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LECTURE 21

PHYSICS 420 SPRING 2006 Dennis Papadopoulos. LECTURE 21. THE HYDROGEN AND HYDROGENIC ATOMS. The Hydrogen Atom. 1 Application of the Schr ö dinger Equation to the Hydrogen Atom 2 Solution of the Schr ö dinger Equation for Hydrogen 3 Quantum Numbers

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LECTURE 21

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  1. PHYSICS 420 SPRING 2006 Dennis Papadopoulos LECTURE 21 THE HYDROGEN AND HYDROGENIC ATOMS

  2. The Hydrogen Atom • 1 Application of the Schrödinger Equation to the Hydrogen Atom • 2 Solution of the Schrödinger Equation for Hydrogen • 3 Quantum Numbers • 4. Energy Levels and Electron Probabilities • 5 Magnetic Effects on Atomic Spectra – The Normal Zeeman Effect • 6 Intrinsic Spin The atom of modern physics can be symbolized only through a partial differential equation in an abstract space of many dimensions. All its qualities are inferential; no material properties can be directly attributed to it. An understanding of the atomic world in that primary sensuous fashion…is impossible. - Werner Heisenberg

  3. 1: Application of the Schrödinger Equation to the Hydrogen Atom The approximation of the potential energy of the electron-proton system is electrostatic: Rewrite the three-dimensional time-independent Schrödinger Equation. For Hydrogen-like atoms (He+ or Li++) Replace e2 with Ze2 (Z is the atomic number).

  4. Application of the Schrödinger Equation Transform to spherical polar coordinates because of the radial symmetry. Insert the Coulomb potential into the transformed Schrödinger equation. The potential (central force) V(r) depends on the distance r between the proton and electron.

  5. Fig. 8-5, p.266

  6. Application of the Schrödinger Equation The wave function ψ is a function of r, θ, . Equation is separable. Solution may be a product of three functions. We can separate the SE into three separate differential equations, each depending on one coordinate: r, θ, or .

  7. ----Radial equation ----Angular equation Solution of the Schrödinger Equation • Eqs (8.11) to eqs (8.15) yield • SE has been separated into three ordinary second-order differential equations each containing only one variable.

  8. Solution of the Radial Equation • The radial equation is called the associated Laguerre equation and the solutions R that satisfy the appropriate boundary conditions are called associated Laguerre functions. • Assume the ground state has ℓ = 0 and this requires mℓ = 0. The radial equation becomes or

  9. Solution of the Radial Equation • Try a solution Take derivatives of R and insert them into the SE equation. • To satisfy it for any r each of the two expressions in parentheses to be zero. Set the second parentheses equal to zero and solve for a0. Set the first parentheses equal to zero and solve for E. Both equal to the Bohr result.

  10. Quantum Numbers • The appropriate boundary conditions to the radial and angular equations leads to the following restrictions on the quantum numbers ℓ and mℓ: • ℓ = 0, 1, 2, 3, . . . • mℓ = −ℓ, −ℓ + 1, . . . , −2, −1, 0, 1, 2, . ℓ . , ℓ − 1, ℓ • |mℓ| ≤ ℓ and ℓ< 0. • The predicted energy level is

  11. Hydrogen Atom Radial Wave Functions First few radial wave functions Rnℓ Subscripts on R specify the values of n and ℓ.

  12. Solution of the Angular and Azimuthal Equations • The solutions for azimuthal equation are . • Solutions to the angular and azimuthal equations are linked because both have mℓ. • Group these solutions together into functions. ---- spherical harmonics

  13. Solution of the Angular and Azimuthal Equations The radial wave function R and the spherical harmonics Y determine the probability density for the various quantum states. The total wave function depends on n, ℓ, and mℓ. The wave function becomes

  14. Normalized Spherical Harmonics

  15. Table 8-2, p.269

  16. Table 8-3, p.269

  17. Probability Distribution Functions • We must use wave functions to calculate the probability distributions of the electrons. • The “position” of the electron is spread over space and is not well defined. • We may use the radial wave function R(r) to calculate radial probability distributions of the electron. • The probability of finding the electron in a differential volume element dτis .

  18. Probability Distribution Functions • The differential volume element in spherical polar coordinates is Therefore, • We are only interested in the radial dependence. • The radial probability density is P(r) = r2|R(r)|2 and it depends only on n and l.

  19. Isotropic States only l=0 Fig. 8-9, p.282

  20. Average vs. most probable distance Fig. 8-10a, p.283

  21. Fig. 8-10, p.283

  22. Probability Distribution Functions • R(r) and P(r) for the lowest-lying states of the hydrogen atom.

  23. Table 8-5, p.280

  24. Fig. 8-11b, p.285

  25. l=2 Fig. 8-11c, p.285

  26. Probability Densities Symmetric about z-axis Fig. 8-12, p.286

  27. 3: Quantum Numbers The three quantum numbers: • n Principal quantum number • ℓ Orbital angular momentum quantum number • mℓ Magnetic quantum number The boundary conditions: • n = 1, 2, 3, 4, . . . Integer • ℓ = 0, 1, 2, 3, . . . , n − 1 Integer • mℓ = −ℓ, −ℓ + 1, . . . , 0, 1, . . . , ℓ − 1, ℓ Integer The restrictions for quantum numbers: • n > 0 • ℓ < n • |mℓ| ≤ ℓ

  28. Principal Quantum Number n • It results from the solution of R(r) in because R(r) includes the potential energy V(r). The result for this quantized energy is • The negative means the energy E indicates that the electron and proton are bound together.

  29. Orbital Angular Momentum Quantum Number ℓ • It is associated with the R(r) and f(θ) parts of the wave function. • Classically, the orbital angular momentum with L = mvorbitalr. • ℓ is related to L by . • In an ℓ = 0 state, . It disagrees with Bohr’s semiclassical “planetary” model of electrons orbiting a nucleus L = nħ.

  30. 4 quantum numbers needed to specify an electron's state The allowed energy levels are quantized much like or particle in a box. Since the energy level decreases a the square of n, these levels get closer together as n gets larger.

  31. Atomic structure

  32. Chemical properties of an atom are determined by the least tightly bound electrons. • Factors: • Occupancy of subshell • Energy separation between the subshell and the next higher subshell.

  33. Pauli principle and Minimum Energy Principle

  34. He Z=2, n=1, l=0, m=0. • Two electron with opposite spin • Zero angular momentum • High ionization energy 54.4 eV • Inert

  35. Li Z=3, n=1 full, go to n=2, L-shell • Bigger atom, 4 times ao (~n2) • Nuclear charge partially screened by n=1 electrons • Low ionization potential • Energy of outer electrons

  36. Hund’s Rule Unpaired spins Fig. 9-15, p.321

  37. Table 9-2a, p.322

  38. Table 9-2b, p.323

  39. Fig. 9-16, p.324

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