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Unformatted text preview: 2 The Relation of Wave and
Particle Viewpoints 2—1 Probability wave amplitudes In this chapter we shall discuss the relationship of the wave and particle
Viewpoints. We already know, from the last chapter, that neither the wave view
point nor the particle viewpoint is correct. We would always like to present things
accurately, or at least precisely enough that they will not have to be changed when
we learn more—it may be extended, but it will not be changed! But when we try
to talk about the wave picture or the particle picture, both are approximate, and
both will change. Therefore what we learn in this chapter will not be accurate in a
certain sense; we will deal with some halfintuitive arguments which will be made
more precise later. But certain things will be changed a little bit when we interpret
them correctly in quantum mechanics. We are doing this so that you can have
some qualitative feeling for some quantum phenomena before we get into the
mathematical details of quantum mechanics. Furthermore, all our experiences
are with waves and with particles, and so it is rather handy to use the wave and
particle ideas to get some understanding of what happens in given circumstances
before we know the complete mathematics of the quantummechanical amplitudes.
We shall try to indicate the weakest places as we go along, but most of it is very
nearly correct—~it is just a matter of interpretation. First of all, we know that the new way of representing the world in quantum
mechanics—the new framework—43 to give an amplitude for every event that can
occur, and if the event involves the reception of one particle, then we can give the
amplitude to ﬁnd that one particle at different places and at diﬁerent times. The
probability of ﬁnding the particle is then proportional to the absolute square of
the amplitude. In general, the amplitude to ﬁnd a particle in different places at
diﬁcrent times varies with position and time. In some special case it can be that the amplitude varies sinusoidally in space
and time like ei("’tTk"), where r is the vector position from some origin. (Do not
forget that these amplitudes are complex numbers, not real numbers.) Such an
amplitude varies according to a deﬁnite frequency w and wave number k. Then
it turns out that this corresponds to a classical limiting situation where we would
have believed that we have a particle whose energy E was known and is related to
the frequency by E = hw, (2.1) and whose momentum p is also known and is related to the wave number by
p = hk. (2.2) (The symbol h represents the number k divided by 271'; h = h/21r.) This means that the idea of a particle is limited. The idea of a particleiits
location, its momentum, etc—which we use so much, is in certain ways unsatis
factory. For instance, if an amplitude to ﬁnd a particle at different places is given
by e“‘”“'"""), whose absolute square is a constant. that would mean that the prob—
ability of ﬁnding a particle is the same at all points. That means we do not know
where it is—it can be anywhere—there is a great uncertainty in its location. On the other hand, if the position of a particle is more or less well known and
we can predict it fairly accurately. then the probability of ﬁnding it in different
places must be conﬁned to a certain region, whose length we call Ax. Outside this
region, the probability is zero. Now this probability is the absolute square of an
amplitude, and if the absolute square is zero, the amplitude is also zero, so that 2—1 2—1 Probability wave amplitudes 2—2 Measurement of position and
momentum 2—3 Crystal diﬁraction 2—4 The size of an atom 2—5 Energy levels 2—6 Philosophical implications Note: This chapter is almost exactly
the same as Chapter 38 of Volume I. Fig. 2—1. A wave packet of length Ax. Fig. 2—2. Diffraction
passing through a slit. of particles we have a wave train whose length is Ax (Fig. 2—1), and the wavelength (the
distance between nodes of the waves in the train) of that wave train is what corre
sponds to the particle momentum. . Here we encounter a strange thing about waves; a very simple thing which has
nothing to do with quantum mechanics strictly. It is something that anybody
who works with waves, even if he knows no quantum mechanics, knows: namely,
we cannot deﬁne a unique wavelength for a short wave train. Such a wave train does
not have a deﬁnite wavelength; there is an indeﬁniteness in the wave number that
is related to the ﬁnite length of the train, and thus there is an indeﬁniteness in
the momentum. 2—2 Measurement of position and momentum Let us consider two examples of this idea—to see the reason that there is an
uncertainty in the position and/or the momentum. if quantum mechanics is right.
