The Quantum Theory of Atoms Schrodinger equation 2009

The Quantum Theory of Atoms Schrodinger equation 2009 - The...

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Unformatted text preview: The Quantum Theory of Atoms and Molecules The Schrdinger equation and how to use wavefunctions Dr Grant Ritchie An equation for matter waves? De Broglie postulated that every particles has an associated wave of wavelength: p h / = Wave nature of matter confirmed by electron diffraction studies etc (see earlier) . If matter has wave-like properties then there must be a mathematical function that is the solution to a differential equation that describes electrons, atoms and molecules. The differential equation is called the Schrdinger equation and its solution is called the wavefunction, . What is the form of the Schrdinger equation ? The classical wave equation 2 2 2 2 2 1 t v x = We have seen previously that the wave equation in 1d is: Where v is the speed of the wave. Can this be used for matter waves in free space? Try a solution: e.g. ) ( ) , ( t kx i e t x - = Not correct! For a free particle we know that E=p 2 /2 m. An alternative. t x = 2 2 ) ( ) , ( t kx i e t x - = t i x m = - 2 2 2 2 Try a modified wave equation of the following type: ( is a constant) Now try same solution as before: e.g. Hence, the equation for matter waves in free space is: For ) ( ) , ( t kx i e t x - = ) , ( ) , ( 2 2 2 t x t x m k = then we have which has the form: (KE) wavefunction = (Total energy) wavefunction The time-dependent Schrdinger equation ) , ( 2 2 t x V m p E + = For a particle in a potential V ( x , t ) then and we have (KE + PE) wavefunction = (Total energy) wavefunction t i t x V x m = + - ) , ( 2 2 2 2 TDSE Points of note: 1. The TDSE is one of the postulates of quantum mechanics. Though the SE cannot be derived, it has been shown to be consistent with all experiments. 2. SE is first order with respect to time ( cf . classical wave equation). 3. SE involves the complex number i and so its solutions are essentially complex . This is different from classical waves where complex numbers are used imply for convenience see later. The Hamiltonian operator = + - = + - H t x V x m t x V x m ) , ( 2 ) , ( 2 2 2 2 2 2 2 T T ) ( 2 ) ( 2 2 2 2 2 x V m p x V x m H x + =...
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This note was uploaded on 04/25/2010 for the course CHEM 112c taught by Professor Dahlquist during the Spring '09 term at UCSB.

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The Quantum Theory of Atoms Schrodinger equation 2009 - The...

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