[試題] 100上 陳智泓 量子物理 期中考
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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