We have also seen before that if there were not such a thing—if it were possible to
measure the position and the momentum of anything simultaneously—we would
have a paradox; it is fortunate that we do not have such a paradox, and the fact
that such an uncertainty comes naturally from the wave picture shows that every
thing is mutually consistent. Here is one example which shows the relationship between the position and
the momentum in a circumstance that is easy to understand. Suppose we have a
single slit, and particles are coming from very far away with a certain energy—so
that they are all coming essentially horizontally (Fig. 2—2). We are going to
concentrate on the vertical components of momentum. All of these particles have
a certain horizontal momentum p0, say, in a classical sense. So, in the classical
sense, the vertical momentum 1),, before the particle goes through the hole, is
deﬁnitely known. The particle is moving neither up nor down, because it came from
a source that is far away—and so the vertical momentum is of course zero. But
now let us suppose that it goes through a hole whose width is B. Then after it has
come out through the hole, we know the position vertically—the yposition—with
considerable accuracy—namely j=B.’r That is, the uncertainty in position, Ay, is
of order B. Now we might also want to say. since we known the momentum is
absolutely horizontal, that A191, is zero; but that is wrong. We once knew the mo
mentum was horizontal. but we do not know it any more. Before the particles
passed through the hole, we did not know their vertical positions. Now that we
have found the vertical position by having the particle come through the hole, we
have lost our information on the vertical momentum! Why? According to the
wave theory, there is a spreading out, or diffraction, of the waves after they go
through the slit, just as for light. Therefore there is a certain probability that
particles coming out of the slit are not coming' exactly straight. The pattern is
spread out by the ditl‘raction effect, and the angle of spread, which we can deﬁne
as the angle of the ﬁrst minimum, is a measure of the uncertainty in the ﬁnal angle. How does the pattern become spread? To say it is spread means that there is
some chance for the particle to be moving up or down, that is, to have a component
of momentum up or down. We say chance and particle because we can detect this
diffraction pattern with a particle counter, and when the counter receives the
particle, say at C in Fig. 2—2, it receives the entire particle, so that, in a classical
sense, the particle has a vertical momentum, in order to get from the slit up to C. To get a rough idea of the spread of the momentum, the vertical momentum
1),, has a spread which is equal to [70 A0, where p“ is the horizontal momentum.
And how big is A6 in the spreadout pattern? We know that the ﬁrst minimum
occurs at an angle A0 such that the waves from one edge of the slit have to travel
one wavelength farther than the waves from the other side—we worked that out
before (Chapter 30 of Vol. 1). Therefore A0 is VB, and so Am, in this experiment
is pox/B. Note that if we make B smaller and make a more accurate measurement T More precisely, the error in our knowledge of y is i[3/2. But we are now only in
terested in the general idea, so we won‘t worry about factors of 2. 2—2 of the position of the particle, the diffraction pattern gets wider. So the narrower
we make the slit, the wider the pattern gets, and the more is the likelihood that we
would ﬁnd that the particle has sidewise momentum. Thus the uncertainty in the
vertical momentum is inversely proportional to the uncertainty of y. In fact, we
see that the product of the two is equal to pot. But >\ is the wavelength and [)0 is
the momentum, and in accordance with quantum mechanics, the wavelength times
the momentum is Planck’s constant 11. So we obtain the rule that the uncertainties
in the vertical momentum and in the vertical position have a product of the order h: Ay Ap, x h. (2.3) We cannot prepare a system in which we know the vertical position of a particle
and can predict how it will move vertically with greater certainty than given by
(2.3). That is, the uncertainty in the vertical momentum must exceed h/Ay, where
Ay is the uncertainty in our knowledge of the position. Sometimes people say quantum mechanics is all wrong. When the particle
arrived from the left, its vertical momentum was zero. And now that it has gone
through the slit, its position is known. Both position and momentum seem to
be known with arbitrary accuracy. It is quite true that we can receive a particle,
and on reception determine what its position is and what its momentum would
