Selection Rules: Selection Rules Each of the spectroscopies have - PowerPoint PPT Presentation

Selection Rules:

Selection Rules Each of the spectroscopies have  associated selection rules. Selection rules originate from the  quantum mechanical description of electromagnetic radiation interaction with matter. Use time-dependent perturbation  theory to derive probability of excitation between two states. Consider a two-level system  Molecular Spectroscopy CEM 484 2

General Properties Wavefunctions are  Normalized  Orthonormal  Wavefunctions are eigenstates of a specific operator  Molecular Spectroscopy CEM 484 3

Iclicker: Eigenstates Which of the following functions are eigenstates of  the hamiltonian, Ĥ o = d/dx A - Ψ = x  B - Ψ = x 2  C - Ψ = e x  D - Ψ = e x^2  Molecular Spectroscopy CEM 484 4

Perturbative Hamiltonian Apply perturbation theory  H = H o + H 1  Ĥ o is independent of time.  Separated time and spatial parts  ψ 1 (r,t) = ψ 1 (r) ψ 1 (t)  Ĥ o ψ 1 = E 1 ψ 1 = ihbar d ψ 1 /dt  Interaction with radiation is represented by perturbative  component Ĥ 1 depends on time and radiation.  Ĥ 1 = - m E = - m E 0 cos2 pn t  Molecular Spectroscopy CEM 484 5

Total wavefunction Total wavefunction a linear combination of eigenstates  ψ = (a 1 ψ 1 + a 2 ψ 2 )  Probability of finding system in state 2 at time t is given  by * (t)a 2 (t) P 1→2 = a 2  Finally get to P 1→2 = |< m > 12 | d (E 2 -E 1 -h n )  Molecular Spectroscopy CEM 484 6

Excitation Probability (1) Evaluate time-dependent Schrodinger equation to  determine excitation probability Molecular Spectroscopy CEM 484 7

Excitation Probability (2) Evaluate time-dependent Schrodinger equation to  determine excitation probability Molecular Spectroscopy CEM 484 8

Excitation Probability (3) Evaluate time-dependent Schrodinger equation to  determine excitation probability Molecular Spectroscopy CEM 484 9

Rotational Selection Rules (1) Selection rule comes from  < m > 12 = ∫ ψ 2 * m z ψ 1 dr  Molecular Spectroscopy CEM 484 10

Rotational Selection Rules (2) Selection rule comes from  Summary  Molecular Spectroscopy CEM 484 11

Harmonic Oscillator Selection Rules (1) Selection rule comes from  < m > 12 = ∫ ψ 2 * m z ψ 1 dr  m z (x) = m 0 + (d m /dx) 0 q + …  Molecular Spectroscopy CEM 484 12

Harmonic Oscillator Selection Rules (2) Selection rule comes from  < m > 12 = ∫ ψ 2 * m z ψ 1 dr  m z (x) = m 0 + (d m /dx) 0 q + …  Summary  Molecular Spectroscopy CEM 484 13