課程名稱︰量子物理上
課程性質︰物理系必帶
課程教師︰高涌泉
開課學院：理學院
開課系所︰物理學系
考試日期（年月日）︰2019/01/10
考試時限（分鐘）：120分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Quantum Physics Final Exam
1.Consider the Hermition matrix M
1 i
M=( -i 1 )
(a)(5%)Find the eigenvalues of M.
(b)(10%)Find a diagonal matrix D and a unitary matrix U such that U^(-1)MU=D
2.(a)(15%)How did Born and Jordan arrive at the canonical commutation relation?
[p,q]=pq-qp=h/(2πi)I ?
(b)(10%)What is the so-called Heisenberg equation of motion for the dynamic-
al variable A in the matrix mechanics? (A'=?)
3.(10%)If a particle of mass m moves with a constant speed v, what is the
associated de Broglie wavelenght and frequency? (The answer should be
consistent with special relativity.)
4.(15%)Let J be the probability current obeying the equation
∂ρ
----- + div(J) = 0
∂t
where ρ≡Ψ^*Ψ is the probability density. What is J if the wave function
Ψ is given by
-(i/hbar)(Et-p．x)
Ψ(x,t)=Ae^
where A is the normalization constant?
5.(a)(10%)Suppose that ψ(p) is the wave function of a 1-dim system in
momentum space at t=0.(ψ(p)≡<p|Ψ(t=0)>) If you are asked to obtain
<x>, the expectation value of x, the position of the particle, at t=0,
what integral should you evaluate?
(b)(10%)If Ψ(x) is the wave function in position space at t=0
(Ψ(x)≡<x|Ψ(t=0)>), and
2α(α)^(1/2)xe^(-αx) ,x>0
Ψ(x)={
0 ,x<0 ,
what is the corresponding wave function ψ(p) in momentum space?
6.(15%)At t=0, a 1-dim simple harmonic oscillator is in a state that is
described by the normalized wave function
Ψ(x,t=0)=(1/5)^(1/2) ψ_0(x)+(1/2)^(1/2) ψ_2(x)+(3/10)^(1/2) ψ_3(x)
where ψ_n(x) is the n-th time-independent eigenfunction for the oscillator
(H ψ_n=E_n ψ_n, E_n=(n+1/2)hbar ω
What is the wave finction at time t? (Ψ(x,t)=?)
7.(10%)Max Born said in his Nobel Lecture that he encountered 2 "dramatic
surprises" with respect to the establishment of quantum mechanics. What are
these 2 dramatic surprises?
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※ 編輯: TunaVentw (140.112.102.148), 01/10/2019 17:01:55