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Overview of QM

Overview of QM. Statics. Translational Motion. Cartesian. P. in Box. M.O. Calculations, Spectroscopy, and Q. Stat. Mech. ex) STM, Devices. Rotational Motion. Spherical Polar. Dynamics. Rigid Rotor. Spin. ex) FTS, NMR. Harmonic Motion. Centre of Mass.

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Overview of QM

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

  2. Quantum Mechanics for Many Particles m1 m3 z1 z3 m4 m2 En – Energy Levels Yn – Wavefuntions Electronic Structure of Mols. z4 z2 (0,0,0)

  3. wo ro The Wavefunction Single Valued Finite and continuous wt Re Im 14_01fig_PChem.jpg

  4. 14_01fig_PChem.jpg The Wavefunction Spherical Polar Coordinates

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

  6. Probability Distribution and Time

  7. Probability Distribution of Wavefunctions Probability of finding a particle in a given interval is independent of time and is determine only by the y(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. wt In most experiments the wavefunctions are incoherent.

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

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

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

  11. 14_01tbl_PChem.jpg

  12. Operator Algebra Linearity Addition Association

  13. Operator Algebra Commutation Commutator Ex) Position and Momentum

  14. Properties of Hermitian Operators For matrices For functions Alternatively

  15. Properties of Hermitian Operators

  16. Properties of Hermitian Operators Orthonormal set Degenerate eigenvalues Not orthogonal

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

  18. The Momentum Operator is Hermitian ? Integration by parts

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

  20. Operators with Simultaneous Eigenfunctions Commute. Order of operations does not matteronlyif A and B commute.

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

  22. Expectation Values Revisited

  23. Expectation Values Revisited

  24. Expectation Values Revisited Consider Repeat k-1 times

  25. Expectation Values Revisited

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

  27. Non Stationary States The states under this new Hamiltonian are 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!!! The act of measurement can cause the system to change state Indeterminacy?? A non-commuting operators can therefore induce the state to change over time. (i.e the state can be influenced externally!!!)

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