1. Quantum Mechanical Model Systems
Erwin P. Enriquez, Ph. D.
Ateneo de Manila University
CH 47

2. Based on mode of motion Translational motion: Particle in a Box
Infinite potential energy barrier: 1D, 2D, 3D
Finite Potential energy barrier
Free particle
Harmonic Oscillator
Rotational motion

3. Harmonic Oscillator

4. Classical Harmonic Oscillator

5. Quantum Harmonic Oscillator (H.O.)
Schrödinger Equation
Potential energy
v = 0,1, 2, 3, …
SOLUTION: Allowed energy levels

6. Solving the H.O. differential equation
Power series method
Trial solution:
Substituting in H. O. differential equation:
Rearranging and changing summation indices:
Mathematically, this is true for all values of x iff the sum of the coefficients of xn is equal to zero. Thus,
rearranging:
2-TERM RECURSION RELATION FOR COEFFICIENTS: Two arbitrary constants co (even) and c1 (odd)

7. General solution Becomes infinite for very large x as x ∞.
This is resolved by ‘breaking off’ the power series after a finite number of ters, e.g., when n = v Thus, our recursion relation becomes:
When n > v, coefficient is zero (truncated series, zero higher terms)
v = 0,1, 2, … also, QUANTIZED E levels

8. Quantum Harmonic Oscillator
Nomalization constant
Hermite Polynomials generated through recursion formula
v=1
v=2
v=3
v=4
Example:
What is 
SOLUTION

9. General properties of H.O. solutions
Equally spaced E levels
Ground state = Eo = ½ hn (zero-point energy)
The particle ‘tunnels’ through classically forbidden regions
The distribution of the particle approaches the classically predicted average distribution as v becomes large (Bohr correspondence)

10. Molecular vibration Often modeled using simple harmonic oscillator
For a diatomic molecule:
In Cartesian system, the differential equation is non-separable. This can be solved by transforming the coordinate system to the Center-of-Mass coordinate and reduced mass coordinates.

11. Reduced mass-CM coordinate system
Separable differential equation

12. Separation of variables (DE)
Particle of reduced mass 'motion' (just like Harmonic oscillator case)
Center-of-mass motion, just like Translational motion case
The motion of the diatomic molecule was ‘separated’ into translational motion of center of mass, and
Vibrational motion of a hypothetical reduced mass particle.

13. H. O. model for vibration of molecule
E depends on reduced mass, m
Note: particle of reduced mass is only a hypothetical particle describing the vibration of the entire molecule

14. Anharmonicity
Vibrational motion does not follow the parabolic potential especially at high energies.
CORRECTION:
ce is the anharmonicity constant

15. Selection rules in spectroscopy
For excitation of vibrational motions, not all changes in state are ‘allowed’.
It should follow so-called SELECTION RULES
For vibration, change of state must corrspond to Dv= ± 1.
These are the ‘allowed transitions’.
Therefore, for harmonic oscillator:

16. The Rigid Rotor Classical treatment Shrödinger equation Energy
Wavefunctions: Spherical Harmonics
Properties

17. The Rigid Rotor 2D (on a plane) circular motion with fixed radius.
3D: Rotational motion with fixed radius (spherical)
The Rigid Rotor

18. Classical treatment Motion defined in terms of Angular velocity
Linear velocity
Linear frequency
Motion defined in terms of
Angular velocity
Moment of inertia
Angular momentum
Kinetic energy
The Rigid Rotor

19. Quantum mechanical treatment
Shrödinger equation
In spherical coordinate system
Laplacian operator in Spherical Coordinate System
The Rigid Rotor

20. l = azimuthal quantum number Degeneracy = 2l+1
Substituting into Schrodinger equation:
Since R is fixed and by separation of variable:
SOLUTION:
SPHERICAL HARMONICS
(Table 9.2: Silbey)
l = azimuthal quantum number
Degeneracy = 2l+1
The Rigid Rotor

21. Plots of spherical harmonics and the corresponding square functions
From WolframMathWorld (just Google ‘Spherical harmonics’

22. Notes:
E is zero (lowest energy) because, there is maximum uncertainty for first state given by
We do not know where exactly is the particle (anywhere on the surface of the ‘sphere’)
The Rigid Rotor

