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Fig. 5-9. Electron micrograph of some virus molecules. The “large”
sphere is for calibration and is known to have a diameter of 2x 10” meter
(2000 Ä).
might #nagine that a highly magnified view of the situation——looking along the
particle beam——would look like Eig. 5-10.
Fig. 5-10. lmagined view through a block of carbon 1 cm thick If only
the nuclei were observed.
'The chance that a very small particle will hit a nuecleus on the trip through is
Just the total area covered by the profiles of the nuclei divided by the total area
in the picture. Suppose that we know that in an area A of our slab of material
there are W atoms (each with one nucleus, of course). Then the fraction of the
--- Trang 117 ---
area “covered” by the nuclei is Nơ/A. Now let the number of particles of our
beam which arrive at the slab be ø+ and the number which come out the other
side be mạ. The fraction which do nø£ get through is (m¡ — m2)/m+, which should
just equal the fraction of the area covered. We can obtain the radius of the
nucleus from the equationF
_.-ˆ....
N T1
trom such an experiment we fnd that the radii of the nuclei are from about
1 to 6 times 1015 meter. The length unit 1015 meter is called the ƒermi, in
honor of Enrico Fermi (1901-1954).
What do we fnd If we go to smaller distances? Can we measure smaller
distances? Such questions are not yet answerable. It has been suggested that
the still unsolved mystery of nuclear forces may be unravelled only by some
modifcation of our idea. oŸ space, or measurement, at such small distances.
Tt might be thought that ít would be a good idea to use some natural length as
our unit o£ length—say the radius of the earth or some fraction of it. 'Phe meter
was originally intended to be such a unit and was defned to be (/2) x 10—7 times
the earth”s radius. I% is neither convenient nor very accurate to determine the
unit of length in this way. For a long tỉme it has been agreed internationally that
the meter would be defined as the distance between two scratches on a bar kept
in a special laboratory in France. More recently, ¡it has been realized that this
defnition is neither as precise as would be useful, nor as permanent or universal as
one would like. It is currently beïng considered that a new defnition be adopted,
an agreed-upon (arbitrary) number of wavelengths of a chosen spectral line.
Measurements of distance and of time give results which depend on the
observer. 'Wwo observers moving with respect to each other will not measure
the same distances and times when measuring what appear to be the same
things. Distances and time intervals have diferent magnitudes, depending on the
coordinate system (or “frame of reference”) used for making the measurements.
W© shall study this subJect in more detail in a later chapter.
* 'Phis equation is right only if the area covered by the nuclei is a small fraction of the total,
1.e., 1Ÿ (mị — 2)/mị is much less than 1. Otherwise we must make a correction for the fact that
some nuclei will be partly obscured by the nuclei in front of them.
--- Trang 118 ---
Perfectly precise measurements of distances or times are not permitted by
the laws of nature. We have mentioned earlier that the errors in a measurement
of the position of an obJect must be at least as large as
Az> h/2Ab,
where ñ is a small fundamental physical constant called the reduced Planck
constant and Ấp 1s the error in our knowledge of the momentum (mass times
velocity) of the object whose position we are measuring. It was also mentioned
that the uncertainty in position measurementfs is related to the wave nature of
particles.
The relativity of space and time implies that time measurements have aÌso a
minimum error, given in fact by
At>h/2AE,
where A is the error in our knowledge of the energy of the process whose tỉme
period we are measuring. lf we wish to know rmore precisely hen something
happened we must know less about +0ha# happened, because our knowledge of
the energy involved will be less. The time uncertainty is also related to the wave
nature of matter.
--- Trang 119 ---
PProberbrlrty
“The true logic of this world is in the calculus of probabilities.”
— James Clerk Maxwell
6-1 Chance and likelihood
“Chance” is a word which is in common use in everyday living. The radio
reports speaking of tomorrow's weather may say: “There is a sixty percent chance
of rain” You might say: ““There is a small chance that I shall live to be one
hundred years old.” Scientists also use the word chance. A seismologist may be
interested ¡in the question: “What ¡is the chance that there will be an earthquake
of a certain size in Southern California next year?” A physicist might ask the
question: “What is the chance that a particular geiger counter will register bwenty
counts in the next ten seconds?” A politician or statesman might be interested
in the question: “What is the chance that there will be a nuclear war within
the next ten years?” You may be interested in the chance that you will learn
something from this chapter.
By chance, we mean something like a guess. Why do we make guesses? We
make guesses when we wish to make a judgment but have incomplete information
or uncertain knowledge. We want to make a guess as to what things are, or what
things are likely to happen. Often we wish to make a guess because we have to
make a decision. For example: Shall I take my raincoat with me tomorrow? For
what earth movement should I design a new building? Shall I build myself a
fallout shelter? Shall I change my stand in international negotiations? Shall I go
to class today?