have had to have been to have gotten there. That is true, but that is not what the
uncertainty relation (2.3) refers to. Equation (2.3) refers to the predictability
of a situation, not remarks about the past. It does no good to say “I knew what
the momentum was before it went through the slit, and now I know the position,"
because now the momentum knowledge is lost. The fact that it went through the
slit no longer permits us to predict the vertical momentum. We are talking about
a predictive theory, not just measurements after the fact. So we must talk about
what we can predict. Now let us take the thing the other way around. Let us take another example
of the same phenomenon, a little more quantitatively. In the previous example
we measured the momentum by a classical method. Namely, we considered the
direction and the velocity and the angles, etc., so we got the momentum by classical
analysis. But since momentum is related to wave number, there exists in nature
still another way to measure the momentum of a particle—photon or otherwise—
which has no classical analog, because it uses Eq. (2.2). We measure the wave
lengths of the waves. Let us try to measure momentum in this way. Suppose we have a grating with a large number of lines (Fig. 2—3), and send
a beam of particles at the grating. We have often discussed this problem: if the
particles have a deﬁnite momentum, then we get a very sharp pattern in a certain
direction, because of the interference. And we have also talked about how accu
rately we can determine that momentum, that is to say, what the resolving power
of such a grating is. Rather than derive it again, we refer to Chapter 30 of Volume
I. where we found that the relative uncertainty in the wavelength that can be
measured with a given grating is l/Nm, where Nis the number of lines on the grat
ing and m is the order of the diffraction pattern. That is, AVA = l/Nm. (2.4)
Now formula (2.4) can be rewritten as
AVA2 : l/Nmtt : l/L, (2.5) where L is the distance shown in Fig. 2—3. This distance is the difference between
the total distance that the particle or wave or whatever it is has to travel if it is
reﬂected from the bottom of the grating. and the distance that it has to travel if
it is reﬂected from the top of the grating. That is, the waves which form the diffrac
tion pattern are waves which come from different parts of the grating. The ﬁrst
ones that arrive come from the bottom end of the grating, from the beginning of
the wave train. and the rest of them come from later parts of the wave train, coming
from different parts of the grating, until the last one ﬁnally arrives, and that involves
a point in the wave train a distance L behind the ﬁrst point. So in order that we 2—3 Fig. 2—3. Determination of momen
tum by using a diffraction grating. Fig. 2—4. crystal planes. Scattering of waves by shall have a sharp line in our spectrum corresponding to a deﬁnite momentum,
with an uncertainty given by (2.4), we have to have a wave train of at least length
L. If the wave train is too short, we are not using the entire grating. The waves
which form the spectrum are being reflected from only a very short sector of the
grating if the wave train is too short, and the grating will not work right—we will
ﬁnd a big angular spread. In order to get a narrower one, we need to use the whole
grating, so that at least at some moment the whole wave train is scattering simul
taneously from all parts of the grating. Thus the wave train must be of length L
in order to have an uncertainty in the wavelength less than that given by (2.5).
Incidentally,
A)\/)\2 = A(l/)\) = Ak/27r. (2.6)
Therefore
Ak = 21r/L, (2.7) where L is the length of the wave train. This means that if we have a wave train whose length is less than L, the un
certainty in the wave number must exceed 27r/L. Or the uncertainty in a wave
number times the length of the wave train*we will call that for a moment Ax—
exceeds 271'. We call it Ax because that is the uncertainty in the location of the
particle. If the wave train exists only in a ﬁnite length, then that is where we could
ﬁnd the particle, within an uncertainty Ax. Now this property of waves, that the
length of the wave train times the uncertainty of the wave number associated with
it is at least 277', is a property that is known to everyone who studies them. It has
nothing to do with quantum mechanics. It is simply that if we have a ﬁnite train,
we cannot count the waves in it very precisely. Let us try another way to see the reason for that. Suppose that we have a
ﬁnite train of length L; then because of the way it has to decrease at the ends, as
in Fig. 2—1, the number of waves in the length L is uncertain by something like 3:1.