23. For a two-particle rigid rotor
The two coordinate system can be Center of Mass and Reduced Mass
since radius is fixed, the distance between the two particles R is also fixed
The kinetic energy for rotational motion is:
The result is the same: Spherical Harmonics as wavefunctions (but using reduced mass)
The Rigid Rotor

24. Angular momentum and the Hydrogen Atom

25. Angular Momentum This is a physical observable (for rotational motion)
A vector (just like linear momentum)
Recall: right-hand rule
L2 =L∙ L=scalar
The Rigid Rotor

26. Angular momentum operators
NOTE: SAME AS FOR RIGID ROTOR CASE

27. Angular momentum eigenfunctions
Are the spherical harmonics:
l =0,1,2,…
m=0, ±1,…, ±l
The z-component is also
solved (Lx and Ly are
Uncertain)
REMINDER: SKETCH ON THE BOARD. FIGURE 9.9 and 9.10 SILBEY

28. Angular momentum and rotational kinetic energy
RECALL
RECALL: HCl rotational energies (l is called J)
The spherical harmonics are eigenfunctions of both Hamiltonian and Angular Momemtum Square operators.

29. Hydrogen Atom

30. H-atom: A two-body problem: electron and nucleus
describes translational motion
of entire atom (center of mass motion)
Page 352 Ball. Note Z = 1 for H atom, otherwise it is called a hydrogen-like or hydrogenic atom. Ex: He+, Li2+, etc.
Write the SE for H-atom written in Cartesian Coordinate system.
When the coordinates are converted to CM and reduced mass, the d. e. separates into ______________________?
To be solved to get
the wavefunction
for the electron
Note that the reduced mass is approx. mass of e-. Thus

31. Shrödinger equation for electron in H-atom
SPHERICAL HARMONICS
RADIAL FUNCTIONS (depends on quantum numbers n and l
3. Write the electronic part of the Schrödinger Equation.
4. The electronic part of the Schrödinger Equation is also separable. These are into two general parts, which are ______________. One component is further separable into two: see Ball p 353 just before Equation
Quantized energy (as predicted by Bohr as well), Ryd = Rydberg constant.
E depends only on n
Degenerary = 2n2
E1s = eV

32. 5. See Table 11.4 of Ball. Write the 2p-1 orbital function and compare with 2px.
See Equation 11.67, p. 360 of Ball = write the complete form of 2px.
NOTE THAT 2px is not an eigenfunction of the Lz operator anymore. Show why.

33. Hydrogen atom wavefunctions
Are called atomic orbitals
Technically atomic orbital is a wavefunction = y
Given short-cut names nl:
When l = 0, s orbital
l = 1 p
l = 2, d

34. Plotting the H-atom wavefunction
Probability density =
Radial probability density (r part only)= gives probability density of finding electron at given distances from the nucleus
Probability =
The spherical harmonics squared gives ‘orientational dependence’ of the probability density for the electron:

35. Radial probability density (or radial distribution function)
Bohr radius, ao
Node for 2s orbital
Nodes for 3s orbital
8. See page 363 Ball. What are the quantum numbers corresponding to the diagrams? What is the simple rule for number of radial in terms of quantum numbers?
See also Figure 10.5 Silbey

36. Electron cloud picture2s 1s 3s 2p
DEMO here.
9. What is an electron cloud exactly?

37. Shapes of y (orbitals) NOTE: This is not yet the y2.
10. Are these also electron clouds? Explain your answer.
NOTE:
This is not yet the y2.

38. Shapes of y (orbitals)

39. Properties for Hydrogen-like atom
H, He+, Li2+
Energy depends only on n
Degeneracy: 2n2 degenerate state (including spin)
The energies of states of different l values are split in a magnetic field (Zeeman effect) due to differences in orbital angular momentum
The atom acts like a small magnet:
Solve Ball
Answer Ball (no need to get E in Joules, just write the corresponding expressions).
Review Example p. 365 and answer Ball (no need to get E in Joules, just write the corresponding expressions).
Answer Ball
Answer Ball
Magnetic dipole moment
Magnetogyric ratio of the electron
Orbital angular momentum

