課程名稱︰物理化學二 課程性質︰系必修 課程教師︰陳逸聰 開課學院：理學院 開課系所︰化學系 考試日期（年月日）︰2012/4/22 考試時限（分鐘）：120 (open book) 是否需發放獎勵金：yes （如未明確表示，則不予發放） 試題 : 1.Consider the transformation from Cartesian coordinates to plane polar coordinates where 2 2 0.5 -1 x=rcosθ y=rsinθ r=(x +y ) θ=tan (y/x) (1) If a function f(r,θ) depends upon the polar coordinates r and θ, then the chain rule of partial differentiation says that (df/dx) = (df/dr) (dr/dx) +(df/dθ) (dθ/dx) (2) y │ y θ y r y │ P(x,y)or │ ／ P(r,θ) (df/dy) = (df/dr) (dr/dy) +(df/dθ) (dθ/dy) (3) │ ／ x θ x r x │ ／ │／ └───────── x For simplicity, we will assume r is constant so that we can ignore terms involving derivatives with respect to r. In other words, we will consider a particle that is constrained to move on the circumference of a circle. This system is sometimes called "a particle on a ring." (a) Using equation 1 and 2, show that (df/dx) = (-sinθ/r)(df/dθ) and (df/dy) = (cosθ/r)(df/dθ) y r x r (r fixed) (b) Now apply equation 2 again to show that 2 2 2 2 2 2 2 (d f/dx ) =(sinθcosθ/r )(df/dθ) +(sin θ/r )(d f/dθ ) (r fixed) y r r Similarly, show that 2 2 2 2 2 2 2 (d f/dy ) =(sinθcosθ/r )(df/dθ) +(cos θ/r )(d f/dθ ) (r fixed) x r r and that 2 2 2 2 2 2 2 2 ▽ f=( d f/dx ) +(d f/dy) → (1/r )(d f/dθ ) (r fixed) r (c) Now show that the Schordinger equation for a particle of mass m constrained to move on a circle of radius r is 2 2 -(h/4πI)(d Ψ(θ)/dθ ) = EΨ(θ) 0≦θ≦2π 2 where I = mr is the moment of inertia 2.As derived above, the Schordinger equation for a particle on a ring is 2 2 -(h/4πI)(d Ψ(θ)/dθ ) = EΨ(θ) 0≦θ≦2π Solve the eigenfunctions and eigenvalues for this system 3.For a particle in a one-dimensional box of a length a, show that <p> = 0 for all quantum states. 4.Many proteins contain metal porphyrin molecules. The general structure of the porphyrin molecule is drawn on the right hand side, This molecule is planar and so we can approximate the π electrons as being confined inside a square. (porphyrin圖省略ˊˇˋ) (a) What are the energy levels and degeneracies of a particle in a square of side a? (b) The porphyrin molecule has 26 π electrons. If we approximate the length of molecule by 1000pm, then what is the predicted lowest energy absorption of the porphyrin molecule (1/cm) ? 5. 12 16 C O is an example of a stiff diatomic molecule, and it has a vibration frequency of 2170 (1/cm). (a) What is the value of the force constant k? (b) What is the value of the standard deviation Δx in the internuclear distance? (c) What is the standard deviation Δpx of the momentum of the vibrational motion? (d) Check that the product ΔxΔpx yields h/4π in accordance with the Heisenberg uncertainty principle. 6. 12 16 The C O molecule has an equilibrium bond distance of 112.8pm. Calculate (a)the reduced mass and (b)the moment of inertia. (c)Calculate the wavelength of the photon emitted when the molecule makes the transition from l=1 to l=0 using equation 9.144 for the energy levels. 7.In a hydrogen like atom in the 1s state, there is a difference between the average distance (<r>) between the electron and the nucleus and the most probable distance (r mp) between the electron and the nucleus. (a)Derive the expression for the average distance <r> between the electron and the nucleus (b)Derive the expression for the most probable distance r mp 8. m Using explicit expressions for Y (θ,φ), l 1 2 0 2 -1 2 (a)Show that │Y (θ,φ)│ + │Y (θ,φ)│ + │Y (θ,φ)│ = constant 1 1 1 l m 2 This is a special case of the general theorem Σ│Y (θ,φ)│ = constant -l l known as Unsold's theorem. (b)What is the physical significance of this result? 9.(a)Calculate the probability that a hydrogen 1s electron will be found within a distance 2a from the nucleus. 0 (b)Calculate the radius of the sphere that encloses a 90% probability of finding a hydrogen 1s electron 10.In chapter 10, we learned that if Ψ1 and Ψ2 are solution of Schordinger equation that have the same energy En, then c1Ψ1+c2Ψ2 is also a solution. Let Ψ1=Ψ210 and Ψ2=Ψ211 (see table 10.1) 2 2 (a) What is the energy corresponding to Ψ=c1Ψ1+c2Ψ2 where c1 +c2 =1 (b) What does this result tell you about the uniqueness of the three p orbitals, px, py, and pz 11.The spin function α and β cannot be expressed in terms of spherical harmonics, but they can be expressed as column matrices: ┌ 1 ┐ ┌ 0 ┐ α=│ │ and β=│ │ └ 0 ┘ └ 1 ┘ The spin operator can be represented by the following Pauli matrix: ^ 1 ┌ 1 0 ┐ ^ ^ Sz = ─│ │ Show that Szα=0.5α and Szβ= -0.5β 2 └ 0 -1 ┘ --

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