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課程名稱︰物理化學下
課程性質︰農化系必修
課程教師︰張大釗
開課學院：理學院
開課系所︰化學系
考試日期（年月日）︰2013.4.16
考試時限（分鐘）：110分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. Who gave the major contribution to the following discovery?(8)
____ 1. hydrogen atom spectrum (a)P.Curie (i)J.J.Thomson
____ 2. blackbody radiation (b)A.Einstein (j)L.de Broglie
____ 3. theoretic model of hydrogen atom
(c)Balmer (k)W.Heisenberg
____ 4. duality of light (d)H.Dalton (l)L.Boltzmann
____ 5. x-ray diffraction pattern (e)N.Bohr (m)E.Schrodinger
____ 6. wave equation (f)L.Pauling (n)Davison & Germer
____ 7. uncertainty principle (g)M.Planck (o)Binnig & Rohrer
____ 8. scanning turning microscopy(h)E.Rutherford(p)Stern & Gerlach
2. What do you observe from this hydrogen oine spectrum? What is your
hypothessis for this spectrum? How do you verigy your model?(8)
410nm 486nm
434nm 656nm
┌─┬──┬───┬──────┬─┐
│ │ │ │ │ │
└─┴──┴───┴──────┴─┘
2 2 ikx -ikx
3. Consider the operators d/dx and d /dx . Is Ψ(x)=Ae +Be an
eigenfunction of these operators? If so, what are the eigenvalues?
Do these operators commute?(10)
4. Use the relation △x‧△P≧(h/2π)/2 to derive △t‧△E≧(h/2π)/2
2
based on E =P /2m, △E=(dE/dP)△P, and △x=v△t. Estimate the spectral
-9 -14
width from the decay of a state of lifetime 1.0X10 s and 1.0X10 s
in inverse centemeters. What is important of this finding?(10)
5. A baseball player tries to convince his manager that he cannot hit a
45m/s baseball that has a mass of 140g and relative momentum uncertainty
of 1.00% because of the uncertainty principle. Is his argument valid?(6)
6. Use scientific method (observation, hypothesis, experiments, and importance)
to descrige the Stern-Gerlach experiment.(10)
ikx -ikx
7. An electron moves from a region with V(x)=0 and Ψ(x)=Ae +Be for x<0
iκx
to a region with V(x)=V and Ψ(x)=Ce (V <E) for x≧0, What is the key to
0 0
derive the transmission probability T? If V =1/2E, calculate "T"?
0
●----------------------------------------------------------
electron ↑ ↑
│ E
↑ │ kinetic 2
│ │ │
│Energy E ↓
│ kinetic 1 ┌───────
│ │ │
│ │ │
│ │ │
↓ ↓ │
──────────────────────┘
8. The tunneling for a chemical reaction from compound A to compound B
2 1/2
through a barreir width a depends on exp[-2[2m(V -E)/(h/2π) ] a].
0
Propose a method and use numerical caculation to verify the tunneling
process.(10)
9. Considering a 3-D box of quantum well structure with alternating layers
of GaAs and Al Ga As , the energy can be estimated as
a 1-a
2 2 2 2 2 2
E=h /8m[(n +n )/b +n /a ], where b~1 μm and a ~ 0.1nm. Compare the
x y z
energy differences of (n ,n ,n ) between (1,1,1) and (1.1.2) to
x y z
(1,1,2) and (10,10,2) . How many energy levels between (1,1,2) and
(10,10,2) What is important of this finding?(14)
10. Using the Schrodinger equation to solve the hydrogen atom, three quantum
numbers (n,l,and m ) are obtianed for the construction of period table.
l
2 2 2
-(h/2π) 2m{1/r ∂/∂r[r ∂/∂r[Ψ(r,θ,ψ)]]
2
+1/r sinθ∂/∂θ[sinθ∂/∂θ[Ψ(r,θ,ψ)]
2 2 2 2 2
+1/r sin θ∂/∂ψ [Ψ(r,θ,ψ)]}-e /4πε rΨ(r,θ,ψ)=EΨ(r,θ,ψ)
0
(a)What is the key to solve this equation?
(b)How to isolate the radial part from angular part (expressed gy equation)?
(c)How the obtain the magnetic quantum numger (m )? The angular quantum
I
number (I) can be obtained frm the Legendre equation. Enentually, we can
obtain the differential equation for the radial part,
2 2 2 2 2 2
-(h/2π) /2mr [d/dr[r d/dr[R(r)]]+[I(I+1)(h/2π) 2mr -e /4πε r]R(r)
0
=ER(r),
to obtain the primciple quantum number (n).
(d)What is the physical picture of each term in the last equation?
(e)What are the relations aming n,l, and m ?(20)
I
11. Cy3(R -C=C-C=R )and Cy5(R -C=C-C=C-C=R ), two fluorescence dyes, are
1 2 1 2
widely used as a pair in monitoring the structual change of biological
system via fluorescence resonant energy transfer (FRET). The FRET
6
efficiency (E) is given as E=l/[l+(r/R ) ], where r is the distance
0
between these two dye molecules and R is the Forster distance of this
0
o
pair (assuming 50 A) . Which one acts as an energy donor and which one
acts as an energy acceptor for FRET experiments? Based on the equation,
give a plot to explain how it works for your porposed biological expample.
(12)
12. Using HΨ=EΨ to calculate resonance energy and electron density of two
delocalized π-electrons of O3 molecule (O =O -O ←→ O -O =O ). Assuming
A B C A B C
O3 is a linear molecule. ΣHi and Ψ=ПΦi , where Φi=Σ(Cij)(ΨCij)
and <Ψij |Hi| Ψij'> = α for J=J',=β for J=J'±1 and =0 otherwise. Using
variation method, we can obtain secular equation as
/ \ / \
│ α-E B 0 │ │C1A│
│ B α-E B │ │C1B│ = 0
│ 0 B α-E│ │C1C│
\ / \ /
(a) Calculate the resonance energy.
(b) Derive the wave functions.
(c) Are these wave functions normalized and orthogonal with each other?
(d) Draw energy diagram.
(e) Calculate electron density of the ground state and first excited state
of O and O atoms in O3 molecule.(20)
A B
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