[試題] 101上 陳智泓 量子物理上 期末考
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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