課程名稱︰近代物理
課程性質︰電機系複選必修
課程教師︰林清富
開課學院：電資學院
開課系所︰電機系
考試日期（年月日）︰2018/01/11
考試時限（分鐘）：180
試題 :
1/2
Final (two pages)
The following formulas or values may be needed for the problems.
2 1 ∂ ∂f 1 ∂ ∂f 1 ∂^2 f
a.▽ f= ── ──(r^2──)+───── ──(sinθ──)+─────── ───
r^2 ∂r ∂r r^2 sinθ ∂θ ∂θ r^2 sin^2(θ) ∂φ^2
b. wave functions for Hydrogen atom:
1 -r/a_0
─────── e for n = 1
√π a_0^(3/2)
1 -r/2a_0 r
───────── e (2 - ──) for n = 2, l = 0
4√(2π) a_0^(3/2) a_0
1 -r/2a_0 r
───────── e (──) cosθ for n = 2, l = 1, m = 0
4√(2π) a_0^(3/2) a_0
1 -r/2a_0 r ±jφ
──────── e (──) sinθe for n = 2, l = 1, m = ±1
8√π a_0^(3/2) a_0
_2 2 -11
c. Bohr radius a_0 = 4πε_0h / me = 5.292× 10 m.
4 2 3 7 -1
d. Rydberg constant R = me / 8ε_0 ch = 1.097×10 m .
_
eh -5
e. Bohr magneton μ = ── = 5.788×10 eV / Tesla;
B 2m
-23
Boltzmann's constant k = 1.381x10 J/K;
-34
Planck's constant h = 6.63x10 J‧s;
-31 -27
electron mass m = 9.1x10 kg; photon mass m = 1.67x10 kg.
1. Consider a particle in a spherical potential well of finite depth:
V(r) = 0 0＜r＜R
V(r) = K r≧R
where K＞0.
(a) Is the orbital angular momentum quantized in this system? If yes, how is it
quantized? (i.e. L^2 or L=?)
(b) If three electrons are in such a spherical potential well, please assign
the possible quantum numbers to each electron as the system is in the ground
state. (Ignore the Coulomb interaction between electrons, but consider that
electrons are Fermions.) (10)
2. A photon with energy in the visible region (between about 400 and 700 nm)
causes the transition n → n+1 in doubly ionized lithium,
++
Li