課程名稱︰量子物理
課程性質︰必修
課程教師︰侯維恕
開課學院：理學院
開課系所︰物理學系
考試日期（年月日）︰3/28/2007
考試時限（分鐘）：150mins
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.
(a)Explain the physical origins of [1/(2 m^2 c^2)][Ｌ．Ｓ]<1/r dV/dr> for
atomic systems with single outer electron, and derive the spin-orbit
splitting for alkali atoms with l≠0 and l=0. ,
(b)Describe general LS coupling and derive the Lande interval rule. A LS
multiplet is found with spacing ratio 7:5:3. Deduce its spectroscopy
notation. ,
(c)Give the Zeeman splitting formula and write out the Lande g factor. For j=0
but l,s≠0, show that μ≠0. Explain why there is no Zeeman splitting.
(d)Draw the energy level diagram in spectroscopic notation for S and P state
sodium in a weak magnetic field. Give the allowed transitions, and give the
reason why.
2.A monolayer structure like graphite can be viewed as a 2-dimensional Debye
solid.
(a)It still has 3N0 modes/mole, but N(ν)dν=2πA/v^2 νdν, where A is the
area. Interpret this in terms of phonons.
(b)Find the maximum frequency νm, then derive the "area" specific heat cA in
terms of the Debye temperature Θ.
(c)Find the T>>Θ and T<<Θ behavior of cA and compare with cV of normal
solids.
3.A laser is typically intense, coherent, monochromatic, and beam-like. Give
the reasons behind each of these qualities. What is the photon perspective?
4.
(a)Using N(E)dE = (4πV/h^3)(2m^3)^0.5 E^0.5 dE, find the total number of
particles N for a spinless Bose gas as an expansion in e^-α, then express
N at leading order in e^-α.
(b)Find the mean energy <E> ≡ Etot / N to first order in e^-α, then replace
e^-α by N.
(c)Interpret the correction term in (b) to the classical result as a quantum
effect.
(d)Conduction electrons inside a metal act like a Fermi gas. Use the T->0+
limit to find the relation between Fermi energy Ef and the electron
density.
(e)Ef for Lithium is 4.72eV. Justify the T->0+ approximation in (d), and
explain why the conduction electrons do not contribute to cV at room
temperature.
(f)A neutron star is ~ 4x10^30 kg in mass (twice m⊙), but is only ~ 10km in
radius. Calculate Ef and compare with the gravitational energy at the
surface.
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