Download Physics 8.04 MIT September 19, 1373 Exercises

Physics 8.04 MIT
September 19, 1373
Exercises for Chapter 2
Due: September 26
PUT ON YOUR PAPER THE HOUR AND INSTRUCTOR'S NAME FOR YOUR SECTION
10. Resolving power of an electron microscope
A fundamental result of physical optics is that no optical instrument
can resolve the structural details of an object that is smaller than
the wavelength of the light by which it is being observed. A similar
analysis applies to the deBroglie wavelength of electrons in an
electron microscope.
It is desired to study a virus of diameter 0.02 micron (= 200 8).
This is.impossible with an .optical microscope (wavelength about
5000 8) but can be done with an electron microscope. Calculate
the voltage through which electrons must be accelerated to give
them a deBroglie wavelength 1000 times smaller than the linear
dimension of the virus, so as to permit formation of a very good
image.
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11. The domains of special relativity and of quantum mechanics
It is generally accepted that the domain of experience in which
non-relativistic mechanics is valid is that in which particles
move with speed v such that v/c « 1 in the frame of observation.
(That this rule has exceptions is witnessed by the fact that^
magnetic effects are essentially relativistic , but may be produced
by electric charges moving at 1 mm/sec in a wire!)' In a similar
manner we can set as a criterion for the validity of non-quantum
mechanics that the deBroglie wavelength of particles in the
system under consideration be much less than some characteristic
dimension of the system:A/r « 1 , where X is the deBroglie
wavelength and r is some characteristic dimension. Use these
criteria to determine which of the following systems are likely
to require relativity and/or quantum mechanics for their description.
(a) Electron in the atom. Calculate the speed of an electron in
the lowest (n = 1) Bohr orbit of hydrogen. What is the ratio of
this speed to the speed of light? Are we justified in using a
non-relativistic analysis of the hydrogen atom? Carry out a similar
analysis for an electron in the inner orbit of mercury (Z = 80).
Use the deBroglie relation to calculate X/r for the electron.in
the lowest energy state of hydrogen. Does your result verify
that we need to use quantum physics to describe the behavior of
electrons in atoms?
Assume that
(b) Proton in the nucleus.
12 one of the protons in a
T--V
cm ) executes an orbit
large nucleus (nuclear radius ££ 10
inside the nucleus with 'kinetic energy 10 MeV. Is special relativity
required to describe this motion? (Compare kinetic energy with
rest energy.)
Calculate A/r for the orbiting proton. Do we need to use quantum
physics to describe this nuclear system?
Physics 8.04 Exercises Chapter 2 page 2
(exercise 11., continued)
(c) Electrons in an accelerator. The Cambridge Electron
Accelerator (CEA) on Oxford Street at Harvard University
accelerates electrons along an approximately circular path
to a terminal kinetic energy of 6 GeV (6 x 10 eV). Does the
analysis of the electron orbits require the use of relativistic
mechanics? The radius of the orbit is 30 meters. What is the
ratio A/r for the electrons at terminal energy? Would you
expect that the engineers who designed the magnets whose field
holds -the electrons in orbit found it necessary to use quantum
physics in their calculations?
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12. The Bohr atom derived from deBroglie's relation
Here is another development of Bohr's results for hydrogen, based
directly on the deBfoglie relation. If a deBroglie wavelength
can be associated with an electron in orbit, then it seems
reasonable to suppose that the circumference of an orbit be equal
to an integral number of wavelengths. Otherwise (one might argue)
the electron would interfere destructively with itself. Leaving
to one side the mongrel-like nature of this argument (which
employs both words like "orbit 11 and words like "wavelength"),
apply it to rederive Bohr's results for hydrogen using the
following outline or some other method.
(a) Calculate the speed and hence the magnitude of the momentum
of an electron in a circular orbit in a hydrogen. (Use the answer
to lla.)
(b) Use the deBroglie relation to convert momentum to wavelength.
(c) Demand that the circumference of the circular orbit be equal
to an integral number of deBroglie wavelengths.
(d) Solve for the radii of permitted orbits and calculate the
permitted energies. Check that they conform to Bohr's results,
as verified by experiments.
(e) As an alternative method, omit all mention of forces and simply
use the deBroglie relation and the condition of integer number of
wavelengths in a circumference to show directly that the angular
momentum rp.(for the circular case) equals an integer times \\/2ir.
This is just the condition that we showed earlier (lecture of 9/17)
leads most simply to the Bohr results. (See also text, page 1-25.)
.
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13. Single-slit diffraction
A sufficiently wide single slit illuminated by light or other wave
casts on a distant screen a "geometrical image" of width equal to
that of the slit* However, close examination reveals that the edges
of this image are fuzzy, and if the slit is made narrow and narrower,
the image ultimately becomes wider than the width of the slit.
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Physics 8.04 Exercises Chapter 2 page 3
i
*
(exercise 13, continued)
Moreover, if the slit is illuminated with monochromatic light
a series of light and dark fringes appear on either side of the
central image, as shown below. This phenomenon is called diffraction
and results from the wave nature of the disturbance being propagated.
A simple analysis accounts for the observed intensity pattern
(see, for example, Vibrations and Waves in this series, page 288).
