# Vibrational Motion of Molecules. SpectroscopicPhenomena Hamiltonian PhysicalModel EigenstatesEigenvalues Dynamics Energy Construction Correspondence Testing.

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Vibrational Motion of Molecules

SpectroscopicPhenomena Hamiltonian PhysicalModel EigenstatesEigenvalues Dynamics Energy Construction Correspondence Testing Solution Model Construction in Quantum Mechanics

An Ultrafast Spectroscopy While illuminating an iodine molecule by a pair of extremely short light pulses (t~100fsec), some of the molecules undergo ionization. The measured ion quantity depends in a periodical modulation in the time gap between the pulses.

Iodine molecules absorb light in the visible range of the spectrum. For long wavelengths the absorption is discrete. The shorter the wavelength the smaller the distance between the absorbing lines. Below a threshold wavelength the absorption becomes continuous and the molecule dissociates into separate atoms. Absorption in the Visible Range

In first approximation let us consider a vibrating molecular system as a one-dimensional harmonious oscillator: The Model and the Hamiltonian Construction The spatial orientation is static (approximation) 1. 2. The system consists of a single particle with a reduced mass, rotating around the center of mass (accurate): 3. The restoring force is linearly proportional to the displacement (approximation):

Eigenstates of the Hamiltonian 1.Fulfill the schrodinger formula:.. 2.Fulfill the boundary conditions: The eigenfunctions are solutions of a differential equation of the second order:

For a coherent superposition of the eigenstates the observable values of the location and the momentum vary in correspondence to the classical equation of motion Dynamics of Wavepackets

In quantum mechanics, due to the uncertainty principle there are additional aspects to the kinetic energy (in addition to kinetic energy being an index for particle motion) Expanding the Concept of Kinetic Energy 1.Kinetic energy of volume: 2.Negative Kinetic energy - the tunneling effect

Expanding the Concept of Kinetic Energy 1.The existence of discrete absorption frequencies 1.accumulation of the energy levels 3.Modulation of the periodical signal The model describes the following phenomena: The model does not describe the following phenomena: The model’s validity range: Describes well small vibrations around the center of mass (low vibration levels) 2.Periodicity in the measurement of dynamic quantities 2.Dissociation of a molecule by radiation with frequency above the threshold frequency

SpectroscopicPhenomena Hamiltonian PhysicalModel EigenstatesEigenvalues Dynamics Energy ImprovementCorrespondence Prediction Solution Improving the Model and New Predictions

The Anharmonic Oscillator The weakening of the bond must be considered: the spring constant decreases as the bond id stretched, eventually causing the bond to break. The model: 1.The distance between the energy levels decreases 2.There is a transition to the continuum of the dissociation New Results:

The Revival Phenomenon After a long period of time, during which the vibrations are modulated and decrease, the observable signal is revived Prediction:

A Mathematical Appendix Operating with the Energy Operators on the Vibrational Basis States

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