Expansion Theorem - Advanced Quantum Chemistry and Spectroscopy - Lecture Slides, Slides of Chemistry

Download Expansion Theorem - Advanced Quantum Chemistry and Spectroscopy - Lecture Slides and more Slides Chemistry in PDF only on Docsity! 1 1.5: The Expansion Theorem Any arbitrary function of coordinates and time ),,,( tzyxΩ can be represented exactly in terms of the eigenfunctions { }iφ (in the same coordinate/time space) of a self-adjoint operator. → Ω can be written as a linear combination of the eigenfunctions which are said to form a complete set, {Φi}. ),,,(),,,( 1 tzyxctzyx i i iφ∑ ∞ = =Ω⇒ The {ci} are coefficients to be determined, and {Φi} are orthonormal with real eigenvalues since the generating operator is Hermitian. 3-D vector correspondence Any vector G r can be written as a linear combination of the orthonormal vectors kji ˆ,ˆ,ˆ kGjGiGG zyx ˆˆˆ ++= r kji ˆ,ˆ,ˆ are the basis vectors for G and span the 3-D vector space. { } { }izyxi cGGGkji ≡≡ ,,;ˆ,ˆ,ˆ φ docsity.com 2 The complete set {Φi}, are said to span a Hilbert space. Can determine the expansion coefficients by using orthonormality properties. i i icφ∑=Ω ik i ik c φφφ ** ∑=Ω⇒ kiikkik i ik dcdcd δτφφτφφτφ ===Ω⇒ ∫∫∫ ∑ *** Q Thus: τφ dc kk Ω= ∫ * docsity.com 5 We can expand any well behaved one-dimensional periodic function as a series of sines and cosines! This is known as a Fourier series. p-p ∑ ∞ = ++= 1 )sincos()( n nno nxbnxaaxf We if we use enough sine and cosine functions can represent any periodic well behaved functions, including those below: p-p complete set! docsity.com 6 xcxcxcxcxf 7sin5sin3sinsin)(~ 7531 +++=xcxf sin)(~ 1= xcxcxf 3sinsin)(~ 31 += xcxcxcxf 11sin...3sinsin)( ~ 1131 +++= docsity.com 7 1.6: A first look at transition probabilities When the Hamiltonian is independent of time, the state of the system and its properties are independent of time Φ1, E1 Φ2, E2 Φ3, E3 Φ4, E4 If at t=0 the system is in state Φ1 with energy E1, it will remain there unless perturbed. When the system interacts with a time-dependent perturbation; for example, light )(ˆˆ)(ˆ tVHtH oldnew += Isolated molecule Hamiltonian → stationary state External perturbation (light, particle collisions, etc) Stationary states will be gone: in fact, there will be exchange of population among these levels = transitions= spectroscopy docsity.com