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CHEM 222 – Physical Chemistr y II
Final Examination (3 hours; Close-book examination)
21 May 2011, Saturday, 12:30-15:30; LG1031
NAME: (English) (Chinese) STUDENT ID#
Tips:
(a) Calculator will be useful
(b) There are a total of 15 questions in 10 sheets, plus 4 blank for scratch in the back
(c) For questions asking you to “ write down ” need no derivations
(d) For questions asking you to “ derive ” need detail steps
(e) Useful constants:
Planck constant: h = 6.626×10
−
34
J s ћ
= h /2
π
= 1.05457
×
10
−
34
J s
Speed of light: c = 2.998 × 10
8
m s
Mass of electron: m e
= 9.11 × 10
−
31
-1
kg
(f) Energy in wavenumber (inverse wavelength) : 1 cm
−
1
<==> 1.986 × 10
−
23
J =
1
8065 .
5 eV
Total
190

Part I: Quantum Mechanics
1.
BASIC CONCEPTS
• Quantization and Wave–Particle Duality
– Photoelectric effect and photons
– de Broglie relation
– Energy vs.
wavenumber and frequency vs.
temperature
– Zero–point energy; Tunneling; Interference
– Uncertainty relations
– Quantization of energy and angular momentum
2.
TIME–DEPENDENT/INDEPENDENT SCHR ¨
• Basic properties of Schr¨odinger equations
– Physical picture of wave functions and Born interpretation
– Properties of acceptable wave functions
• Operators for coordinate (ˆ p )
– ˆ p in coordinate representation
– Basic commutator relation
– Relation to the uncertainty principle
• Hamiltonian operator ˆ and Laplacian operator ∇ 2
• Wave functions vs.
wave vectors; Operators vs.
matrices
• Dirac bra-ket notation
3.
BASIC KNOWLEDGE
3.1.
Hermitian operators versus dynamics variables
• Eigenequation of a Hermitian operator
– Property of the resulting eigenvalues and mathematical proof
– Property of the resulting eigenvectors and mathematical proof
– Eigenvalue and eigenvector in relation to measurement
• Uncertainty relation versus commutator
• Implication of [ ˆ
ˆ
] = 0 (in terms of uncertainty principle and the existence of common set of eigenvectors)
• Expectation value h A i = h ψ | ˆ | ψ i h ψ | ψ i
• Variation principle for a test wave function for Schr¨odinger equation
1

3.2.
Fermion versus Boson particles
• What are they referred to? Give one example for each type
• The particle occupation number in an orbital with spin
• Symmetry requirement of the total many–particle wave functions
4.
IMPORTANT SYSTEMS
4.1.
Particle in a box
Solve the SE and memorize the solutions (including normalization)
4.2.
Particle in a ring z
= ~ i
∂
∂φ
, and memorize solutions
4.3.
Vibration motion
• Eigenenergy expression
• Hamiltonian of vibration motion in ( x, p ) and in (ˆ † , ˆ )
• About the creation (ˆ † ) and annihilation (ˆ ) operators
– In relation to coordinate and momentum operators
– Basic relation [ˆ ˆ † ] a † ˆ ˆ † a † ˆ a ], etc.
a † | n i ∝ ?, ˆ | n i ∝ a | 0 i = ?
4.4.
Angular momentum theory
• Basic relations: [ ˆ x
,
= ⇒ [ ˆ 2 , J
ˆ z
J
ˆ y
] = i ~ J
ˆ z
, and cyclic permutations
] = 0, and also the permutations
= ⇒ Implications?
• Important results in eigenvalues and quantum numbers
• Angular momentum theory applied in different cases
– The s, p, d, · · · electron orbitals
– Spin angular momentum
– Molecular rotational energies and degeneracies
– The term symbol 2 S +1 L
J for many-electron atomic systems
2

