QM

The Schrödinger equation gives the evolution over time of the wave function in a quantum mechanical system.

i: Imaginary number

h: reduced Planks constant

Psi: Wave function

H: Hamiltonian operator (total Energy of the system)

Fermis Golden rule

Die Gleichung ergibt die theoretische Voraussage für die Übergangsrate (Übergangswahrscheinlichkeit pro Zeit), mit der ein Anfangszustand unter dem Einfluss einer Störung in einen anderen Zustand übergeht.

The Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy.

Transition moment describes the energy between two states n and k.

n: ground state

k: excited state

^H(1)(t): hamiltonian of light

psi: wave function of state n/k

Einstein coefficients are quantities describing the probability of absorption or emission of a photon by an atom or molecule. The Einstein A coefficients are related to the rate of spontaneous emission of light, and the Einstein B coefficients are related to the absorption and stimulated emission of light.

Matter is interacting with a field of electromagnetic radiation. The radiation is absorbed with a transition rate of:

(Bfi: Einstein coefficient, rho: density of states in the respective energy region)

Emission occurs at the same transition rate:

As this does not fulfill a Boltzmann occupation of energy levels, there is additional spontaneous emission:

This is about electronic and vibronic transitions of electrons (y-axis). Electronic transitions happen fast, transitions of the nuclei (R, x-axis)are much slower (only move after the electronic transition is done).

(Which means: A transition from E0 to E1 must take place at a point (R) where both curves are as high as possible, as seen in case of the red arrow. In case of the blue arrow, there is no intensity and therefore no transition).

Formula:

The intensity of an electronic-vibronic transition is roughly proportional to the overlap integral between the two vibronic states.