Chem 4502 Quantum Mechanics & Spectroscopy (Jason Goodpaster) By 1900, many advances in physics Chap. 1: Dawn of Quantum Theory Text: McQuarrie and Simon's "Physical Chemistry: A Molecular Approach", 1997. • 1897 - JJ Thomson discovered (All figures and tables are from our text unless otherwise noted.) electron, measured its mass/charge ratio Lecture 2: pp. 1-6 Blackbody radiation; Planck's constant, h • Maxwells equations completed in pp. 7-10 Photoelectric effect; 1873-4 partial differential equations that Einsteins explanation; Ephoton = h ν Lecture 3: can solve all problems in classical pp. 10-14 Hydrogen atom spectrum electricity and magnetism Rydberg formula But there were a few experimental results pp. 15-18 de Broglie waves that could not be explained... 1 2 Blackbody Radiation Fig. 1.1 p. 3 Measured frequency distribution of emitted blackbody radiation Ideal Blackbody: As T increases: Absorbs and emits radiation without • maximum shifts to higher favoring particular frequencies frequency (ν) E.g., pinhole in sealed container • total radiated energy (area Radiation emitted is characteristic of the under curve) increases temperature of the body. Figure: Engels Quantum Chem 3 4 Problem.... Blackbody radiation: Plancks theory Dashed line: theoretical prediction (Rayleigh – Jeans Law) Planck (unwillingly) introduced the concept of energy quantization into physics. Ultraviolet Catastrophe: E = n h ν energy density of radiation -1 predicted to diverge as ν2 ν frequency of light (sec ) absorbed or emitted Predicts infinite total by an electron oscillating in the blackbody energy emitted! n positive integer (0, 1, 2, ...) E energy of an oscillating electron; for a given ν, the energies of the oscillators are discrete h a new constant; units must be energy x time 5 Note: Planck didnt quantize the light 6 Blackbody radiation: Plancks theory Using statistical thermodynamics, he derived a formula for the energy density of the radiation. The value of h was fit to the data. Planck was able to fit the data, using the value: h = 6.63 x 10-34 J·sec This success alone did not convince people (including Planck) that energy is really quantized. 7 8 Photoelectric Effect Early measurements of the photoelectric effect: Schematic diagram of a typical apparatus Ejection of an electron (e-) from a metal by radiation. Experiment is: Shine light on a metal surface. light Measure the kinetic energies of the electrons ejected (if any). See how their energies vary with the intensity and frequency of light. 9 10 from Karplus & Porter Atoms and Molecules p. 53 Photoelectric Effect: Fig. 1.4 p. 8 Experimental Observations: Predictions of Classical Physics Photoelectric Effect (Sodium) Electron kinetic energy (eKE): eKE : • increases linearly • should increase with intensity of light with frequency of 2 (proportional to Amplitude of electric field) light does not depend • should not depend on frequency of light • on intensity of light As amplitude of oscillating electric field increases, • no photoelectrons electrons on metal surface should oscillate more detected below a and escape with more energy. certain ν 11 12 Photoelectric Effect: Einsteins Theory Photoelectric Effect: Einsteins Theory One electron in the metal absorbs one photon, Particle model of light: light exists as small in a sort of collision. packages of energy, now called photons Conservation of energy: Ephoton = hν 2 Ephoton = hν = Φ + ½ mev Ephoton energy in one photon E energy in one photon absorbed by one ν frequency of the photon photon electron in the metal Φ work function of the metal This applies to light, not to matter (minimum energy needed to free an electron) (not to electrons, atoms, etc.) 2 ½ mev kinetic energy of the photodetached electron 13 14 Quantum Quote of the Day Chap. 1: Dawn of Quantum Theory (continued) Lecture 3: pp. 15-18 de Broglie waves The next revolution in physics pp. 10-14 Hydrogen atom spectrum will occur when the properties of Rydberg formula mind will be included in the equations of quantum theory. Lecture 4: Eugene Wigner pp. 18-23 Bohr theory of the H atom 1963 Physics Nobel Laureate pp. 23-25 Uncertainty Principle 15 16 de Broglies Hypothesis (1924) After the first World War, I gave a great deal Einstein showed that the energy of a of thought to the theory relativistic particle moving freely in space is: of quanta. It was then that I had a sudden E = (m2c4 + p2c2)½ where m = mass, p = momentum inspiration. Einsteins For a photon, m = 0 and E = hν, so wave-particle dualism h = ( 0 + p2c2)½ = p c for light was an ν absolutely general p = hν / c photon momentum (in the direction of phenomenon extending Louis de Broglie propagation), another particle aspect of radiation 1892-1987 to all physical nature. Also, ν = c / λ and 1 / λ = ν / c, where λ is the wavelength of light and is its frequency, so have: (Fig. 18.2 p. 38) ν 17 p = h / λ λ = h / p for light 18 de Broglies Hypothesis de Broglies Hypothesis How to test this strange hypothesis? de Broglie generalized this equation to matter as well: Recall 2-slit experiment with light: λ = h / p But: to detect where p = classical momentum = m v (mass x velocity) interference effects, must detect λ = de Broglie wavelength of the material particle differences on the Matter waves ?!? order of λ between the lengths of He [de Broglie] has lifted one corner of the great veil. alternate paths. Einstein Maxima on detector occur where Is this possible?? 