Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Quantum mechanics provides us with an understanding of atomic structure and atomic properties. Lasers are one of the most important applications of the quantum-mechanical properties of atoms and light. Chapter Goal: To understand the structure and properties of atoms.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Topics today: • Two dimensional waves • The Hydrogen Atom: Angular Momentum and Energy • The Hydrogen Atom: Wave Functions and Probabilities • The rest on WednesdayCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. • 48 Iron atoms assembled into a circular ring. • The ripples inside the ring reflect the electron quantum states of a circular ring (interference effects). D. Eigler (IBM)Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Particle in box: 2 dimensions For a rectangular box, the x and y motions are independent. For a general box, the classical motion may be chaotic. Motion in x direction Motion in y directionCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Particle in box: 2 dimensions Solution is a product of standing wave solutions, one for each dimension: The energy is the sum of energies along each dimension: Schrodinger’s equation:Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Quantum Wave Functions for 2-D box Ground state: same wavelength (longest) in both x and y Need two quantum #’s, one for x-motion one for y-motion Denote a q# pair (nx, ny) Ground state: (1,1) Probability (2D) Wavefunction Probability = (Wavefunction)2 One-dimensional (1D) caseCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. 2D excited states (nx, ny) = (2,1) (nx, ny) = (1,2) These have exactly the same energy for a square box, but the probabilities look different. The different states correspond to ball bouncing in x or in y direction.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Particle in a box What quantum state could this be? A. nx=2, ny=2 B. nx=3, ny=2 C. nx=1, ny=2Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Waves confined by a circular box vanish at the edge radius and are products of a radial wave and an azimuthal wave: The two quantum numbers n and m count nodes in the radial and azimuthal probability densities.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Three dimensions • Object can have different velocity (hence wavelength) in x, y, or z directions. – Need three quantum numbers to label state • (nx, ny , nz) labels each quantum state (a triplet of integers) • Each point in three-dimensional space has a probability associated with it. • Not enough dimensions to plot probability • But can plot a surface of constant probability.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Particle in 3D box • Ground state surface of constant probability • (nx, ny, nz)=(1,1,1) 2D caseCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. (211)!(121)!(112)!All these states have the same energy, but different probabilitiesCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. (221)!(222)!Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. 3-D particle in box: summary • Three quantum numbers (nx,ny,nz) label each state – nx,y,z=1, 2, 3 … (integers starting at 1) • Each state has different motion in x, y, z • Quantum numbers determine – Momentum in each direction: e.g. – Energy: for a cube • Some quantum states have same energy for a cube. The degeneracy is related to the symmetry of the cube. px=hλnx= nxh2L E =px22m+py22m+pz22m= Eonx2+ ny2+ nz2( )Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Use spherical coordinates matched to symmetry Must write the Laplacian in spherical coordinates and solve. Solutions are products of waves in each coordinate Solutions specified by three quantum numbers n, l, m withCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Use spherical coordinates matched to symmetry Solutions are products of waves in each coordinate Solutions specified by three quantum numbers n, l, m withCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Solutions to the Schrödinger equation for the hydrogen atom potential energy exist only if three conditions are satisfied: 1. The atom’s energy must be one of the values where aB is the Bohr radius. The integer n is called the principal quantum number. These energies are the same as those in the Bohr hydrogen atom. Note that the energy does not depend on l and m. Thi is a quirk of the Coulomb potential.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. 2. The angular momentum L of the electron’s orbit must be one of the values The integer l is called the orbital quantum number. 3. The z-component of the angular momentum must be one of the values The integer m is called the magnetic quantum number. Each stationary state of the hydrogen atom is identified by a triplet of quantum numbers (n, l, m).Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The quantum number ‘n’ is the number of nodes in radial direction. The quantum number ‘l’ is the number of nodes in theta. The quantum number ‘m’ is the number of nodes in azimuth. Subtle classical correlation: Orbital angular momentum is conserved and can be related to l and m. The wave factor exp(-im phi) implies spatial variation around the z axis and current density proportional to m.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Wave representing electron!Wave representing electron! Electron wave extends around circumference of orbit. Only integer number of wavelengths around orbit allowed. Bohr’s wave is the azimuthal factor in the complete wave mechanics solution.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The Bohr model computes energy independent of orbit orientation. The full wave theory of spherically
View Full Document
Unlocking...