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Chemistry 330 The Quantum Mechanics of Simple Systems.

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Presentation on theme: "Chemistry 330 The Quantum Mechanics of Simple Systems."— Presentation transcript:

1 Chemistry 330 The Quantum Mechanics of Simple Systems

2 Properties of an Acceptable Wavefunction The wavefunction must be Continuous Single-valued No singularities Continuous first derivatives

3 The Curvature of a Wavefunction The average kinetic energy of a particle can be ‘determined’ by noting its average curvature.

4 The Wavefunction The wavefunction  is a probability amplitude Square modulus (  *  or  2 ) is a probability density. The probability of finding a particle in the region dx located at x is proportional to  2 dx.

5 Particle Wavefunctions The wavefunction for a particle at a well-defined location is a sharply spiked function Zero amplitude everywhere except at the particle's position.

6 Postulates of quantum Mechanics There exists a wavefunction that is the solution of the Schrödinger equation  *  - probability density function

7 Postulates #2 The expectation value of any observable is defined as follows - the expectation value of the operator A

8 Postulates #3 The wavefunction must satisfy the relationship - the Hamiltonian operator

9 The Spatial and Temporal Functions Consider the complete wavefunction (  (x,y,z,t)) to be a product of a Spatial function –  (x,y,z) Temporal function – f(t)

10 The Hamiltonian and the energy The eigenvalues for the Hamiltonian operator are the total energy of the system The temporal function describes the variation of the potential energy with time

11 Commutators and Expectations Values Two operators that commute Observables corresponding to those operators can have precise values simultaneously Two operators that don’t commute Observables corresponding to those operators can’t be determined simultaneously

12 Hermitian Operators For an operator An operator that satisfies this condition is said to be Hermitian

13 Superposition and Expectation Values A wavefunction that is written as a linear combination The probability of measuring a particular eigenvalue  {c n } 2

14 Superposition and Wavefunctions The wavefunction for a particle with an ill- defined location Superposition of several wavefunctions of definite wavelength An infinite number of waves is needed to construct the wavefunction of a perfectly localized particle.

15 The General Approach Solve the Schrödinger equation for the physical description of the system Obtain expectation values for the observables Obtain the probability density function at various points in space

16 The Free Particle The particle moves in the absence of an external force x Choose V = 0

17 The Schrödinger Equation The Schrödinger equation The wavefunctions

18 The Particle in a ‘Box’ A particle in a one- dimensional region with impenetrable walls. Potential energy is zero between x = 0 and x = L, Rises sharply to infinity at the walls.

19 The Schrödinger Equation The Schrödinger equation The wavefunctions n – energy level L – length of box

20 The Energy Expression The energy of the particle depends directly on the value of n!

21 The Solutions for the Particle in a ‘Box’ The first five normalized wavefunctions of a particle in a box. Successive functions possess one more half wave and a shorter wavelength.

22 The Particle in a ‘Box’ The allowed energy levels for a particle in a box. Note that the energy levels increase as n 2, and that their separation increases as the quantum number increases.

23 The Probability Distributions The first two wavefunctions and the corresponding probability distributions The probability distribution in terms of the darkness of shading.

24 Orthogonal wavefunctions A graphical illustration of orthogonality for two wavefunctions The integral is equal to the total area beneath the graph of the product, and is zero.

25 Tunnelling Suppose that the energy at the walls dos not rise abruptly to infinity!! V

26 Tunneling Probability The probability that the particle will tunnel through the wall Transmission probability

27 The Particle in a 2D ‘Box’ A two-dimensional square well. Potential energy is zero between x = 0 and x = L 1 and y= 0 and y = L 2, Rises sharply to infinity at the walls.

28 The Schrödinger Equation The Schrödinger equation The wavefunctions

29 The Energy Expression The energy of the particle depends directly on the values of n x and n y and n y !

30 The Quantum Mechanical Harmonic Oscillator Examine a particle undergoing harmonic motion.

31 The Schrödinger Equation The Schrödinger equation The wavefunctions H v - Hermite Polynomials v – vibrational quantum number

32 The Energy Expression The energy of the oscillator depends v – vibrational quantum number The force constant – k Particle mass – m

33 The First Wavefunction of the QM Oscillator The normalized wavefunction and probability distribution (shown also by shading) for the lowest energy state of a harmonic oscillator.

34 Harmonic Oscillator The normalized wavefunction and probability distribution (shown also by shading) for the first excited state of a harmonic oscillator.

35 The Wavefunctions The first five normalized wavefunctions of the QM harmonic oscillator

36 The Probability Distributions The probability distributions for the first five states of a harmonic oscillator Note – regions of highest probability move towards the turning points of the classical motion as v increases.

37 Angular Momentum of A Particle Confined to a Plane Represented by a vector of length |m l | units along the z- axis Orientation that indicates the direction of motion of the particle.

38 Quantization of Rotation Examine a particle undergoing rotation in a plane

39 The Schrödinger Equation The Schrödinger equation

40 The Wavefunctions The wavefunctions are dependent on the quantum number m l

41 The Momenta and their Operators The angular momentum operators are written as follows Eigenvalue - J z Eigenvalue – J 2

42 Wavefunctions of the Particle on a Ring The real parts of the wavefunctions of a particle on a ring. For shorter wavelengths, the magnitude of the angular momentum around the z-axis grows in steps of ħ.

43 3-D Rotation Suppose we allow the particle to move on the surface of a sphere. Two angles  - the azimuthal angle  - the colatitude

44 The Schrödinger Equation The Schrödinger equation  2 – the Laplacian Operator

45 The Wavefunctions The wavefunctions are dependent on the angles  and . The Schrödinger equation is simplified by the separation of variables technique.

46 The Solutions The solutions to the SE for this systems are the spherical harmonics lmlml Y l,ml 00 1/(4  ) 1/2 10 3/(4  ) 1/2 Cos  1±1± /(8  ) 1/2 Sin  e ±i 

47 The Probability Distributions A representation of the wavefunctions of a particle on the surface of a sphere.

48 Space Quantization Represent the vectors for the angular momenta as a series of cones!

49 The Stern-Gerlach Experiment a) The experimental arrangement for the Stern-Gerlach experiment: the magnet provides an inhomogeneous field. b) The classically expected result. c) The observed outcome using silver atoms.

50 The Spin Functions of Electrons An electron spin (s = 1 / 2 ) can take only two orientations with respect to a specified axis.  electron (top) - electron with m s = + 1 / 2 ;  electron (bottom) is an electron with m s = - 1 / 2.


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