課程名稱︰物理化學二 課程性質︰必修 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）︰98/10/24 考試時限（分鐘）：120 是否需發放獎勵金：yes （如未明確表示，則不予發放） 試題 : 1. Calculate the de Broglie wavelengths of the following: (a) (6%) A 10^-6 g particle with velocity 10^-6 m/s (b) (6%) An H2 molecule with energy of (3/2)kT at T=20K (c) (6%) An electron that has been accelerated through a potential difference of 100V 2. What is the commutators for the two operators: ^ ^ (a) (6%) x and Px ^ ^ (b) (6%) y and Px (c) (8%) Show that the function ψ=xe^-a(x^2) is an eigenfunction of the operator (d^2/dx^2)-4(a^2)(x^2). What is the eigenvalue? 3. (a) (12%) For a particle in a one-dimensional box, the ground-state wavefunction is ψ=(2/a)^1/2 sin(πx/a) What is the probability that the particle is in the (I) right-hand half of the box and (II) middle third of the box (a/3~2a/3) ? (b) (8%) For a helium atom in a one-dimensional box (1nm long), calculate the value of the quantum number of the energy level for which the energy is equal to (3/2)kT at 25度C (c) (12%) Calculate <Px> and <Px^2> for a particle in a one-dimensional box (for all n value). (d) (6%) What is the minimum uncertainly in the velocity of an electron if the uncertainly in its position is 100pm? 4. (a) (8%) For the problem of particle-on-a-line, where = 0 for 0≦x≦a V(x) = 無限大 for x＞a = 無限大 for x＜0 h有加一撇 d是偏微分 the Hamiltonian of the system is written as -(h^2)/2m (d^2)/dx^2 A student argues that because the above Hamiltonian operator ^ commutes with the momentum operator Px = -ih d/dx, therefore the wave function ψ=(2/a)^1/2 sin(nπx/a) must also be the eigen ^ function of Px. Do you agree with this argument? Explain your answer in two or three sentences. (b) (8%) Using the bra-ket notation, prove that for a skew-Hermitian operator, its eigenvalues must be zero or purely imaginary. (c) (8%) In our treatment of the harmonic oscillator, we difine two non-Hermitian operators a = (mω/2h)^1/2 (x+ip/mω) a+ = (mω/2h)^1/2 (x-ip/mω) and a Hermitian operator N = (a+)a What are the effects of the operators a and a+ on ∣n ﹥? Hints: You can take the following relations for granted. [a,a+] = 1 [N,a] = -a [N,a+] = a+ N∣n ﹥= n∣n Boltzmann constant k = 1.38 X 10^-23 J/k Mass of electron = 9.1 X 10^-31 Kg Planck constant = 6.626 X 10^-34 Js Velocity of light = 3 X 10^8 m/s --

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