1. The Quantum Mechanics of Simple Systems
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.
The observed kinetic energy of a particle is an average of contributions from the entire space covered by the wavefunction. Sharply curved regions contribute a high kinetic energy to the average; slightly curved regions contribute only a small kinetic energy.

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

8. - the Hamiltonian operator
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. An operator that satisfies this condition
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  {cn}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.
These wavefunctions interfere constructively in one place but destructively elsewhere. As more waves are used in the superposition (as given by the numbers attached to the curves), the location becomes more precise at the expense of uncertainty in the particle's momentum.

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 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 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 n2, 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!!
A particle incident on a barrier from the left has an oscillating wavefunction, but inside the barrier there are no oscillations (for EV). If the barrier is not too thick, the wavefunction is nonzero at its opposite face, and so oscillations begin again there. (Only the real component of the wavefunction is shown.) V

26. Tunneling Probability
The probability that the particle will tunnel through the wall
Transmission probability
Tunneling is dependent on the thickness of the barrier and the mass of the particle. It is important for electrons and muons, less so for protons, and not important for heavy particles. Speed of proton transfer reactions – dependent on tunneling.

27. The Particle in a 2D ‘Box’
A two-dimensional square well.
Potential energy is zero between x = 0 and x = L1 and y= 0 and y = L2,
Rises sharply to infinity at the walls.
Note that the particle is confined to the surface.

28. The Schrödinger Equation
The wavefunctions

29. The Energy Expression
The energy of the particle depends directly on the values of nx and ny and ny !

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

31. The Schrödinger Equation
The wavefunctions
Hv- 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
The normalized wavefunctions for the first five states of a harmonic oscillator. Even values of v are black; odd values are green. Note that the number of nodes is equal to v and that alternate wavefunctions are symmetrical or antisymmetrical about y = 0 (zero displacement).

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 |ml| units along the z-axis
Orientation that indicates the direction of motion of the particle.
The direction is given by the right-hand screw rule.

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

39. The Schrödinger Equation

40. The Wavefunctions
The wavefunctions are dependent on the quantum number ml

41. The Momenta and their Operators
The angular momentum operators are written as follows
Eigenvalue - Jz
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
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 l ml
Yl,ml
1/(4)1/2 1
3/(4)1/2 Cos ±1
-+ 3/(8)1/2 Sin e±i

47. The Probability Distributions
A representation of the wavefunctions of a particle on the surface of a sphere.
Note that the number of nodes increases as the value of l increases. All these wavefunctions correspond to ml = 0; a path around the vertical z-axis of the sphere does not cut through any nodes.

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

49. The Stern-Gerlach Experiment
The experimental arrangement for the Stern-Gerlach experiment: the magnet provides an inhomogeneous field.
The classically expected result.
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 ms = +1/2;
 electron (bottom) is an electron with ms = - 1/2.
The vector representing the magnitude of the spin angular momentum lies at an angle of 55 to the z-axis (more precisely, at arccos (1/3 1/2)).