# Aug 29 - Homework set Due September 5 p 311 Problems 7B1...

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Homework set Due September 5 p 311 Problems 7B1, 7B3, 7B5, 7C1, 7C3, 7C5, 7C11

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The Schrodinger Equation 7B.1 Second order differential equation  wavefunction Why is this important? The Schrodinger Equation shows us how to determine the allowed (quantum) energies of particles (e.g., electrons) in nano- scale systems (e.g., atoms and molecules)
This term comes from the kinetic energy of the particle This term comes from the potential energy of the particle The second derivative with respect to position is called the Laplacian Operator E is the total energy of the system Is the wavefunction H-bar is h/ 2 m = mass of the particle

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H = Hamiltonian Operator = Total Energy of the System, T + V Table 7B.1 Notice that we can write the Schrodinger Equation in any coordinate space; not just in one dimension ! In chemistry, it is often more convenient to work with spherical polar coordinates as opposed to linear coordinates Toolkit 7B.1 [more about that later]
What is an Operator ? Operator Algebra An operator is an instruction for manipulating a function in some defined manner The Hamiltonian Operator calculates the total energy of a wavefunction Other operators that you may be familiar with:

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The Schrodinger Equation is an Eigenvalue Equation Take the second derivative of the wavefunction…… …and we get the wavefunction returned, multiplied by a constant ! Textbook, Justification 7B.1  cos kx
The wavefunction contains all of the possible dynamical information about the system it describes NOTE the use of the word " possible " The implication is that it is NOT possible to know everything !!! [More about that later]

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What is the physical interpretation of the wavefunction ? Born Interpretation | | 2 = *  the square modulus of  If the wavefunction has the value of  x at some point x, then the probability of finding the particle between x and x + dx is proportional to | | 2 dx Note: Points at which | | 2 dx = 0 are called nodes ; at such points there is zero probability of locating the particle! Note: The wavefunction itself does not have a physical meaning—it is simply a mathematical function and can be either a real or an imaginary function. It is the square modulus of  that has a physical meaning—it represents the probability density Familiar example: Orbitals represent the probability of locating an electron in a region of space with respect to the nucleus (molecular framework). Most orbitals have nodes.
Normalizing the wavefunction Normally we work with normalized wavefunctions | | 2 represents the probability of finding the particle at a specific position We make assumptions that (1) the particle exists, and (2) at any instant it must be someplace within the system.

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