課程名稱︰量子物理上 課程性質︰物理系大三必修 課程教師︰陳智泓 開課學院：理學院 開課系所︰物理學系 考試日期（年月日）︰2013/01/08 考試時限（分鐘）：120 min 是否需發放獎勵金： 是 因為打字技術的關係，想取用這份的人請注意以下兩點： 1.整份試卷的 h 都是 h-bar 2.角動量量子數 l 長的很像數字 1 ，我會用黃色表示 l 來和數字 1 區別 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~分隔線~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 試題 : Phys309,Final Exam 1/8/2013 1.A particle of mass m is placed in a finite spherical well -V_0 , if r <= a V(r) = { 0 , if r > a Find the ground state (10%) , by solving the radial equation wuth l = 0 . 2 2 2 Show that there is no bound state if (V_0)a < πh / 8m . (10%) 2.A hydrogen atom states out in the following linear combination of the stationary states n=3,l=2,m=1 , and n=3,l=1,m=-1 , and n=2,l=0,m=0 : Ψ(r,0) = ( ψ_321 + ψ_31-1 + ψ_200 ) / √6 (a) (5%) Construct Ψ (r,t) (b) (10%) Find the expectation value (L^2) and (L_z) 3.The electron in a hydrogen atom occupies the combined spin and position state 1 R Y χ 21 1 + Let J ≡ L + S (a) (5%) If measured (J^2) , what values might you get ? (b) (5%) If measured (J_z) , what values might you get ? 4.Construct the matrix S_r representing the component of spin angular momentum ^ along an arbitrary direction r . Use spherical coordintes , for which ^ ^ ^ ^ r = sinθcosφ i + sinθsinφ j + cosθ k Find the eigenvalues (10%) and normalized eigenspinors (10%) of S_r . 5.(10%) Quarks carry spin 1/2 . A quark and its antiparticle bind together to make a meson . Assume the orbital angular momentum in a meson between the quark and the anti-quark is l=2 . Find all possible angular momentum states of the meson . 6.(10%) Two identical spinless fermions of mass m are attached to the ends of a massless rigid rod of length a . The system is free to rotate in three dimensions about the center (but the center point itself is fixed) . Find allowed energies of this rigid rotor . 7.Consider two identical boson of mass m attached by a spring , and the potential energy is given by 1 2 2 V(r) = --- m ω r 2 where r is the relative distance between them . Compute the first excited state , and the second excited state energy (10%) , and the degeneracy for these two states . (10%) --

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