19 20 r2 – r1 = nλ (n = 1, 2, 3, ...) de Broglies Hypothesis de Broglies Hypothesis Can we observe a de Broglie wavelength? • Problem: What is the de Broglie wavelength of Problem: What is the de Broglie wavelength of a golf an electron (m = 9.11 x 10-31 kg) accelerated ball (50 grams) traveling at 30 meters/second? e through a potential of 54.0 Volts? λ = h / p h = 6.63 x 10-34 J∙s λ = h / p h = 6.6 x 10-34 J∙s e = 1.60 x 10-19 C λ = 4.4 x 10-34 meters = 4.4 x 10-22 picometers (pm) λ = 1.67 x 10-10 m = 1.67 Å = 167 pm This wavelength is on the scale of atomic diameters, -15 The size of a nucleus of an atom is about 10 meters. and it is in the x-ray region. Conclude: golf ball displays only particle-like properties. (What if its at rest? For this, need uncertainty Slits: spacings between atoms in a crystal. 21 22 principle.) Davisson-Germer experiment, 1927: For single-plane diffraction: n λ = a sin θ Scattering of an electron beam from a nickel a = distance between Ni atoms = 2.15 x 10-10 m crystal shows constructive and destructive (known from x-ray diffraction experiments) interference (Atkins Physical Chemistry 6th Ed. p. 293) θ = angle at which intensity maximum is observed Constructive n = 1 for the strong 1st order peak interference occurs at angles at which For electrons accelerated to 54.0 V, D&G measured the electron waves the maximum scattering angle (n=1) to be 50.1o. scattered by different Ni atoms What is the wavelength, λ, of the electrons ? have path lengths λ = a sin θ / n = 2.15 x 10-10 m (sin 50.1o) / 1 differing by nλ. λ = 1.65 x 10-10 m Destructive: nλ/2, n odd → nodes. 23 Same as we calculated from the de Broglie formula ! 24 George Thomson - used ≈ 10 keV electrons to These electron diffraction experiments confirmed penetrate thin metal films, recorded diffraction patterns the de Broglie wavelength: Fig. 1.8 p. 17 Similarity between X-ray and λ = h / p Electron Diffraction Patterns (Aluminum Foil) Nobel Prizes in Physics: 1929 de Broglie 1937 G. P. Thomson and C. J. Davisson 1906 JJ Thomson (Georges Dad) discovery of the electron as a subatomic particle (1987) X-rays Electrons 25 26 What would we see if we passed http://www.charfac.umn.edu/instruments/ a single electron through Davisson and Germers or Thomsons experiments ? IT Characterization Facility 12 Shepherd Labs Transmission electron The electron would land at a localized spot on the microscope (TEM) with an detector (no interference pattern). operating voltage range of 20 to 120 kV. Allows magnifications of thin samples (<500 nm) up to 700,000 times. Point-point resolution: 0.34 nm This is like the single photon 2-slit experiments – Line resolution: 0.2 nm 27 28 wave-particle duality applies to both light and matter. SEE DOWN TO 0.6 Å Chemical and Engineering News Electron microscope Sept. 2004 achieves direct sub-angstrom imaging of a crystal A milestone in electron microscopy--the first direct sub- angstrom imaging of a crystal lattice--has been reported by researchers at Oak Ridge National Laboratory and Nion, a company in Kirkland, DUMBBELLS STEM image Wash., that specializes in of a silicon crystal in the advanced electron-microscope [112] orientation reveals optics [Science, 305, 1741 pairs of atom columns in (2004)]. The researchers fitted a which the intrapair 300-kV scanning transmission electron microscope (STEM) at separation is 0.78 Å. ORNL with a Nion aberration 29 30 corrector... Emission spectra of atoms (e.g., hydrogen): another experiment that baffled classical physics. Fig. 1.7 p. 13 Hydrogen Atom Emission Spectrum 31 32 Emission spectrum of H atom Rydberg formula for H atom ∼ -1 2 2 ν = 1 / λ (cm) = 109,680 cm ( 1 / n1 – 1 / n2 ) What are n1 and n2 for the shortest wavelength The observed wavelengths emitted (lines) were emission line of the H atom? found to fit the Rydberg formula: What is this wavelength, in nm? In what region of the electromagnetic spectrum is it? ∼ -1 2 2 ν = 1 / λ (cm) = 109,680 cm ( 1 / n1 – 1 / n2 ) n1=1, n2 = ∞ wavenumbers ν∼ = 109,680 cm-1 where n1 and n2 are integers (n2 > n1) λ = 1 / ν∼ = 1 / 109, 680 cm-1 = 91.2 x 10-7 cm λ = 91.2 x 10-9 m = 91.2 nm UV region Why would H atom care about integers?33 34 Note on use of wavenumbers (cm-1) Chap.