09.12.2011 - Chem 260/261 Monday, January 10, 2011...

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Unformatted text preview: Chem 260/261 Monday, January 10, 2011 Chemistry 260/261 Today in Chemistry 260/261 •  Wave- particle duality and the deBroglie wavelength •  The birth of the quantum theory •  Instructor Of@ice Hour in SLC Satellite in USB 11am- 12pm – Team Room D (MWF) This Week in Chemistry 260/261 •  •  •  •  •  •  •  Reading: (Wed) 4.5- 4.6, (Fri) 4.7 5.1 169- 174. The Fourier Theorem and quantum mechanical operators Focus on energy: Schrödinger’s equation Particle in a Box Harmonic Oscillator Hydrogen Atom Problem set 1 due Today by 3:00 pm Lecture 3 Sept. 12, 2011 2. Blackbody Radiation This is a familiar concept: Rayleigh- Jeans law (classical mechanics) = ultraviolet catastrophe! 2. Blackbody Radiation This is a familiar concept: Electric Stove Electric Stove Light Bulb Incandescent lamp T = 2500 K White Hot Black Red Orange Cold Hot Hotter A B C D $1' ρ T (λ ) = 8 πkB & ) % λT ( 4 Light Bulb Incandescent lamp T = 2500 K White Hot Black Red Orange Cold Hot Hotter Thermal Imaging Which of these is hottest? Temperature °C Planck’s Suggestion Max Planck (1858- 1947) Nobel Prize 1918 ρT (ν ) = 8 π hν 3 c3 Planck’s Suggestion Max Planck (1858- 1947) Nobel Prize 1918 1 e hν kB T ρT (ν ) = −1 known constants c = speed of light kB = Boltzmann constant 1 e hν kB T −1 known constants c = speed of light kB = Boltzmann constant h = bitting parameter = 6.626 × 10- 34 J s h = bitting parameter = 6.626 × 10- 34 J s The ‘blackbody’ is composed of oscillators. Energy The ‘blackbody’ is composed of oscillators. Lecture 3 8 π hν 3 c3 classicallyallowed energies 1 Chem 260/261 Monday, January 10, 2011 3. The Photoelectric Effect Max Planck (1858- 1947) Nobel Prize 1918 8 π hν 3 c3 1 e hν kB T ee- -eee- λ ρT (ν ) = −1 known constants c = speed of light kB = Boltzmann constant Metal Surface kinetic energy of ejected electron Planck’s Suggestion y = a + bx Φ frequency of light h = bitting parameter = 6.626 × 10- 34 J s Energy The energy of each oscillator of the black body is limited to discrete values and cannot be varied arbitrarily. 1. No electrons are ejected, regardless of the intensity of the radiation, unless the frequency exceeds a threshold value characteristic of the metal. 2. The kinetic energy of the ejected electron varies linearly with the frequency of the incident radiation, but is independent of its intensity. E = nhν n = 0,1, 2, ... Einstein’s First Paper 3. Even at low light intensities, electrons are ejected immediately if the frequency is above the threshold value. Planck’s constant permitted energy levels EK = hν − Φ QUANTIZATION Extended Planck’s quantum hypothesis The Quantum Stone Age: Bohr’s Atomic Model •  Light of frequency ν consists of quanta of energy hν (photons) E = hν = hc •  Light transfers energy and momentum only in these particle- like photon bundles h Photons are massless particles, but still carry momentum p = In 1910, Ernest Rutherford’s “planetary model” of the atom was widely accepted λ In 1913, Niels Bohr added a few assumptions to Rutherford’s theory based on Planck’s idea of quantization. λ •  An electron in the metal gains enough energy from a photon to overcome the forces that holds it in the metal. $ hν − Φ when hν > Φ 1 mv 2 ax = % m 2 when hν < Φ &0 free electron hν Φ bound electron The Quantum Stone Age: Bohr’s Atomic Model v −e r e +Ze 1. Attraction between the proton and electron v −e r 2. Electron’s motion is a circular orbit e +Ze 3. Forces are balanced m v2 Ze2 = r 4 πε 0 r 2 4. Angular momentum of electron is quantized and must be an integral multiple of h/2π Le = me vr = n h n = 1,2,3... 2π Bohr’s model explained atomic emission spectra!... 5. The electrons can only attain certain stable states (orbits): rn = ε 0n2h2 n2 = a0 π Ze2 me Z With certain quantized energy levels: En = hν=ΔE − Ze4 me Z2 = −(2.18 × 10 −18 J) 2 222 8ε 0 n h n RH 6. Absorption (or emission) of light corresponds to a transition between stable states excited state Atoms absorb and emit light at a discrete set of frequencies which is characteristic of the corresponding element. Energy levels of atoms are quantized. Lecture 3 ground state ΔE = hν 2 Chem 260/261 Monday, January 10, 2011 … but only that of hydrogen… Using ΔE = hν 24 ν= Z e me $ 1 1' &−) 2 8ε 0 h 2 % nf2 ni2 ( Z2(3.29 × 1015 s-1) Bohr’s model was little more than quantization grafted upon classical mechanics… still it was a step in the right direction… What kind of Physics can give rise to quantization? Circular standing wave: Bohr’s hypothesis: nλ nλ = 2πr , n = 1,2,3,… ⇒ r = 2π nh me vr = , n = 1,2,3,… 2π nλ nh ⇒ me vr = me v = 2π 2π Wave Particle Duality: p= What kind of Physics can give rise to quantization? Standing