課程名稱︰近代物理 Modern Physics
課程性質︰大二必修
課程教師︰熊怡 Prof. Yee B. Hsiung
開課學院：理學院
開課系所︰物理系
考試日期（年月日）︰96.06.22
考試時限（分鐘）：210min
是否需發放獎勵金：yes thx
試題 :
1. A gold foil(Z=79) of thickness 3.0μm is used in a Rutherford expirement
to scatter α particles with energy 7.7Mev.
a) What fraction of the α particles will be sacttered at an angles greater
than 90deg.?(5 pts.)
b) What's the ratio of those scattered at 30deg.to those at 5deg.?(5 pts.)
c) Suppose tat 1000 of those particles suffer a deflectoin of more than
25deg., how many are deflected between 75deg. and 90deg,?(5 pts.)
(gold densityρ=19.3g/cm3, atomic mass M=197 g/mole, ke^2=1440eV*nm)
2. De Broglie's relation was dveloped initially for photons, assuming that
they had a small but finite mass. Now assume that RF waves with λ=30m
travelled at a speed of at least 99.5 percent of that of visible light
with λ=500 nm. Beginning with the relativistic expression hf=γmc^2,
verify de Broglie's calculation and find the upper limit of the rest mass
of a photon in units of gram as well as eV/c^2.
(Hint: Find an expression for v/c in terms of hf and mc^2, and then let
mc^2<<hf, note that h=6.626*10^-34Js, hc=1240eV*nm, 1 eV=1.6022*10^-19J)
(15pts.)
3. Under the uncertainty principle, the proton can emit and reabsorb a π+
according to p -> n + π+ where n is a neutron. The π+ has a mass of
139.6 MeV/c^2, Mp=938.27 Mev/c^2, Mn=939.56 Mev/c^2. The reabsorption
must occur within a time Δt consistent with the uncertainty principle.
a)Considering that example by how much ΔE is energy conservation violated?
(ignore kinetic energy)(5 pts.)
b)For how long Δt can the π+ exist?(5 pts.)
c)Assuming that the π+ is moving at nearly the speed of light, how far
from the nucleus could it get in the time Δt ?(5 pts.)
4. Consider a barrier potential
V(x)= 0 for x < 0 (region I) and x < a (region III)
Vo for 0 < x < a (region II)
and a particle with E < Vo and wave number k1 is traveling to the right
from region I.
a) Derive the barrier penetration or tunneling effect which is the
coefficient of transmission T from region I into region III in terms of
αa and E/Vo, where α=√2m(Vo-E)/h_bar.(15 pts.)
b) Find also the approximated expression of T for the case of αa >> 1,
so you can use the expression to estimate the transmission of a beam of
electrons, each with kinetic energy E = 3.0 eV, incident on a potential
barrier with Vo = 6.0 eV and width 1.0 * 10^-10 m. What fraction of the
electrons in the beam will be transmitted through the barrier?(7 pts.)
c)Under what condition the Ramsauer-Townsend effect(100% transmission
and no reflection) would occur for an electron scattering with a noble
gas atom in terms of a simple 1-D potential well model
(V=-Vo in region II)?(8 pts.)
5. Consider a hypothetical hydrogen atom in which the electron is replaced
by K- particle. The K- is a meson with spin 0, hence, no intrinsic
magnetic moment. The mass of K- is 493.7 Mev/c^2. The only magnetic
moment for this atom comes from the orbital angular momentum. If this
atom is placed in a magnetic field with Bz = 3.0 T,
a) what is th effect on the 1s and 2p states and into how many lines
does the 2p -> 1s spectral line split?(7 pts.)
b) What is the fractional separation Δλ/λ between adjacent lines?
(8 pts.)
6. The density of the electron states in a metal can be written as
g(E)=AE^1/2, where A is a constant and E is measured from the bottom of
the conduction band.
a) Show that the total number of states is (2/3)A(Ef)^3/2 and what is the
average energy <E>?(5 pts.)
b) Find the expression for the fraction of the conduction electrons that
are within kT of the Fermi energy and evaluate it for silver(Ef = 5.53 eV)
at T = 300K, and kT = 0.0259 eV.(5 pts.)
c) How much the conduction electrons in the aluminum contribute to the
molar heat capacity in R at ordinary temperature?(5 pts.)
7. Consider a system of N particles which has only two possible energy
states, E1 = 0 and E2 = ε. The distribution function is Fi=Cexp(-Ei/kT)
a) What is C for this case?(5 pts.)
b) Compute the average energy <E> and what is <E> as T -> 0 and also
as T -> ∞.(5 pts.)
c) Derive the expression of heat capacity Cv. and sketch Cv versus T.
(5 pts.)
8. a)Explain the Stern-Gerlach experiment and what is the important discovery?
(5 pts.)
b)At what temperature are the amounts of superfluid helium and normal
liquid helium equal?(Note the lambda point is 2.17K for helium)(5 pts.)
c)Explain the Anomalous Zeeman effect? Give the expression of anamolous
magnetic moment and energy splitting, as well as the Lande g factor.
(10 pts.)
(Note that Bohr magneton μB = e*h_bar/2Me = 9.27 * 10^-24 J/Tesla
= 5.79 * 10^-5 eV/Tesla and Me = 0.511 Mev/c^2.)
Eight problems in total.(140 points)
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