Last Build Date: Fri, 10 May 2013 19:01:48 PDTCopyright: c. 2005 Michelle M Francl
Mon, 13 Nov 2006 19:25:54 PSTWhy does MRI require high magnetic fields? Why is it such a low energy technique compared to X-ray?
Mon, 13 Nov 2006 19:25:54 PSTA Mathematica exercise to review the finer points of orthonormality. We explore these concepts by comparing the behaviors of Slater type orbital basis functions and Gaussian basis functions (the latter are widely used in quantum calculations of molecular wavefunctions).
Mon, 13 Nov 2006 19:25:55 PSTIn which we say good-bye...and consider how a laser "amplifies" light.
Mon, 13 Nov 2006 19:25:54 PSTPopulation inversion is a key feature of a system which be used to construct a laser. A system in thermal equilibrium follows Boltzmann's statistics, in which the number of molecules in higher energy states is smaller than the number in the lowest energy state. Lasers require that you have a non-equilibrium situation established, in which more molecules are "stuck" in an excited state than are currently in a lower energy state. This phenomenon is called population inversion. A second feature of lasers is that the emission process(the release of a photon when a molecule or atom relaxes from an excited state to a lower energy state) can be stimulated, or enhanced by the emissions from other molecules. This is where the "se" in the name comes from! (LASER = Light Amplification by Stimulated Emission of Radiation).
Mon, 13 Nov 2006 19:25:54 PSTWe wrap up NMR and begin to consider the quantum mechanics behind lasers. Lasers are magic wands for chemists, making it possible to explore what happens in chemical processes on very short time scales. Lasers are ubiquitous tools in everyday life, too. Grocery store scanners and CD players use lasers to read information, an when you "burn" a CD, a laser is used to literally score the material.
Mon, 13 Nov 2006 19:25:54 PSTCould you build an NMR that could fit in your pocket? The effect of magnetic field on the splitting between nuclear spin states. What would happen if you walked through a very strong magnetic field? Say a million Tesla field? Are there such fields? We propose building a pocket-sized NMR from a cow magnet. It could be done, if you're not interested in very high resolution.
Mon, 13 Nov 2006 19:25:54 PSTThe quantum mechanics of nuclear spins. How a magnetic field splits degenerate spin states of at nuclei, setting the stage for NMR.
Mon, 13 Nov 2006 19:25:53 PSTThe vibrational spectra of most molecules is very complex. We considered how additional lines arise in diatomic spectra including isotopic substitution and "hot bands". There are many more vibrational modes available to polyatomic molecules. How many? 3N-6!
Mon, 13 Nov 2006 19:25:53 PSTThe rotational spectra of polyatomic molecules depend on the moments on inertia about the principal axes. We considered 4 cases: linear molecules, spherical tops, oblate symmetric tops and prolate symmetric tops.
Mon, 13 Nov 2006 19:25:53 PSTWe noted in our demonstration on Friday that rotation affected vibration. We quantified this, including a term in the energy to account for centrifugal distortion. The effect is small, but noticeable, as we saw with HCl. We consider the appearance of overtones in the vibrational spectrum, and the shifts in equilibrium bond length that occur as a result of the anharmonicity of the vibrational potential.
Mon, 13 Nov 2006 19:25:53 PSTWhy are there all those lines in the HCl spectrum? Why is there no line at the fundamental frequency? We consider the interplay of rotation and vibration and their respective selection rules to see why.
Mon, 13 Nov 2006 19:25:53 PSTUsing the harmonic oscillator to model vibrational energy transitions can be done, but has its limits. Consider the observed high resolution spectrum of gaseous HCl.
Mon, 13 Nov 2006 19:25:53 PSTWhy does your white shirt glow under a blacklight? What makes the glow in the dark stars glow? How does a glow stick work? We look at the absorbtion and emission of light by molecules. This is an appropriate lecture for Halloween since the first meaning of "spectrum" is "ghost".
Mon, 13 Nov 2006 19:25:52 PSTElectron spin is generally viewed as an ad hoc development in wave mechanics (though it arises naturally in other forumations, such as Dirac's). Using a general statement of the Pauli Exclusion Principle, we showed that Slater's suggestion of using wavefunctions constructed from determinants would insure that the Pauli's principle was satisfied.
Mon, 13 Nov 2006 19:25:52 PSTLinear combinations of functions are a good way to build wavefunctions. The goal is to have "off the shelf" sets of functions that we can use to build wavefunctions for molecules. We show how linear variation theory can be used to find the variational energy of the ground AND excited states and how to find the coefficients for the linear expansion of the wavefunction.
Mon, 13 Nov 2006 19:25:52 PSTDifferent trial functions yield different energies, the quality of the energy doesn't necessarily predict the quality of other properites predicted from the wavefunction (such as average position). We looked at the framework for linear variation theory, since this is the backbone of one of the standard methods for computational molecular quantum chemistry.
Mon, 13 Nov 2006 19:25:52 PSTA Mathematica exercise based on the one-dimensional particle in the box explores the variational principle. Does a function with a lower energy necessarily do better at predicting other quantities, such as the average position of the particle within the box?
Mon, 13 Nov 2006 19:25:52 PSTThough the Schrodinger equation cannot be solved exactly, robust approximate techniques exist for finding solutions to problems of interest to chemist. The variational theorem is the foundation for much of computational chemistry. Using a Mathematica notebook we explore how a simple gaussian function can be used to find an approximation to the wavefunction and the energy.
Mon, 13 Nov 2006 19:25:52 PSTTime to decide on a class t-shirt! Send your ideas to email@example.com and I'll post them up here. Comments and suggestions?
Mon, 13 Nov 2006 19:25:51 PST(image)
Mon, 13 Nov 2006 19:25:51 PSTWe look at several common ideas about atomic orbitals:
Mon, 13 Nov 2006 19:25:51 PSTWe rewrote the Hamiltonian for a one-electron atom in terms of the operator L2. Knowing that linear operators that commute share at least one set of eigenfunctions, we tested to see if the Hamiltonian and L2 did commute. They do, and so there must exist a common set of eigenfunctions. Since we already know one set of eigenfunctions for angular momentum operator, the the spherical harmonics or Yl,m, we tried a solution to the one-electron atom Schrodinger equation of the form R(r)Yl,m(θ,φ). Such solutions do work and allow us to derive a differential equation in a single variable, r, to solve for the radial part of the wavefunction.
Mon, 13 Nov 2006 19:25:51 PSTAs a step along the path to creating a quantum mechanical model of an atom, we considered the solution to the problem of a single particle moving on the surface of a sphere. We saw that the Hamiltonian for the motion could be written simply in terms of the angular momentum operator, L2. The eigenfunctions of this operator are well known and called the spherical harmonics or Yl,m. We noted that the solutions depended on two quantum numbers, l and ml.
Mon, 13 Nov 2006 19:25:51 PSTNOTE: The answer in the back of the book is computed assuming that the fundamental line is at 2559 cm-1, not the 2630 cm-1 the authors give. With thanks to Jennifer Gerfen who noticed this! The actual value is 2648.97 cm-1, as reported in the NIST database, so the authors' value is closer than the 2559 value Jennifer found in other texts.
Mon, 13 Nov 2006 19:25:51 PSTWe consider one more model problem, this one concerning the rigid rotation of a diatomic molecule. Though the problem is simple compared to most molecular systems chemists are interested in, it yielded our first example of a wavefunction that was complex and not real valued. It will also provide a basis for an atomic model problem - the hydrogen atom.