課程名稱︰量子力學一
課程性質︰必修
課程教師︰高涌泉
開課學院：理學院
開課系所︰物理所
考試日期（年月日）︰2014.01.07
考試時限（分鐘）：130 min
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
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.
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.252.204