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W2 - A Few Derivations from Week 2 Changyi Li The...

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Unformatted text preview: A Few Derivations from Week 2 Changyi Li January 17, 2010 The Analytical Form of Wavefunctions In case the in-line fractions on the lecture slides are a bit too confusing for you, here it is in a bit greater detail. There are two forms of the Schr®dinger's equation, the time-independent equation, H Φ = E Φ (1) and the time-dependent equation, H Φ = i ~ ∂ Φ ∂t (2) In order to solve the time-dependent equation, we assume the wavefunction Φ is separable and can be written as Φ = Ψ( r ) φ ( t ) (3) Substitute this expression into the time-dependent equation and perform separation of variables, we obtain H Ψ Ψ = i ~ ∂φ ∂t φ = C (4) Since this equation holds for all time and all locations, both sides of the equation cannot be equal to an expression involving either time or location. Therefore it has to be a constant. This yields two di erential equations H Ψ = C Ψ (5) which is the time-independent Schr®dinger's equation, and i ~ ∂φ ∂t = Cφ (6) From our previous experience with time-independent Schr®dinger's equation, we know that the con- stant must be the energy associated with the wavefunction, which we will denote as E n , assuming the time-independent part is an eigenfunction of the system. With that, we can solve the time-dependent part φ = exp (- i E n ~ t ) (7) From the Planck's relation E n = ~ ω n (8) we can rewrite the time-dependent part as φ = exp (- iω n t ) (9) Since any linear combination of the eigenfunctions of the equation is a solution to the same equation, we can write the wavefunction, in a general sense, as Φ = X n Ψ n exp (- iω n t ) (10) Eistein Coe cients Spectroscopy is an application of light-matter interaction. In other words, we utilize electric eld and magnetic eld to interrogate the molecules. For the earlier part of the course, we will focus on the interaction between the electric eld and the chemical species, and we start this with the introduction of the oh-so-ubiquitous Beer's Law. For the following derivation, I will be using the bra-ket notation. This notation is widely used in quantum mechanics and the basics of it can be found in most introductory quantum mechanics books, such as the textbook for Ch21, Molecular Quantum Mechanics. Since molecular excitations are temporal processes, we need to use the time-dependent Schr®ndinger's equation. Each peak on the spectrum corresponds to a transition from a lower energy level (A) to a higher energy level (B). Ignore all other energy levels for now and focus on these two....
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W2 - A Few Derivations from Week 2 Changyi Li The...

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