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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