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Lecture 3 Notes 53750.53760

Lecture 3 Notes 53750.53760 - Lecture 3 Notes B A Rowland...

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Lecture 3 Notes B. A. Rowland 53750/53760 Coulomb’s Law Charged particles (protons [+] and electrons [-]) can exert a force between each other known as electrostatic forces . A knowledge of electrostatics is very helpful in chemistry, as one can explain things like dipoles in water, ionic bonds, and ionization energies, amongst others. Remember the expression: opposites attract . This is true in electrostatics, as oppositely charged particles (think one proton and one electron) will exert an attractive force on one another. The converse of the above statements is also true: likes repel . Similarly charged particles (think two protons or two electrons) will exert a repulsive force upon each other. We can write expressions for attractions and repulsions, Attractions (proton/electron) Repulsions (proton/proton and electron/electron) Where r is the distance between the two charged particles in questions. You would write a V(r) expression for each pair of charged particles. Practice Problem: Write expressions for the electrostatic interactions in the lithium atom. The Hamiltonian The Hamiltonian is the operator appearing as H in Schrödinger’s equation. It has the form The first term is a second derivative, and represents the kinetic energy of the wave function. The second term is a potential energy term. This is where we customize the Hamiltonian for each system of interest (for atoms, we will put attractive and repulsive Coulomb terms V(x) , for the Particle in a box, V(x) = 0), thus generating the appropriate Schrödinger equation. Practice Problem : Which (neutral) atom will have 6 electron-electron repulsion terms in V(x) ? Particle in a Box There are three types of motions atoms and molecules may undergo. There are translations (movement through space), rotations , and vibrations . The particle in the box is the model 2 2 ˆ ( ) d H V x dx ψ ψ ψ = + 1 ( ) ~ V r r 1 ( ) ~ V r r -
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for quantum translations—basically, how does an object with wave-particle duality (think electrons) move through space in a confined potential? We will explicitly model the wave through Schrödinger’s equation. The set up for the particle in the box is simple. Basically, a “particle” is placed in a canyon with infinitely high walls (to prevent the particle from leaving the box). The
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