CHEM 1310 A/B Fall 2006CHAPTER 16: Quantum Mechanics and the Hydrogen Atom•Waves and Light•Paradoxes in Classical Physics•Planck, Einstein, and Bohr•Waves, Particles, and the Schrödinger equation•The Hydrogen AtomCHEM 1310 A/B Fall 2006Questions• What is quantum mechanics?• When do we need it?• What does it do?• How does it apply to the H atom?CHEM 1310 A/B Fall 2006Quantum Mechanics (QM)Quantum mechanics is…• The set of rules obeyed by small systems (molecules, atoms, and subatomic particles)• One of the two greatest achievements of 20thcentury physics• The basis for new research into smaller electronic devices (e.g., quantum dots)• Required to understand chemistryCHEM 1310 A/B Fall 2006The two-slit experiment• Fire very small particles at a barrier with two tiny slits in it… expect a result like this:R. Shankar, Principles of Quantum MechanicsCHEM 1310 A/B Fall 2006The two-slit experiment• For very small particles, actually get something more like this…R. Shankar, Principles of Quantum MechanicsAn interference pattern! Wavelike properties!CHEM 1310 A/B Fall 2006Actual experiment with electronsResults of a double-slit experimentsending one electron through at atime. Numbers of electrons are(a) 10, (b) 200, (c) 6000, (d) 40000(e) 140000Strange: the wave-like interferencepattern happens even when wesend through only one electronat a time!!!CHEM 1310 A/B Fall 2006Even stranger…• If we watch to see which slit a particle goes through, the interference pattern disappears and we see the “expected” pattern! The experiment changes depending on how we observe it!• Richard Feynman (Nobel Prize in Physics, 1965): “I think it is safe to say that no one understands quantum mechanics. Do not keep saying to yourself, if you can possibly avoid it, 'but how can it be like that?' … Nobody knows how it can be like that." (The Character of Physical Law, 1965, p.129).CHEM 1310 A/B Fall 2006QM: Historical Background• Near the end of the 19thcentury, physicists thought they knew everything• Several key experiments showed something really unknown was going on• QM developed to explain these unusual experiments in early 1900’s (~1900-1930’s)• Developed around same time as theory of relativityCHEM 1310 A/B Fall 2006Electromagnetic spectrumCHEM 1310 A/B Fall 2006“Ultraviolet catastrophe”CHEM 1310 A/B Fall 2006Planck to the rescue• In 1900, Planck postulated that the blackbody is made of tiny oscillators with energies proportional to the frequency of oscillation, E = n h ν, where h is a constant (Planck’s constant, 6.626E-34 J s)• The equation means not just any energies are allowed. Only certain values are allowed. Energy is quantized.• Using this hypothesis, blackbody radiation curves can be predicted accuratelyMax PlanckNobel Prize in Physics, 1918CHEM 1310 A/B Fall 2006The Photoelectric Effect• Light can cause electrons to be ejected from a metal surface• Would expect electrons to be ejected with greater KE if greater light intensity• Problem: KE of electrons does not depend on intensity, but does depend on frequency νCHEM 1310 A/B Fall 2006Einstein to the rescue• Borrowed Planck’s “quantum” idea --- maybe light might have quantized energy levels, too!• Light comes in “packets” of energy E = hν, called “photons”• Explains the photoelectric effect --- higher ν, more energy in each light packet (photon), kicks out electron with more KE Albert EinsteinNobel Prize in Physics, 1905, for explaining the Photoelectric EffectCHEM 1310 A/B Fall 2006Photoelectric effect explainedMinimum energy to remove an electron is hν0, the “work function” of the metalCHEM 1310 A/B Fall 2006Atomic/molecular SpectraIndividual lines. Why???CHEM 1310 A/B Fall 2006H atom spectrum• The lines follow a particular pattern…• Lines fit the “Rydberg formula”ν = (1/n2–1/m2)(3.29 x 1015s-1)where n and m are integers. Amazing!CHEM 1310 A/B Fall 2006Bohr to the rescue• Bohr (1913) borrowed ideas of quantization from Planck and Einstein and explained the H atom spectrum• Bohr argued that angular momentum was quantized ---leads to quantization of H atom energy levels• Bohr frequency condition: ∆E = hν• Equations match the Rydbergformula to an accuracy not seen previously in all of scienceNiels BohrNobel Prize in Physics, 1922, for explaining H atom spectrumCHEM 1310 A/B Fall 2006Bohr’s solution• Quantization of angular momentum…• Leads to quantization of radii (“Bohr orbits”)• Leads to quantization of energies• Assume the “Bohr frequency condition”• Yields the same “Rydbergformula” for allowed energy levels!!!a0= 1 bohr (0.529 Å), Ry= 1 Rydberg = 2.17987 x 10-18JCHEM 1310 A/B Fall 2006H atom spectrum explainedMcQuarrie, “Quantum Chemistry”CHEM 1310 A/B Fall 2006“New quantum theory”• The “quantization” idea was groundbreaking, but it did not have a firm foundation• De Broglie (1924) realized that if light can act as a wave and a particle, then maybe particles like electrons can also act like waves! (Recall 2-slit experiment…) “Wave/particle duality” also works for matter!• Can relate momentum (particle property) to wavelength (wave property) via the de Broglierelationλ = h / p (p = mv)CHEM 1310 A/B Fall 2006Proof of de Broglie relation for a photon (Einstein)CHEM 1310 A/B Fall 2006The Schrödinger Equation• 1925: Schrödinger developed new mechanics for “matter waves” shown by de Broglie. Quantum mechanics!Erwin SchrödingerNobel Prize in Physics, 1933, for the Schrödinger equation, the foundation of quantum mechanicsAustrian 1000 Schilling bank noteCHEM 1310 A/B Fall 2006The Schrödinger Equation• Nuclear kinetic energy• Electron kintetic energy• Nuclear/electron attraction• Nuclear/nuclear repulstion• Electron/electron repulsionCHEM 1310 A/B Fall 2006The Schrödinger Equation• Ψ is the wave function. It gives the amplitude of the matter wave at any position in space (for more than 1 electron, need the coordinates xi= {xi, yi, zi} for each particle i)• Ψ(x1, x2, …, xn) for n particles• Focus on wave function for a single particle (like an electron) for now…CHEM 1310 A/B Fall 2006Classical standing waves• String tied to the wall at both ends (x=0 and x=L)• Have to fit a half-integer number of wavelengths λ in the length L• Number the standing waves n=1, n=2, …• Max amplitude for standing wave n is