Here we develop only one result of that analysis the width of
the central image on a distant screen and apply it to diffraction
of light (part b below) and diffraction of deBroglie waves (next
exercise).
The fundamental technique is to divide the slit analytically into
sub-slits, treat each sub-slit as a line source in phase with all
the others, and add the effects of these sub-slits at a distant point
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The left-hand figure shows this subdivision for a propagation
direction in which we expect to observe the zero intensity nearest
to the central maximum. This direction is such that the sub-slit
I 1 just below the center of the opening is a distance X/2 more
distant from the screen than the sub-slit 1 just below the top of
the opening. If this condition is satisfied, the wave from I 1 will
cancel the wave from 1 at the screen. Similarly the wave from 2 1
will cancel the wave from 2, and so forth. As a result, no net
propagation takes place in the direction indicated by 9 in the
figure. This analysis assumes*that the observing screen is a distance
D away much greater than the slit width d (official name:
Fraunhofer diffraction). *
.
(a) Complete the analysis to show that the first zero of. intensity
occurs at an angle 9 from the forward direction given by
'''
sin
=
X/d
(slit)
and that the assumptions in the problem lead to the approximation
Physics 8.04 Exercises Chapter 2 page 4
(exercise 13 continued)
sin 0
23
0 £J
tan 9
so that the central maximum in the diffraction pattern has total
width W given by
t
W £
2 A D/d
(slit)
!
.
x
For a circular aperture the analysis is more complicated but leads
to results that differ only slightly in numerical value
sin 0 = 1.22 A /d
(circular hole)
W 2
2.44 A D/d
where d is the diameter of the hole.
. '
.
(b) Pinhole camera design To see, how diffraction effects enter
design problems, think about how to obtain the sharpest image with
a pinhole camera a camera with a pinhole for a lens. The pinhole
images each point on the object as a small circular disk on the
film. If the hole is too large, the disk-image will be geometrical
and too large. On the other hand, if the hole is too small,
diffraction effects will make the disk-image large in this case
also. For what pinhole size will, the image have the sharpest
features? Assume the pinhole-to-film qistance is 10 centimeters
and visible light has wavelength 5000 Angstroms « Define some
clearly-stated approximate measure of the disk-image size, for
example (geometric image diameter d) plus (diffraction-disk
width W) . Minimize your expression to yield a numerical pinhole
diameter for greatest image sharpness. Is the result larger
or smaller than you expected? Could you fabricate a pinhole of
this size, say with a pin?
14. Slit width in an atomic beam apparatus
In an atomic beam apparatus, potassium metal (atomic weight 39) is
heated to its boiling point (T = 760° C ^ 1000° Kelvin) and
streams out of the oven aperture as individual atoms (at average
energy 53 kT ££ 1/10 eV, where k is Boltzmann f s constant =
-16
1.38 x 10
erg/degree Kelvin and T is degrees Kelvin) into a
vacuum chamber. The experimenter wishes to make the beam narrower
and masks it down with a slit made of mounted razorblades.
The detector is located 50 centimeters downstream from the slit.
Is there any danger than the slit will be so narrow that
dif f araction effects make the beam significantly wider at the
detector than the geometrical image of the slit? Justify your
answer with an approximate numerical calculation. Use the
results of exercise 13.
t..
Physics 8.04 Exercises Chapter 2 page 5
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15 . Hydrogen; a Structure in Equilibrium
A simple but sophisticated argument (due to Victor Weisskopf)
holds that the hydrogen atom has its observed size because this
size minimuzes the total energy of the system. The argument rests
on the assumption that the lowest energy state corresponds to
physical size comparable to a deBroglie wavelength of the electron.
Larger size means larger deBroglie wavelength/ hence smaller momentum
and kinetic energy. In contrast/ smaller size means lower potential
energy/ since the potential well is deepest near the proton. The
observed size is a compromise between kinetic and potential energies
that minimizes total energy of the system. Develop this argument
explicitly/ for example as follows:
(a) Write down the classical expression for the total energy of
the hydrogen atom with an electron of momentum p in a circular
orbit of radius'r. Keep kinetic and potential energies separate.
(b) Failure of classical energy minimization Use the force law
to obtain the total energy as a function of radius. What radius
results if this expression is set to yield a minimum energy?
(c) For the lowest energy state, demand that the orbit circumference
be one deBroglie wavelength (recall exercise 12). Eliminate
wavelength and substitute the result for momentum into the energy
relation of part (a) to .yield an expression that is a function of
radius. Note how a larger radius decreases the kinetic energy
and increases the potential energy, and vice versa.
(d) Take the derivative of the energy vs. radius function and find
the radius that minimizes total energy,. Show that the resulting
radius is the Bohr radius a and that the resulting energy is that
calculated by Bohr for the
lowest energy state. Comment; This
exact result depends on setting the deBroglie wavelength equal to
the circumference of the orbit. In fact the argument we are
folltfv/ing specifier only an approximate correspondence between
system size and deBroglie wavelength. We could have chosen
"system size 11 to mean radius or diameter and then derived a size
differing from that calculated above by factors of 2 or IT , which
is as near to the observed values as one has any right to expect.