40. Electron spin ge = 2.002322, electron g factor
Spin is purely a relativistic quantum phenomenon (no classical counterpart)
Shown by Dirac in 1928 as a relativistic effect and observed by Goudsmit and Uhlenbeck in 1920 to explain the splitting (fine structure) of spectroscopic lines
There is a intrinsic SPIN ANGULAR MOMENTUM, S for the electron which also generates a spin magnetic moment:
The spin state is given by s=1/2 for the electron, and with two possible spin orientations given by ms = +1/2 or -1/2 (spin up or spin down)
Spin has no classical observable counterpart, thus, the operators are postulated, and follows closely that of the angular momentum (Table 10.2 of Silbey)
ge = , electron g factor

41. Pauli Exclusion Principle
The wavefunction of any system of electrons must be antisymmetric with respect to the interchange of any two electrons
The complete wavefunction including spin must be antisymmetric:
In other words, each hydrogen-like state can be multiplied by a spin state of up or down
Thus, “No two electrons can occupy the same state = otherwise, each must have different spins” or “no two electrons in an atom can have the same 4 quantum numbers n, l, ml, ms.”
Spatial part
Spin part

42. More complicated systems…
MANY-ELETRON ATOMS

43. He atom Three-body problem (non-reducible) Not solved exactly!
Use VARIATIONAL THEOREM to find approximate solutions +2 - r2
Kinetic energy of e-s
Electrostatic repulsion between e’s and attraction of each to nucleus

44. Variational Theorem (or principle)
One of the approximation methods in quantum mechanics
States that the expectation value for energy generated for any function is greater than or equal to the ground state energy
Any function f is a “Trial Function” that can be used, and can be parameterized (f = f(a)) wherein a can be adjusted so that the lowest E is obtained.

45. He-atom approximation
As a first approximation, neglect the e-e repulsion part.

46. Applying variational principle
Calculating the ‘expectation value from this trial function’ yields:
2E1s=8(-13.6) eV= eV
Subtracting repulsive energy of two electrons by evaluating:
Total energy is eV versus experimental eV.

47. Parameterization of the trial function
The trial wavefunction may be ‘improved’ by parameterization
For the He atom, the ‘effective nuclear charge’ Z is introduced in the trial wavefunction and its value is adjusted to get the lowest variational energy:

48. Going back to Pauli exclusion… implications…

49. Permutation operator Permutation operator Permutation operator squared
Eigenvalues
f is symmetric function
f is antisymmetric

50. Including spin states SINGLE ELECTRON SYSTEM
TWO- ELECTRON SYSTEM (e.g., He atom:
SEATWORK 1: Which of the functions above are antisymmetric, symmetric?

51. Linear combination of spin functions
SEATWORK TWO: Are these antisymmetric functions?

52. Therefore for the ground state He atom
SPATIAL PART
SPIN PART

53. Fermions and Bosons
Quantum particles of half-integral spins are called FERMIONS
S = ½, 3/2, etc. (two spin states, plus or minus)
-requires antisymmetric functions
-follows Fermi-Dirac statistics
Quantum particles with integral spins are called BOSONS
S = 1, 2, etc.
-requires symmetric functions
-follows Bose-Einstein statistics
SEATWORK 2b: What are electrons? Protons?

54. Slater determinants and STO
Slater in 1929 proposed using determinants of spin functions for the spin part
Atomic wavefunctions that use hydrogenic functions in a Slater determinant are called Slater-type Orbitals (STO)

55. First excited state of He
Triply degenerate because of the spin states
Antisymmetric spatial part
This time, symmetric spin part
MULTIPLICITY = 2S +1
S = total spin angular momenta
TRIPLET M = 3 parallel spins for 2 e’s
SINGLET M =1 opposite spins for 2 e’s
SEATWORK 3: What is that principle called, when the lower energy state consists of parallel spins in separate degenerate orbitals? (No erasure please… on your answer)

56. Hartree-Fock Self-Consistent Field (HF-SCF) Method
Variational method
Trial function for electronic wavefunction:
V is a ‘smeared’ out potential due to all the electrons

57. Spin-Orbit coupling Coupling of the spin angular momentum S
and orbital angular momentum L

58. Atomic units
Short-cut way to write Shrodinger equation is to not include constants… values obtained are generic ‘atomic units’ which can be converted back…
Table 10.7 Silbey
E.g., 1a .u. of length is = 1 Bohr radius = Å
SEATWORK 4: What is the length (in Angstroms) equivalent to 3.5 atomic unit of length?
-13.6 eV is how much in a. u.? (look up in Silbey)