Chemistry 4531 Spring 2009 QUANTUM MECHANICS 1890's I. CLASSICAL MECHANICS General Equations LaGrange Hamilton

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1 Chemistry 4531 Spring 2009 QUANTUM MECHANICS 1890's I. CLASSICAL MECHANICS General Equations LaGrange Hamilton Light: II. ELECTRICITY & MAGNETISM Maxwell's Equations III. THERMODYNAMICS Gibbs Helmholz -- under control IV. OPTICS Diffraction Geometrical Optics All of these developments were considered exceedingly rigorous, elegant and complete. Predictions of all of these theories had been tested extensively on macroscopic scale objects. NO VIOLATIONS OF ANY HAD BEEN FOUND! Central theses of the time: No real conceptual issues remain unresolved. Computations on real systems were unbelievably hard, however! 1

2 Blackbody Radiation Spectrom eter Tem perature T Intensity T 1, T 2,... λ THEORY Rayleigh and Jeans derived a famous result for the spectral density of black-body radiation using rigorous classical statistical mechanics and rigorous electricity and magnetism theories. It is: 8π kt ρ ( T, λ) dλ 4 λ =. How does it compare with experiment? Temperature T 2 The Ultraviolet Catastrophe"

3 Max Planck "fixed it up" by trial and error, in the process incorporating a "non-physical" assumption: Photons can behave like particles under some conditions, and have a relationship between their frequency and energy given by E = h ν = h c/λ, where h = 6.6 x erg-s, later known as Planck's constant. Substituting this hypothesis into the Rayleigh - Jeans theory, we obtain ρ( T, λ) 8π hc 1 = 5 hc λ kt e λ 1 It fitted the experimental data!!! But h is an ad hoc quantity. No one took this concept seriously until it was shown to do something else: The Photoelectric Effect ν, (λ) φ ν φ 3

4 ν There now appear to be conditions where waves can act like particles. Is the converse true?? If particles have waves associated with their motion, what is the wavelength, λ? m v p = mv PARTICLE E = mv 2 /2 = m 2 v 2 /2m = p 2 /2m PHOTON m =? v = c E = mc 2 = pc also E = hν = hc/λ so we conclude λ = h/p where p is the linear momentum MATTER WAVES (?) λ D = h/p for PARTICLES! (debroglie) 4

5 Proof of Existence of Matter Waves X-ray Diffraction d λ x ray d Au X-ray diffraction used known wave properties of x-rays and the Bragg diffraction law to measure the distance d in a single crystal sample of gold. With d now known, we can scatter electrons off of the same surface, and compare classical expectations with those we would deduce if there was such a thing as a matter wave. Electron Diffraction e- me V λ d Au There is some sort of wave-particle duality. It is manifest only when λ D is the order of a dimension of the object under study (Just as with light, where diffraction is important only when light interacts with objects with a dimension ~λ). How large is λ D for a macroscopic object? 20 g ball moving at 10 cm sec -1 h = 6.6 x g cm 2 sec -1 λ D = 5

6 ATOMIC LINE EMISSION from discharges suggested that atoms possess some discrete (i.e., quantized) energy states and that light is emitted in transitions between these states. This suggests that stationary (stable) states of matter are characterized by standing debroglie waves - the essential origin of quantization. e.g., the Bohr atom 2πr = nλ D, When r is the radius of the circular electron orbit, and n = 1, 2, 3,..., then there is constructive interference. However, IF nλ D 2πr, (see right hand illustration) then the waves eventually cancel out or spread out (unstable). 4λ D = 2πr 4λ D 2πr 6

7 SUMMARY 1. Matter waves exist and manifest themselves when λ D is near the size of the interacting objects. 2. Boundary conditions and standing waves relate to quantized states. 3. All of this suggests that we might (and Schrödinger did) try to modify the wave equations and apply it to matter waves. This process will require some background in complex numbers, complex functions, linear differential equations, linear operators, and operator mathematics. For more complex problems, matrix manipulations and machine computation are absolutely essential. We will assign some problems for which a computational mathematics program (Mathematica or Mathcad) will be of great value. We will use Mathcad in class, and it will prove to be of real value both on problem sets and in other courses. We will also have some small groups in the class carry out some real computations (>100,000 integrals evaluated!) using a commercial quantum chemical code, Gaussian 98W. As noted in the prospectus, the University has a site license for the PC version of Gaussian 98W, so we can make a CD available for you to install it on your computer for the duration of your time at CU. Before seriously beginning, let's get a feeling for what's to come by an arm-waving solution of the energy levels for the Particle in a Box. 7

8 The One Dimensional Particle in a Box l L M v The particle of mass M is confined to move in one dimension over a distance L. It is shown moving with a velocity v to the right. If we impose a standing wave condition for the matter waves associated with M, then the standing wave condition for the box becomes nλ L = 2 with n = 1,2,3,... Substituting the debroglie condition gives nh L = with n = 1,2,3, 2p rewriting, nh p = 2L Using the relation that E = p 2 /2M, we obtain E n p n h = = with n = 1, 2, 3, M 8ML Quantized Energies!!! The exact result! Thus we obtain the following energy level picture. 8

9 Example of Application - Butadiene H H H C C C C H H H The four π electrons are relatively free to move along the carbon chain. The sigma bonds make up the molecular framework. If we know the length of the molecule (we do: call it L), we can treat the π electrons as though they were constrained in a box of length L. Calculate the energy levels of an electron (mass m e ) in a box of length L. Add in the Pauli Principle, allowing no more than two electrons per level. Then the following picture emerges. The first optical absorption corresponds to E 3 - E 2. E3 - E2, the lowest abs energy The agreement with experiment is better than 15%!! Pretty amazing for a model with such simplicity! 9