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Overview of QM Translational Motion Rotational Motion Vibrations Cartesian Spherical Polar Centre of Mass Statics Dynamics P. in Box Rigid Rotor Spin Harmonic Motion ex) STM, Devices ex) FTS, NMR ex) IR, Raman Mol. dynamics, Q. Comp., Laser Pulse Methods,2D NMR, and SS NMR, and spectroscopy. M.O. Calculations, Spectroscopy, and Q. Stat. Mech.

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Quantum Mechanics for Many Particles (0,0,0) m1m1 m3m3 m2m2 m4m4 z1z1 z3z3 z4z4 z2z2 E n – Energy Levels n – Wavefuntions Electronic Structure of Mols.

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14_01fig_PChem.jpg The Wavefunction Single ValuedFinite and continuous ImIm ReRe tt oo roro

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14_01fig_PChem.jpg The Wavefunction Spherical Polar Coordinates

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Probability Distribution Since Probability of finding the particle at exactly r, as a function of time. Probability of finding the particle between r i and r j, defining the region R, as a function of time

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Probability Distribution and Time

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Probability Distribution of Wavefunctions Probability of finding a particle in a given interval is independent of time and is determine only by the r Measurements are usually an average over a long time on the quantum mechanical time scale and often reflect an average over a large number of particles. tt In most experiments the wavefunctions are incoherent.

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Normalization of Wavefunctions The probability of finding a particle in all space, S, must be 100 %. Therefore wavefunctions must be normalized. If is a solution to the Schrödinger equation it must be normalized. N is the normalization constant.

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Probability Distributions and Averages Observed Distribution of Measurements Normal Distribution N measurements, x i, with c i repeats, of k possible outcomes. P(x) For continuous variables

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Expectation Values Measurements are averages in time and large number of particles of observables. Every observable has a corresponding operator Expectation values of x.

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14_01tbl_PChem.jpg

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Operator Algebra Linearity Addition Association

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Operator Algebra Commutation Commutator Ex) Position and Momentum

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Properties of Hermitian Operators Alternatively For matrices For functions

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Properties of Hermitian Operators

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Orthonormal set Degenerate eigenvalues Not orthogonal

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Superposition Principle Eigen Relationship Eigen Value Set of Eigenfunctions Any linear combination of eigen functions of degenerate eigenvlaues is an eigenfunction: Consider share the same eigenvalue E n = E m =E

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The Momentum Operator is Hermitian Integration by parts ?

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The Momentum Operator is Hermitian wavefunctions are finite and therefore converge to zero as infinity

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Operators with Simultaneous Eigenfunctions Commute. Order of operations does not matter only if A and B commute.

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Description of a Quantum Mechanical System Energy LevelState Energy levels are independent of time. Eigenfunctions are stationary states. The system stays in the same state, even though the phase of the function is time dependent. Ground State 1 st excited State 0 n Quantum number

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Expectation Values Revisited

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Repeat k-1 times Consider

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Expectation Values Revisited

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Non Stationary States Which means that the observable is time dependent. Consider that an additional interaction is introduced modifying the Hamiltonian: where

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Non Stationary States The Energy Levels become time dependent The state can change quantum number with time under the influence of a non-commuting operator. Non-stationary states!!! A non-commuting operators can therefore induce the state to change over time. (i.e the state can be influenced externally!!!) Indeterminacy?? The states under this new Hamiltonian are The act of measurement can cause the system to change state

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