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L4 - Biophysical Chemistry Chemistry 24a Winter Term...

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Biophysical Chemistry Chemistry 24a Winter Term 2009-10 Instructor: Sunney I. Chan Lecture 4 January 11, 2010 Interaction of light with matter: Quantum mechanical treatment The time-dependent Schroedinger equation H ψ = ( i ħ ) ( ψ /∂ t) where ħ = h / 2 π and H is the Hamiltonian for the problem that we are trying to solve, and it could be time-dependent. What happens if H is not time-dependent? Then the wavefunction ψ ( r 1 ,r 2 ,. ; t ) could be written as a product of two functions, one that depends on the spatial coordinates of the system and a second function that depends on time. ψ ( r 1 ,r 2 ,. ; t ) = Φ (r 1 , r 2 ,..) T(t) Substituting into time-dependent Schroedinger equation H [ Φ (r 1 , r 2 ,..) T(t) ] =( i ħ ) [ Φ (r 1 , r 2 ,..) T(t) ] / t or T(t) [ H Φ (r 1 , r 2 ,..) ] =( i ħ ) Φ (r 1 , r 2 ,..) [ T(t) / t ]
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Divide by Φ (r 1 , r 2 ,..) T(t) : T(t) [ H Φ (r 1 , r 2 ,..) ] / [ Φ (r 1 , r 2 ,..) T(t) ] = ( i ħ ) Φ (r 1 , r 2 ,..) [ T(t) / t ] / [ Φ (r 1 , r 2 ,..) T(t) ] or [ H Φ (r 1 , r 2 ,..) ] / [ Φ (r 1 , r 2 ,..) ] = ( i ħ ) [ T(t)) / t ] / [ T(t) ] Left-hand side is a function of the spatial coordinates of the system only; and the right-hand side is a function of time only. The equality applies for all r 1 , r 2 , … and all times. So the two sides must be equal to some constant. I pick a real number and call it λ . So we have [ H Φ (r 1 , r 2 ,..) ] / [ Φ (r 1 , r 2 ,..) ] = ( i ħ ) [ T(t)) / t ] / [ T(t) ] = λ We end up with 2 differential equations! (1) [ H Φ (r 1 , r 2 ,..) ] = λ [ Φ (r 1 , r 2 ,..) ] which is just the time-independent Schroedinger equation if we set λ = E n and Φ = Φ n .
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