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The Quantum Theory of Atoms Schrodinger equation 2009

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

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The Quantum Theory of Atoms and Molecules The Schrödinger equation and how to use wavefunctions Dr Grant Ritchie

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An equation for matter waves? De Broglie postulated that every particles has an associated wave of wavelength: p h / = λ 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 Schrödinger equation and its solution is called the wavefunction, Ψ . What is the form of the Schrödinger equation ?
The classical wave equation 2 2 2 2 2 1 t v x Ψ = Ψ We have seen previously that the wave equation in 1–d 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 ϖ - = Ψ Not correct! For a free particle we know that E=p 2 /2 m.

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An alternative…. t x Ψ = Ψ α 2 2 ) ( ) , ( t kx i e t x ϖ - = Ψ t i x m Ψ = Ψ - 2 2 2 2 Try a modified wave equation of the following type: ( α 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 ϖ - = Ψ ) , ( ) , ( 2 2 2 t x t x m k Ψ = Ψ ϖ then we have which has the form: (KE) × wavefunction = (Total energy) × wavefunction
The time-dependent Schrödinger 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 Ψ = Ψ + Ψ - ) , ( 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.

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The Hamiltonian operator Ψ = Ψ + - = Ψ + Ψ - H t x V x m t x V x m ˆ ) , ( 2 ) , ( 2 2 2 2 2 2 2 T T ) ( ˆ 2 ˆ ) ( 2 ˆ 2 2 2 2 x V m p x V x m H x + = + - = x i p x - = ˆ LHS of TDSE can be written as: where Ĥ is called the
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The Quantum Theory of Atoms Schrodinger equation 2009 - The...

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