Overview of QM Translational Motion Rotational Motion Vibrations Cartesian Spherical Polar Centre of Mass Statics Dynamics P. in Box Rigid Rotor Angular Mom. &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.

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.

14_01fig_PChem.jpg Properties of the Wavefunction Single ValuedFinite and continuous ImIm ReRe tt oo roro Complex Valued

14_01fig_PChem.jpg The Wavefunction in 3D Spherical Polar Coordinates Cartesian Coordinates

Probability Distribution Recall 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

Probability Distribution and Time Probability is independent of time!

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. tt In most experiments the wavefunctions are incoherent.

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.

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

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

14_01tbl_PChem.jpg

Operator Algebra Linearity Addition Association Analogy

Commutation Commutator Order matters !!! Operators Commute Operators Do Not Commute Translation then rotation or rotation then translation Commutation

Ex) Position and Momentum

Properties of Hermitian Operators RHS For matricesFor functions LHS

Properties of Hermitian Operators Consider two eigenfunctions of A with different eigenvalues: If A is Hermetian then: RHS LHS

Properties of Hermitian Operators Orthonormal set Degenerate eigenvalues Not orthogonal

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

The Momentum Operator is Hermitian Integration by parts ?

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

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

Description of a Quantum Mechanical System Energy Level State 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

Expectation Values Revisited

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Expectation Values Revisited Repeat k-1 times

Expectation Values Revisited

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

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 operator 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