課程名稱︰量子力學一
課程性質︰必修
課程教師︰高涌泉
開課學院：理學院
開課系所︰物研所
考試日期（年月日）︰2013.01.10
考試時限（分鐘）：110分
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. (a) What is the formula for the hydrogen atom energy levels in terms of
α (the fine structure constant), m_e (the electron mass) and
c (the speed of light)?
(b) What are the selection rules for the (electromagnetic) transitions in
the hydrogen atoms?
→ →
(c) What are the commutation relations [Li,Pj] where L and P are the
angular momentum operator and momentum operator respectively.
→
(d) It is said that the operator L/h is the generator of rotations. Why?
(h 為 h bar)
(e) Consider the problem of a particle of mass m confined to move on a
sphere of radius r_0. What are the possible energies that the particle
can have?
2 2 2
2. It has been proved in the class that <Ψ|(Δx) |Ψ><Ψ|(Δp) |Ψ>≧(h/2) .
Show that the wave function <x|Ψ>=Ψ(x)=N*exp(ip_0 x/h)*exp(-x^2 β^2/h^2)
(N is normalization constant) saturates the above inequality.
(h 為 h bar)
3. Consider a potential energy step as shown in Fig.1 with a beam of particles
incident from the left. Calculate the reflection coefficient as a function
of the incident energy E for the case where the incident energy of the
incident particles is greater than the height of the step.
V(x)
│ ╔═════V_0
│ ║
│ ║
│ ║
│ ║
╪════╝─────→x Fig.1
x=0
4. Consider a special case of the one-dimensional Kronig-Penney model in which
∞
the periodic potential is of the form V(x)= Σ V_0*δ(x-na) where δ(x) is
n=-∞
the Dirac δ function. According to the Bloch's theorem, we can assume that
the wave function Ψ(x) is of the form
Ψ(x)=A*exp(iqx)+B*exp(-iqx)=exp(ikx)*u_k(x) in the range 0<x<a.
The energy spectrum E(k) of the model is determined by an equation of the
sin(qa)
form cos(ka)=cos(qa)+C*---------. What is C?
qa
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