Waves n λ 2 = L , n = 1, 2,3,K nλ = 2π r , n = 1, 2,3,K Bohr developed a theory of the atom involving electrons in orbits - with a discrete number of wavelengths per orbit, if matter has wave- like properties. Why we thought light was a wave… The “two slit experiment” Instead, we If light were particle like, what might observe many we observe? lines… h ⇒ me v = p = λ h λ ⇔ λ= h h = p mv The only way to explain this phenomenon is diffraction. Bohr’s theory can be justibied. All you have to do is assume that particles behave like waves …just like Einstein’s assumption that light (waves) came in discrete photons (particles). Comment on diffraction: Diffraction of sea water waves Sea waves passing through slits in Tel Aviv, Israel The Physics of Waves (λ) wavelength λν = velocity amplitude (A) time (t) (x) position one oscillation frequency (ν) = # oscillations per sec We will deal primarily with harmonic waves , / % 2π ( A( x, t ) = A0 cos . 2πν t − ' * x + φ 1 &λ) 0 amplitude Diffraction occurs for all waves, whatever the phenomenon. Lecture 3 phase φ varies from π to - π and gives the relative location of the peaks of two waves 3 Chem 260/261 Monday, January 10, 2011 Interaction of two waves Interaction of two waves + % 2π ( . A( x, t ) = A0 cos - 2πν t − ' * x 0 &λ) / , + % 2π ( . A( x, t ) = A0 cos - 2πν t − ' * x 0 &λ) / , wave amplitude is diminished when two waves are out of phase - 0<|φ|<π wave amplitude increases + . % 2π ( A( x, t ) = A0 cos - 2πν t − ' * x + 0 0 &λ) , / If φ = 0, peaks coincide and waves constructively interfere wave amplitude is abolished when two waves are 180º (π) out of phase & # - 2π * A( x, t ) = Ao cos $2πνt − + (x − π ! ,λ) % " If φ = π, - π, the peaks of one wave coincide with the troughs of the other and waves destructively interfere Can matter be wave- like? The wave- particle duality Electromagnetic radiation has properties of waves (diffraction) Electromagnetic radiation has properties of particles (photoelectric effect) Electromagnetic radiation is neither particles nor waves… at the same time… its BOTH! Can matter be wave- like? v −e r λ= Bohr’s “stone age” quantum theory of the atom suggested just this. nλ = 2π r , n = 1, 2, 3,… λ= e- eee Nickel crystal e +Ze 1927: Clinton Davisson and Lester Germer Diffraction of electrons by a Ni crystal à༎ electrons’ wave behavior à༎ Same diffraction pattern from X ray! QUANTUM PARTICLES HAVE WAVE- LIKE PROPERTIES h h = p mv Consider an x- ray incident on an aluminum bilm What about electrons? λ= o h 6.626×10-34 Js = = 2.4×10-10 m = 2.4A -31 6 mv 9.109×10 kg 2.9979×10 m/s ? Diffraction Pattern Diffraction Pattern – depends on arrangement of atoms. This is used to determine the structure of the crystal. Lecture 3 h h = p mv Electron Gun 4 Chem 260/261 Monday, January 10, 2011 What about electrons? Matter waves o h 6.626×10-34 Js λ= = = 2.4×10-10 m = 2.4A mv 9.109×10-31 kg 2.9979×106 m/s de Broglie 1924 Comparing electron diffraction and x- ray diffraction through a bilm of aluminum. photons (x- rays) electrons Electrons behave like waves – experimentally demonstrated How does one combine both the particle- and wave- like properties of matter in a single description? Einstein’s Theory of Special Relativity: E The de Broglie wavelength: λ h h ⇒λ= = = 0.73 nm p me v Davisson and Germer, 1927 atomic size! When should we care about the duality? Speed Rel. QM Quantum Mechanics c 10- 15 m 1 fm momentum Matter waves have the form: h h = p mv # 2π x & f ( x ) = A sin % $λ( ' Orders of magnitude (or when should we care about the duality?) Electron with velocity 1×106 ms- 1 0.01c = = cp Relativity Classical Physics 10- 8 m 10 nm Size Baseball of mass 0.145 kg with velocity 30 ms- 1 ⇒λ= h h = = 1.5 × 10−34 m = 1.5 × 10−25 nm p mv too small to be observed with any known experiment! Wave/Particle Duality Classical Physics: Macroscopic Objects Ordinary Speeds Waves give the distribution, but often particles are detected one- by- one Simulation of Young’s double- slit experiment, one particle at a time: Chemical Systems: Bond Lengths - 1 Å to 5 Å or 0.1 nm to 0.5 nm Orbital Diameters - 0.5 Å to several Å or 0.05 nm to a nm Nuclear Diameters - ~fm (10- 14 - 10- 15 m) Inherently Quantum Mechanical We cannot separate the particle- wave properties at the quantum level! Double slit experiments – a shock introduction to QM ANIMATION: Check out: “Dr quantum double slit” On youtube States: A.  only (a) open B.  only (b) open C.  both (ab) open. Regimes of (ab) event: 1.  