作者Derver (木律)
看板NTU-Exam
标题[试题] 102-1 高涌泉 量子力学一 期末考
时间Tue Jan 7 16:57:36 2014
课程名称︰量子力学一
课程性质︰必修
课程教师︰高涌泉
开课学院:理学院
开课系所︰物理所
考试日期(年月日)︰2014.01.07
考试时限(分钟):130 min
是否需发放奖励金:是
(如未明确表示,则不予发放)
试题 :
1.
a) A spin-1/2 particle in the state │+z〉 goes through a Stern-Gerlach device
^ ^ ^
having orientation n = cosθ z - sinθ x. What is the probability of
finding the outgoing particle in the state │+n〉?
b)Express the total-spin S = 0 state of two spin-1/2 particles
^ ^ ^ ^ ^ ^
│0,0〉= 1/√2 │+z, -z〉-1/√2 │-z, +z〉 in terms of the states │+n, -n〉
^ ^ ^ ^
and │-n, +n〉 where│+n〉and │-n〉 are defined with respect to the
^
direction n in (a).
2.Prove that the 1-dimensional wave function for which ΔxΔp=h_bar/2 must be
a Gaussian.
3.Calculate the reflection coefficient R and the transmission coefficient T for
scattering from the potential energy barrier
2m λ
──── V(x) = ─ δ(x)
h_bar^2 b
where δ(x) is the dirac delta function.
2
4. p 1 2 2
H = ── + ── m ω x is the Hamiltonian for the 1-dimensional simple
2m 2
harmonic oscillator, i.e., a│α〉= α│α〉 where a is the lowering operator
mω i
a = √(────) (x + ── p) . Show that │Ψ(t)〉 is also an eigenstate of
2h_bar mω
a, i.e., show that a│Ψ(t)〉= λ│Ψ(t)〉 and find λ.
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