課程名稱︰物理化學二
課程性質︰必帶
課程教師︰陳振中
開課學院：理學院
開課系所︰化學系
考試日期（年月日）︰98/01/16
考試時限（分鐘）：120 (有延長30分鐘，所以共150分鐘)
是否需發放獎勵金：yes
（如未明確表示，則不予發放）
試題 :
You may or may not find the following identities useful.
sinαsinβ = 0.5cos(α+β) - 0.5cos(α-β)
cosαcosβ = 0.5cos(α+β) + 0.5cos(α-β)
α+β α-β
cosα+cosβ = 2cos(───)cos(───)
2 2
α+β α-β
cosα-cosβ = -2sin(───)sin(───)
2 2
1. Is the following wavefunction Ψ(sp2) normalized or not? Show your
calculation in details.
Ψ(sp2) = [(1/3)^0.5]ψ(2s) + [(2/3)^0.5]ψ(2pz)
where ψ(2s) and ψ(2pz) are normalized orbitals of H atoms.
2. Derive the de Broglie equation 1/λ = p/h
^ h δ δ
3. Given that Lx = i──(sinψ── + cotθcosψ──),
2π δθ δψ
^ h δ δ ^ h δ
Ly = -i──(cosψ── - cotθsinψ──), and Lz = -i── ──; show that
2π δθ δψ 2π δψ
^ ^ h ^
[Lx,Lz] = -i──Ly based on the above relations.
2π
4. Derive the energy expression for a particle freely moving in a ring by the
de Broglie wquation. (You must solve this problem in a semi-classical way,
starting from the de Broglie equation)
5. Consider the case of particla in a two dimension square box of size equal
to a.
(a) Write down the solutions of Ψj,k(x,y) and Ej,k, where j and k are the
relevant quantum in the x and y dimensions, respectively. Your solutions
must fulfill all the boundary conditions. You don't have to provide any
mathematical proofs.
(b) Draw the patterns of Ψ2,1(x,y), Ψ1,2(x,y), Ψ1,3(x,y), and Ψ3,1(x,y).
Use solid and dashed lines for positive and negative phases, respectively.
(按︰畫成等高線的圖形)
30^0.5
6. Consider the trail function of Ψ(x) = ───*x(a-x). Use the variational
a^2.5
method to obtain an upper bound to the ground-state energy of a particle in
a one-dimensional box of length a.
7. In a simple LCAO-MO treatment of a homonuclear diatomic molecule the two
solutions for the energy are E± =(H11±H12)/(1±S). When the two atoms
are infinitely apart, the total energy of the system will be 2H11, the
energy of the two separated atoms. Using this as the reference state, the
potential energy of a two-electron homonuclear diatomic molecule is given by
V(r) = [(2H11 + 2H12)/(1+S)] - 2H11
(a) Express the potential energy V(r) as a function of S by making use of
Cusachs' approximation for the exchange integral: H12 = S(2-S)(H11+H22)/2
Sketch the graph of V(r)/|H11| against S, keeping in mind that H11 is
negative.
(b) Calculate the value of S at the minimum in the potential. Hence express
V(re) in terms of H11. The equilibrium bond distance is denoted by re.
(c) Koopman's theorem states that the first ionization potential (IP) of a
molecule is the negative value of the energy of the highest filled
molecular orbital. Use this to express IP in terms of H11.
(d) Calculate the energy of the antibonding molecular orbital. Hence obtain
the energy of the first spectrum transition (△) in terms of H11.
(e) For the hydrogen molecule H11 has the value -13.6eV. Calculate the bond
dissociation energy, IP, and △. The experimental values are 4.75, 15.42,
and 11.4eV, respectively.
8. A particle is placed in a two-dimensional triangular box as shown below.
(0.a)
y↑↙
◣
█◣ (a,0)
██◣↙
└─────→x
(a) Write down the boundary conditions for the wave function.
(b) Consider Ψj,k(x,y) in question 5. Explain whether or not Ψ2,2(x,y) is an
acceptable solution in the present case.
(c) Determine the general form of the acceptable wave functions and then write
down one particular solution as an illustration.
[Hints: The solutions can be obtained in terms of Ψj,k(x,y)]
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