Part II: Molecular/Atomic Structure and Spectroscopy
5.
ATOMIC SYSTEMS
5.1.
Hydrogen–like atoms and atomic orbitals
• The potential function and the Hamiltonian
• Energy expression:
E n,l
= −
Z n 2
2
R ; n = 1 , 2 , · · ·
What is R called and what is its value?
; l = 0 , 1 , · · · , ( n − 1) .
• Wave function factorization: ψ n,l,m
( r, θ, φ ) = R n,l
( r ) Y l,m
( θ, φ )
– Further factorization: Y l,m
( θ, φ ) = Θ l,m
( θ )Φ m
( φ )
– Normalization expressions for R n,l
( r ), Θ l,m
( θ ), and Φ m
( φ )
• The Schr¨odinger equation for the radial wave function: h
−
2
~ 2 m e
∂
∂r
2
2
+ V eff
( r ) i
Π( r ) = E Π( r ) where Π( r ) ≡ rR ( r )
• What is the V eff
( r ) here? Write it down
• Wave function | n, l, m i versus 2 p x
, 3 d x 2 − y 2
, etc
• Electron spin quantum number s and m s
5.2.
Many–electron atoms
• Pauli principle and the total spin–orbital wave function of multiple electrons: Using a two–electron wave function Ψ(1 , 2) as an example
• Explain in terms of V eff
( r ) the penetration and shielding phenomenons that lead to the energy order of 3 s < 3 p < 3 d
• Term symbol: 2 S +1 L
J
.
– Derive it for one- and two-electron atomic systems
– Identical vs.
non-identical (such as 2 p 2 vs.
2 p 1 3 p 1 ) cases
– Hund’s Rules (for the order of energy)
– Selection Rules (for optical transition)
– Explain the yellow double lines of Na atom emission spectrum
3

6.
MOLECULAR SPECTROSCOPY – ABSORPTION AND EMISSION
• General consideration: Peak position and intensity
• Wavelength region: Microwave, infrared, or visible/UV
• What the information can be obtained
• Optically active or not
• Selection rules
6.1.
Pure rotational spectroscopy
• Rotational line series, average spacing between neighboring lines, and the
B -parameter
• Determine the bond length of diatomic molecule
4
6.2.
Vibrational spectroscopy and ro-vibrational spectroscopy
• Selection rule: ∆ n = ± 1 and ∆ J = ± 1 (may also ∆ J = 0)
• IR absorption spectrum and P , Q , R branches
7.
MOLECULAR SYSTEMS
7.1.
Born–Oppenheimer Approximation
• Physical origin / consideration
• Impact to (i) electron motion; (ii) nuclear motion; (iii) electronic optical transition
7.2.
Electronic spectroscopy
• Vertical transition principle
• Fluorescence (also called emission) versus absorption
• Solvent reorganization (also called solvation) effect and Stokes shift
• Information on vibrational frequencies associated with the electronic absorption or emission

8.
VALENCE–BOND THEORY
• The valence–bond (VB) wave function; bonding and anti-bonding, with including spin parts
• Hybrid orbitals such as sp 2 and sp 3
– Atomic nature
– Orthogonality & normalization; Spatial orientation
5
9.
MOLECULAR ORBITAL THEORY
• Comparison between the MO wave function and the VB wave functions for the σ -bond in A–B molecule
• LCAO–MO: ψ = c
1
φ
1
+ · · · + c n
φ n
• The secular equation ( H − E S ) ψ = 0
– What are the H , S , and ψ here?
– Coulomb integral, resonance integral, and overlap integral
– Relate Coulomb integral α to the AO’s ionization energy
• Hermite relations implied in H and S matrixes
9.1.
Simple MO theory in Diatomic (A–B) Molecules neglecting the overlap integral
(Note: The expressions, λ = p
η 2 + 1 − η and η =
¯
α
A
− α
B
2 β
¯
, will be given)
• Discuss the bonding MO, anti-bonding MO, bonding energy, and the charge in each atom
9.2.
H¨
• Write down the H¨uckel Hamiltonian for a given molecule
• H¨uckel MO results for linear and cyclic conjugate molecules (also for such
H
3 molecules; see HW#5: P11.26 & P11.27/ P14.24 & P14.25
)
– Binding energy, bonding energy, and delocalization energy
– Charge in each atom