Classical particles – Slow emission, No interaction 2. Classical wave – Diffraction pattern 3. microscopic system (QM) – Short times one by one emission – Random frame NOT wave Long times – Fringe develop - NOT particle è༎  The appearance of interference builds up one quantum particle at a time Each electron makes a single spot on the bilm The sum of many single spots produces an interference pattern identical to that for light For each electron, we know where it can go, but not where it will go. Another diffraction experiment: The Airy Pattern θ = 70( L / d ) Light or electrons passing through a circular aperture is diffracted and forms a pattern of light and dark regions on a screen some distance L away from the aperture. First ring occurs at (Const* L/d) When d is small (small uncertainty in position)… the Airy pattern has a large diameter. è༎  As long as the ‘slit’ is comparable to the size of the quantum object Lecture 3 5 Chem 260/261 Monday, January 10, 2011 Orders of magnitude (or when should we care about ‘small’?) Another diffraction experiment: The Airy Pattern Electron with velocity 1×106 ms- 1 Light or electrons passing through a circular aperture is diffracted and forms a pattern of light and dark regions on a screen some distance L away from the aperture. θ = 70( L / d ) ⇒λ= Baseball of mass 0.145 kg with velocity 30 ms- 1 h h = = 0.73 nm p me v atomic size! ⇒λ= h h = = 1.5 × 10−34 m = 1.5 × 10−25 nm p mv too small to be observed with any known experiment! 7 cm = 0.07 m First ring occurs at (Const* L/d) When d is small (small uncertainty in position)… the Airy pattern has a large diameter. The wavelength of an electron is much larger than its size. If you shoot an electron through a small hole it will diffract. When d is large (great uncertainty in position) … the Airy pattern has a small diameter. Small uncertainty in position – large uncertainty in momentum. Large uncertainty in position – small uncertainty in momentum. The wavelength of a baseball is inbinitesimal compared to its size. If you throw a ball through a hole the diffraction is insignibicant. The uncertainty principle matters when the wavelength is comparable to the certainty in position The Indeterminacy Principle Facing the (quantum) facts Like it or not, all experiments tell us that we can only describe quantum systems probabilistically. Classical physics was a world of matter and ?ields (i.e. particles and waves). Matter was particulate; light was wave- like. quantitative form: 1 ΔxΔp x ≥ 2 Δp Δx position = h 2π … where x and p are along the same axis momentum Prob. Prob. Heisenberg showed that for certain quantities, such as position and momentum, the two probability distributions are not independent. momentum Prob. Prob. position position momentum Birth of the Quantum Theory A “quantum theory” needed three things: 1)  some mathematical quantity that stands for quantum “stuff” 2)  A law that describes how this quantum stuff goes through changes 3)  A rule of correspondence that tells how to translate the theory’s symbols into activities in the world •  Planck reasoned that energies must be quantized in order to make a reliable physical model of blackbody radiation. •  Einstein and Compton showed that light (electromagnetic radiation) was particle- like (photoelectric effect, Compton scattering). •  deBroglie predicted that matter particles have wave properties •  Davisson and Germer later proved it by diffracting electrons. •  Heisenberg noted a strange experimental relationship between pairs of experimental observables… things were interrelated via probability. The wave- particle distinction of classical physics was forever dissolved… more for “?ields” or “matter” … everything was quantum “stuff”. What was this quantum “stuff” and how could we model it? The wave function y = A sin( Bx + C ) + D Wave mechanics: a particle is spread through space like a wave… We can no longer use the classical trajectory precise location of particles at any time trajectory Three (!) quantum theories were put forward around 1925… 1)  matrix mechanics (Werner Heisenberg) •  each matrix represents a different observable (e.g. momentum, position) •  matricies followed a law of motion similar to Newton’s laws (with notable differences!) 2)  wave mechanics (Erwin Schrödinger) •  •  •  represents quantum “stuff” as a wave- form wrote quantum laws of motion (Schrödinger equation) Wave imagery is very dubious… these aren’t classical waves! 3)  transformation theory (Paul Dirac) •  Lecture 3 connects wave and matrix mechanics We must use wave functions describes where the particle is most likely to be found (Ψ2 is the probability) wave function wave function Ψ is the wave function - the modern term for de Broglie’s “matter wave” Wave mechanics concerns itself with calculating and interpreting Ψ 6 ...
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This note was uploaded on 01/19/2012 for the course CHEM 260 taught by Professor Staff during the Fall '08 term at University of Michigan.

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