considered an effective current loop with a corresponding magnetic moment

36. Physical Interpretation of ml, 2 • The direction of the angular momentum vector relative to an axis is quantized • Once an axis is specified, the angular momentum vector can only point in certain directions with respect to this axis • Therefore, Lz, the projection of along the z axis, can have only discrete values

37. Physical Interpretation of ml, 3 • The orbital magnetic quantum number ml specifies the allowed values of the z component of orbital angular momentum • Lz = ml h • The quantization of the possible orientations of with respect to an external magnetic field is often referred to as space quantization

38. Physical Interpretation of ml, 4 • does not point in a specific direction • Even though its z-component is fixed • Knowing all the components is inconsistent with the Uncertainty Principle • Image that must lie anywhere on the surface of a cone that makes an angle q with the z axis

39. Physical Interpretation of ml, final • q is also quantized • Its values are specified through • mlis never greater than l, therefore q can never be zero • can never be parallel to the z-axis

40. Spin Quantum Number, ms • Electron spin does not come from the Schrödinger equation • Additional quantum states can be explained by requiring a fourth quantum number for each state • This fourth quantum number is the spin magnetic quantum number, ms

41. Electron Spins • Only two directions exist for electron spins • The electron can have spin up (a) or spin down (b) • In the presence of a magnetic field, the energy of the electron is slightly different for the two spin directions and this produces doublets in spectra of certain gases

42. Electron Spins, cont • The concept of a spinning electron is conceptually useful • The electron is a point particle, without any spatial extent • Therefore the electron cannot be considered to be actually spinning • The experimental evidence supports the electron having some intrinsic angular momentum that can be described by ms • Sommerfeld and Dirac showed this results from the relativistic properties of the electron

43. Spin Angular Momentum • The total angular momentum of a particular electron state contains both an orbital contribution and a spin contribution • Electron spin can be described by a single quantum number s, whose value can only be s = ½ • The spin angular momentum of the electron never changes

44. Spin Angular Momentum, cont • The magnitude of the spin angular momentum is • The spin angular momentum can have two orientations relative to a z axis, specified by the spin quantum number ms = ± ½ • ms = + ½ corresponds to the spin up case • ms = - ½ corresponds to the spin down case

45. Spin Angular Momentum, final • The allowed values of the z component of the spin angular momentum is Sz = msh = ± ½ h • Spin angular moment S is quantized

46. Spin Magnetic Moment • The spin magnetic moment is related to the spin angular momentum by • The z component of the spin magnetic moment can have values

47. Bohr Magneton • The quantity eħ/2meis called the Bohr magneton • It has a numerical value of 9.274 x 10-24 J/T

48. Stern-Gerlach Experiment • A beam of neutral silver atoms is split into two components by a nonuniform magnetic field • The atoms experienced a force due to their magnetic moments • The beam had two distinct components in contrast to the classical prediction

49. Stern-Gerlach Experimental Results • The experiment provided two important results: • It verified the concept of space quanitization • It showed that spin angular momentum exists even though the property was not recognized until long after the experiments were performed

50. Wolfgang Pauli • 1900 – 1958 • Important review article on relativity • At age 21 • Discovery of the exclusion principle • Explanation of the connection between particle spin and statistics • Relativistic quantum electrodynamics • Neutrino hypothesis • Hypotheses of nuclear spin