課程名稱︰量子力學二
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）︰2014/04/22
考試時限（分鐘）：120
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.The ratio of the number of molecules in the rotational level l, with energy
E_l, to the number in the l= 0 ground state, with energy E_0, in a sample of
molecules in equilibrium at temperature T is given by
(2l+1) exp(-(E_l-E_0)/kT)
a)Explain how one obtains this ratio, i.e., explain the meanings of the two
factors (2l+1) and exp(-(E_l-E_0)/kT).
b)Determine the l_max which maximizes this ratio by treating l as if it were
continuous.
20%
2.The wave function for a particle is of the form Ψ(r) = (x+y+z)f(r). What are
→
the values that a measurement of L^2 can yield?
20%
3.It is known that there is at least one bound state for the 1-dimensional
finite square well no matter hoe shallow or narrow the well is. Yet, it is
possible that there are no bound states for the 3-dimensional finite
spherical well. Explain why this is so.
20%
4.The spin Hamiltonian of a spin-1 ion in a crystal is given by
a 2 b 2 2
H = ── S_z + ── (S_x - S_y)
hbar hbar
Assume b << a and treat
b 2 2
── (S_x - S_y)
hbar
as a perturbation. Calculate the unperturbated energies and the first-order
corrections using perturbation theory.
20%
5.Explain
a)Aharonov-Bohm effect.
b)Lamb shift.
20%
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