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. --

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