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Unformatted text preview: 4/27/2004 H133 Spring 2004 1 Chapter 10 We want to complete our discussion of quantum mechanics this week by considering the Schr dinger Equation. Mathematical equation which tells us how to solve for the energy eignenfunctions of a quantum system if we know the potential energy, V. This is only appropriate for nonrelativistic quantum mechanics. We will only consider the time independent Schr dinger Equation. This implies that the potential energy function, V, must be timeindependent. It can be generalized to include time dependencethat will need to wait for a future class. We will also only consider the Schr dinger Equation in one dimension. The potential energy function, V(x). It can be generalized to three dimensionsbut that will have to wait for a future class. We will develop/motivate the Schr dinger Equation by generalizing the de Broglie Relation. Once we have the Schr dinger Equation we can then begin to look at its properties and predictionsthat will be mostly a topic for Chapter 11. p h = 4/27/2004 H133 Spring 2004 2 Generalized de Broglie Relation The de Broglie relation which started as a hypothesis seemed to stand up to experimental testing e.g. Diffraction with monoenergetic electrons. However as it was first presented, the de Broglie relation: was only for a free particle . We saw that this wavelength could be matched to the wavelength of the wave function that encoded the probability of finding the particle in a particular location. We have seen that as a particle moves through a potential, V(x), that is changing with position, as long as the particle is in a classically allowed region, the wave function still had the general shape of a oscillating wavebut the form was not a simple sine wave. Consider the function below. As the particle moves to the right, it slows downso the wavelength should become longer (at least qualitatively). p h = ) ( x V x E m p V E K 2 2 = = 4/27/2004 H133 Spring 2004 3 Generalize de Broglie Relation For a particle at the fixed energy E, we have This implies as the particle moves to the right and EV gets smaller, the momentum gets smaller. If we somehow want to hold onto the de Broglie relation even for quanta that are not free. The wavelength must be getting larger as E V gets smaller (i.e. smaller p). The wave function, which is an energy eigenfunction, must look something like the following: However, at this point we are faced with a problem....
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This note was uploaded on 06/03/2011 for the course H 133 taught by Professor Furnstahl during the Spring '11 term at Ohio State.
 Spring '11
 Furnstahl
 Energy, Potential Energy, Quantum Physics

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