Datasets:
vnpthack
/
vnpt_data

Dataset card Data Studio Files
xet
 Community
1
text
stringlengths
0
6.73k
for each increment. “The probability that D lands somewhere between #ø and za,
which we may write P{z„ < D < za), is equal to the shaded area in Eig. 6-8.
The smaller we take the increments Az, the more correct is our result. We can
write, therefore,
P(œạ< D< z:) = À`p(z) Ax= J p(ø) da. (6.18)
The area under the whole curve is the probability that Ð lands somewhere
(that is, has sormme value between ø = —œ and # = +œc). That probability is
--- Trang 133 ---
X1 X2 x
Fig. 6-8. The probability that the distance D traveled in a random
walk is between xị and xa is the area under the curve of p(x) from xị
to Xa.
surely 1. We must have that
J p(z) dz = 1. (6.19)
Since the curves in Fig. 6-7 get wider in proportion to W, theïr heights must
be proportional to 1/WN to maintain the total area equal to 1.
The probability density function we have been describing is one that is
encountered most commonly. It is known as the n=ormal or gausstøn probability
density. It has the mathematical form
p(&) = —— 9/2, (6.20)
where ø is called the s¿øndard deuiafion and is given, in our case, by ơ = VN or,
1f the rms step size is diferent from 1, by ơ = VN mạ.
We remarked earlier that the motion of a molecule, or of any particle, in a
gas is like a random walk. Suppose we open a bottle of an organic compound
and let some of its vapor escape Into the air. If there are air currents, so that
the air is circulating, the currents will also carry the vapor with them. But even
in perfeclu si azr, the vapor will gradually spread out—will difuse—until it
has penetrated throughout the room. We might detect it by its color or odor.
The individual molecules of the organic vapor spread out in still air because of
the molecular motions caused by collisions with other molecules. If we know the
average “step” size, and the number of steps taken per second, we can fnd the
--- Trang 134 ---
probability that one, or several, molecules wiïll be found at some distance from
their starting point after any particular passage of time. Äs time passes, more
steps are taken and the gas spreads out as in the successive curves of Fig. 6-7.
In a later chapter, we shall fnd out how the step sizes and step Ífrequencies are
related to the temperature and pressure of a gas.
Barlier, we said that the pressure of a gas is due to the molecules bouneing
against the walls of the container. When we come later to make a more quanti-
tative description, we will wish to know how fast the molecules are going when
they bounce, since the impact they make will depend on that speed. We camnot,
however, speak oŸ #he speed of the molecules. Ït is necessary %o use a probability
description. A molecule may have any speed, but some speeds are more likely
than others. We describe what is going on by saying that the probability that
any particular molecule will have a speed between 0 and ø + Ao is p(o) Ao,
where Øø(0), a probability density, is a given funection of the speed ø. We shall see
later how Maxwell, using common sense and the ideas of probability, was able to
ñnd a mathematical expression for ø(0). The form of the function ø(0) is shown
in Fig. 6-9. Velocities may have any value, but are most likely to be near the
most probable value 0;.
N-p(v)
Vp VỊ V2 V
Fig. 6-9. The distribution of velocities of the molecules In a gas.
We often think of the curve of Fig. 6-9 in a somewhat different way. lf
we consider the molecules in a typical container (with a volume of, say, one
liter), then there are a very large number of molecules present ( + 1022).
Since ø(0) Ao is the probability that øøe molecule will have its velocity in Áo,
* Maxwell's expression is (0) = Cu2c—*°Ỷ, where œ is a constant related to the temperature
and Πis chosen so that the total probability is one.
--- Trang 135 ---
by our defnition oŸ probability we mean that the ezpecfed number (AN) to be
found with a velocity in the interval Au is given by
(AM) = Np(o) Ao. (6.21)
We call N p(o) the “distribution in velocity.” The area under the curve bebween
two velocitles 0 and 0a, for example the shaded area in Fig. 6-9, represents [for
the curve ý ø(ø)| the expected number of molecules with velocities bebween 0
and 0a. 5ince with a gas we are usually dealing with large numbers of molecules,
we expect the deviations from the expected numbers to be small (like 1/v `), so
we often neglect to say the “expected” number, and say instead: “The number oŸ
mmolecules with velocitles between 0 and 0a 2s the area under the curve.” We should
remember, however, that such statements are always about probable numbers.
6-5 The uncertainty principle
The ideas of probability are certainly useful in describing the behavior of
the 1022 or so molecules in a sample of a gas, for it is clearly impractical even to
attempt to write down the position or velocity of each molecule. When probability
was first applied to such problems, 1 was considered to be a conwuenience—a
way of dealing with very complex situations. We now believe that the ideas of
probability are essenfial to a description of atomie happenings. According to
quantum mechaniecs, the mathematical theory of particles, there is always some
uncertainty in the specifcafion of positions and velocities. We can, at best, say
that there is a certain probability that any particle will have a position near some
coordinate z.
We can give a probability density ø+(#), such that ø+(#) Az is the probability
that the particle will be found between + and z-+ Az. T the particle is reasonably
well localized, say near zoọ, the function ø1(z) might be given by the graph of
Eig. 6-10(a). Similarly, we must specify the velocity of the particle by means of
a probability density pa(), with pa(0) Ao the probability that the velocity will
be found between 0 and ø + Áo.
lt is one of the fundamental results of quantum mechanics that the two
functions ø¡(#) and øa(0) cannot be chosen independently and, in particular,
cannot both be made arbitrarily narrow. lf we call the typical “width” of the
ø1(z) curve [Az], and that of the øa(ø) curve [Aö| (as shown in the figure),
nature demands that the product of the two widths be at least as big as the
--- Trang 136 ---
pa(v) '
Fig. 6-10. Probability densities for observatlon of the position and
velocity of a particle.
number #/2m, where mm is the mass of the particle. We may write this basic
relationship as
[Az] - [Aul > h/2m. (6.22)