課程名稱︰量子物理 課程性質︰ 課程教師︰陳智泓 開課學院：理學院 開課系所︰物理系 考試日期（年月日）︰11/17/11 考試時限（分鐘）：120 min 是否需發放獎勵金：yes （如未明確表示，則不予發放） 試題 : Phys309 期中考, 2011年11月17日 1. Let Pab(t) be the probability of finding a particle in the range (a<x<b), at time t. Define J(x,t) so that satisfies the following equation dPab ---- = J(a,t)-J(b,t) dt Compute J(x,t )for the following wavefunction 1 Ψ(x,t)= ---------------------- exp[-(x-vt)^2/(2λ^2)] (π)^(1/4) (λ)^(1/2) (20%) 2. A particle of mass m is in the ground state of the infinite square well located between 0 to a. Suddently the well expands to five times its original size, namely the right wall moving from a to 5a and leaving the wave function undistribute. The energy of the particle is now measured. (a) What is the most probable eigen-energy of the new system? What is the probability of getting that result? (12%) (b) What is the expectation value of the energy? (8%) 3. Suppose a quantum state Ψis described by Ψ=A[Ψ0+Ψ1+Ψ2], where Ψ0,Ψ1,Ψ2 are the ground state, the first excited state, and the se- cond excited state of the following potential ╭ (1/2)m w^2 x^2, for x>0 V(x)=╡ ╰ ∞, for x<0 Find A,<x>, <p>, <p^2/2m>, and <m w^2 x^2 /2>. (30%) (Hint: a+=(2hmw)^(-1/2)*(-ip+mwx) and a-=(2hmw)^(-1/2)*(ip+mwx)) 4. Consider the δ-function potential V(x)=-αδ(x) (a) How many bound states are there in the system? (2%) If potential str- ength α becomes 2α, how many bound states are there in the system? (2%) (b) Suppose that a localized wave package is located at somewhere x<0, de- scribed by: 1 ∞ Ψ(x)=---- ∫ A(k) e^ikx dk √2 0 Consider that it is scattered by the δ-function potential. Write down the wave function of the reflected wave (8%) and the transmitted wave (8%) after scattered for a long time. (simplify yhe mathematical form as best as you can) 5. Consider the "triple square well" potential which is reflection symmet- ric with resect to x=0 shown in Fig.(1). For b>>a, sketch the ground state wave function Ψ1 (5%) and the first excited state Ψ2 (5%) ↑V(x) │ │ b │ b x ────┬───┬─┬─┼─┬─┬───┬───→ │ │ │ │ │ │ │ │ a │ │a │ │ │ a │ └───┘ └─┼─┘ └───┘ │-Vo │ --

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