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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