1. Consider a particle in the one-dimensional region O ::S x ::S a. The wave function is given by
'tjJ(x) = N x (a-x)
where N is a constant to be determined. Outside the region, the wave function is zero.
(a) Find N.
(b) Where is the most probable location for the particle?
(c) Calculate the expectation value of the position of the particle.
,., The wave function for a simple harmonic oscillator with n = 2 and energy (5/2)nooc is
(a) Verify that this wave function satisfies Schrodinger's equation.
(b) Show that
Recall the useful integrals in Appendix B of Taylor and Zafiratos, but note that In= \dln_2/d/\.\.
(The absolute value sign is missing in the text version.)
4. Recall the one-dimensional semi-infinite box problem we treated in class:
.th R /2m(U~ - E) d /2m£
WI /J= 1/ _ an a="\;-
\ 11- ~ t?
By solving Schrodinger's equation in the regions O < x < a and x ~ a and invoking continuity of
both the wave function and its first derivative at x= a, we derived the following equation relating
a, f3 and a:
.Btan (a a)= -a (1)
Consider the bound state problem, with E < Uo and Uo = 9Eo, where Eo =1i23t2/(2ma2
) , the
ground state energy for the perfectly rigid box. The allowed energy levels for the semi-rigid
box are En = x/Eo.
By solving eqn. (1), find all the allowed bound state energies; i.e., find all the allowed Xn's.
6. (a) Use your knowledoe of th l . f h S hr'"'-1 · .
. e e o uuon o t e c vumger equation for the h drol!en atom to
est1mate the quantum number n for the Moon orbiting the Earth. Ignore ;edu~d mass effects
and the Moon's rotation on it axis.
(b) _Estimate the quantum numbers / and m for the Moon. (The Moon orbit the :.arth essentially
10 the Earth's equatorial plane. so the lunar angular momentum vector is perpendicular to the
plane of its orbit.)
(mass of Earth = 5.98 x 1024 kg: mass of Moon = 7.35 x 1011 k~: mean distance from Earth to
Moon (centerto center) = 3.84 x 108 m; G = 6.67 x 10·11 N-m /kg2
9. Consider the effect of the rotation of the Sun on the wavelen~s that would be observed in a
spectroscopy experiment. The radius of the Sun is 6.96 x 10 km, and its period of rotation at the
equator is 25.4 days. For the 486.l run line of the hydrogen spectrum (measured in the lab on
the earth). what are the wavelengths that would be emitted from opposite edges of the Sun's
equator because of the SW1 's rotation?
11. , -#1 _,-_:/\ 8_ _
A particle of rest mass Mo is at rest in the laboratory when it decays into three identical
particles, each of rest mass 111o. Two of the particles (labeled #1 and #2) have velocities as
shown in the figure.
( a) Calculate the direction and speed of particle #3.
(b) Find the ratio MJ 111o.
(c) Compare your results with the classical (non-relativistic) solution.
2.9 • Muons are subatomic particles that are produced
· several miles above the earth's surface as a result of
collisions of cosmic rays ( charged particles, such as
protons, that enter the earth's atmosphere from
space) with atoms in the atmosphere. These muons
rain down more-or-less uniformly on the ground, although
some of them decay on the way since the
muon i~ unstable with a proper half-life of about
1. 5 μs. l 1 μs = 1 o-6 s.) In a certain experiment a
muon detector is carried in a balloon to an altitude of
2000 m, and in the course of 1 hour it registers 650
muons traveling at 0.99ctoward the earth. If an identical
detector remains at sea level, how many muons
would you expect it to register in 1 hour? (Re~ember
that after n half-lives the number of muons surviving
from an initial sample of N0 is N0 /2n.)
( ,~. • A traveler in a rocket of length 2d sets up a coordiV
nate system S' with origin 0' anchored at the exact
middle of the rocket and the x' axis along the rocket's
length. At t' = 0 she ignites a flashbulb at 0'.
(a) Write down the coordinates x~, t~ and x~, t~ for
the arrival of the light at the front and back of the
rocket. (b) Now consider the same experiment as observed
in a frame S relative to which the rocket is
traveling at speed v (with Sand S' arranged in the
standard· configuration). Use the Lorentz transformation
to find the coordinates Xp, tp and xB, tB of the
arrival of the two signals. Explain clearly why the two
times are not equal in frame S, although they were in
S'. (This illustrates how two events that are simultaneous
in S' are not necessarily simultaneous in S.)
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