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Physics 249 Final Exam
Dec 21st 2012
Constants
Proton mass: 1.6726x10-27 kg or 938.27 MeV/c2
Electron mass: 9.1094x10-31 kg or 0.51100 MeV/c2
Electron charge: e=1.60x10-19 C
ke = 8.99x109 Nm2/C2
𝑘𝑒 𝑒 2 = 1.44𝑒𝑉𝑛𝑚
ℏ = 1.0546x10-34 J*s, 6.5821x10-16 eV*s
c = 3.00x108 m/s
ℎ𝑐 = 1240𝑒𝑉𝑛𝑚
ℏc = 197.3eVnm
The mass energy of 1 atomic mass unit, u is 931.5 MeV
Integrals
1) a) What is the minimum energy(in eV) and wavelength(in nm) of a photon needed to ionize the n=3 orbit
of He+.
b) If you use a photon of wavelength 100nm to ionize the electron in this system what is the stopping
potential (in V) for the ionized electron.
(15 points)
2) A gold foil is used in a Rutherford experiment to scatter alpha particles with energies 10 MeV. What
fraction of the particles are backscattered to angles within 5 degrees of the incoming direction.
For the foil. Z=79, thickness = 1um, n = 5.90e28 atoms/m 3
(15 points)
3) a) For the Harmonic oscillator wave functions n=0, 1, 2 prove for all cases the equation.
+∞
∫
𝜓𝑛∗ 𝑥𝜓𝑚 = 0
−∞
unless 𝑛 = 𝑚 ± 1
b) Interpret what this result means for transitions between different energy states of the harmonic oscillator
𝜓0 = 𝐴0 𝑒 −𝑦
2 /2
𝜓1 = 𝐴1 𝑦𝑒 −𝑦
2 /2
𝜓2 = 𝐴2 (2𝑦 2 − 1)𝑒 −𝑦
𝑦 = √𝛼𝑥,
(25 points)
𝛼=
ℏ𝜔
2
2 /2
4) Use the Heisenberg uncertainty principle to estimate the size of the hydrogen atom. Calculate the
expectation value of r to determine the average radius of the ground state and compare this to the estimate
from the uncertainty principle.
1 2 −𝑟/𝑎
0
𝑒
4𝜋 √𝑎03
(20 points)
𝜓=√
19
5) Consider 10
𝑁𝑒 and 199𝐹
a) What orbitals are filled for the protons and neutrons in these two nuclei? List how many neutrons or
protons are in each energy level. Use the notation that lists energy levels by primary quantum number,
orbital shell type and total angular momentum, rather than shell number. (Hint – all of the levels that are
filled have n=1)
b) Which is the unstable isotope and why.
c) Write down the decay mechanism that converts the unstable to the stable isotope.
d) Calculate the nuclear radius for these isotopes given
For these isotopes
∆𝑈 = (19.001880 − 18.998404)𝑢 ∗ (
931.5𝑀𝑒𝑉
𝑢
)
(15 points)
6) Annihilating electrons and positrons to photons followed by pair production has been one of High
Energy physics basic tools for producing and studying new particles.
Write Feynman diagram for the following process
a) One diagram for 𝑒 + 𝑒 − → 𝜇 + 𝜇 −
In some cases more diagrams are possible and you have to add the amplitudes of both interactions.
b) Write two diagrams for 𝑒 + 𝑒 − → 𝑒 + 𝑒 −
c) Give one example of a particle antiparticle pair that cannot be produced this way.
(10 points)