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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 find 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( r, t ) + V ( r )Ψ( r, t ) = i ∂t Ψ( r, t ) (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( r, t ) = E Ψ( r, t ) , (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( r, t ) = ψ ( r ) T ( t ) , (4.3) where ψ ( r ) satisfies 2 2 m 2 ψ ( r ) + V ( r ) ψ ( r ) = E ψ ( r ) (4.4) and the time dependence, T ( t ), satisfies i ∂T ( t ) ∂t = E T ( t ) . (4.5) The spatial equation is just the time independent Schr¨

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notes04_qm4_time_dependence - 4 Time dependence So far we...

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