notes04_qm4_time_dependence

notes04_qm4_time_dependence - 4 Time dependence So far, we...

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4 Time dependence So far, we have used the time-independent Schr¨odinger equation to Fnd stationary states. These stationary states do not change with time. We can see this by looking at the time-dependent Schr¨odinger equation H Ψ( r, t )= ~ 2 2 m 2 Ψ( )+ V ( r )Ψ( i ~ ∂t Ψ( ) (4.1) which relates the Hamiltonian operator to the time-derivative of the wavefunction. I’ve used the spatial variable r to indicate that this can be a multidimensional wavefunc- tion. In the case of a stationary state, the wavefunction is an eigenfunction of the Hamiltonian, H Ψ( E Ψ( ) , (4.2) and the solutions are separable in time and space (in the same way that the solutions to a two dimensional box separated in x and y ). Using the same separation of variables technique that we used for the two dimensional box, we can try a solution of the form Ψ( ψ ( r ) T ( t ) , (4.3) where ψ ( r )sat isFes ~ 2 2 m 2 ψ ( r V ( r ) ψ ( r ( r ) (4.4) and the time dependence, T ( t ), satisFes i ~ ∂T ( t ) = ET ( t ) . (4.5) The spatial equation is just the time independent Schr¨ odinger equation, and the time
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This note was uploaded on 08/30/2008 for the course CH 354L taught by Professor Henkelman during the Fall '06 term at University of Texas at Austin.

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notes04_qm4_time_dependence - 4 Time dependence So far, we...

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