But the number of waves in L is kL/27r. Thus k is uncertain, and we again get the
result (2.7), a property merely of waves. The same thing works Whether the waves
are in space and k is the number of radians per centimeter and L is the length of
the train, or the waves are in time and w is the number of oscillations per second
and T is the “length” in time that the wave train comes in. That is, if we have
a wave train lasting only for a certain ﬁnite time T, then the uncertainty in the fre
quency is given by Aw = 27r/T. (2.8) We have tried to emphasize that these are properties of waves alone, and they are
well known, for example, in the theory of sound. The point is that in quantum mechanics we interpret the wave number as
being a measure of the momentum of a particle, with the rule that p = hk, so
that relation (2.7) tells us that Ap z h/Ax. This, then, is a limitation of the classi
cal idea of momentum. (Naturally, it has to be limited in some ways if we are
going to represent particles by waves!) It‘is nice that we have found a rule that
gives us some idea of when there is a failure of classical ideas. 2—3 Crystal diffraction Next let us consider the reﬂection of particle waves from a crystal. A crystal
is a thick thing which has a whole lot of similar atoms—we will include some com
plications later—in a nice array. The question is how to set the array so that we
get a strong reﬂected maximum in a given direction for a given beam of, say, light
(xrays), electrons, neutrons, or anything else. In order to obtain a strong reﬂection,
the scattering from all of the atoms must be in phase. There cannot be equal num
bers in phase and out of phase, or the waves will cancel out. The way to arrange
things is to ﬁnd the regions of constant phase, as we have already explained;
they are planes which make equal angles with the initial and ﬁnal directions
(Fig. 2—4). If we consider two parallel planes, as in Fig. 2—4, the waves scattered from the
two planes will be in phase, provided the difference in distance traveled by a wave 24 front is an integral number of wavelengths. This difference can be seen to be 2d sin 6, where d is the perpendicular distance between the planes. Thus the
condition for coherent reﬂection is 2dsin 6 = m (n = 1,2,. . .). (2.9) If, for example, the crystal is such that the atoms happen to lie on planes obey
ing condition (2.9) with n = 1, then there will be a strong reﬂection. If, on the
other hand, there are other atoms of the same nature (equal in density) halfway
between, then the intermediate planes will also scatter equally strongly and will
interfere with the others and produce no eﬂect. So (1 in (2.9) must refer to ad
jacent planes; we cannot take a plane ﬁve layers farther back and use this formula! As a matter of interest, actual crystals are not usually as simple as a single
kind of atom repeated in a certain way. Instead, if we make a twodimensional
analog, they are much like wallpaper, in which there is some kind of ﬁgure which
repeats all over the wallpaper. By “ﬁgure” we mean, in the case of atoms, some
arrangement—calcium and a carbon and three oxygens, etc., for calcium carbonate,
and so on—which may involve a relatively large number of atoms. But whatever
it is, the ﬁgure is repeated in a pattern. This basic ﬁgure is called a unit cell. The basic pattern of repetition deﬁnes what we call the lattice type; the lattice
type can be immediately determined by looking at the reﬂections and seeing what
their symmetry is. In other words, where we ﬁnd any reﬂections at all determines
the lattice type, but in order to determine what is in each of the elements of the
lattice one must take into account the intensity of the scattering at the various
directions. Which directions scatter depends on the type of lattice, but how strongly
each scatters is determined by what is inside each unit cell, and in that way the
structure of crystals is worked out. Two photographs of xray diﬂraction patterns are shown in Figs. 2—5 and
2—6; they illustrate scattering from rock salt and myoglobin, respectively. Incidentally, an interesting thing happens if the spacings of the nearest planes
are less than V2. In this case (2.9) has no solution for n. Thus if A is bigger
than twice the distance between adjacent planes, then there is no side diffraction
pattern, and the light—or whatever it is—will go right through the material with
out bouncing oif or getting lost. So in the case of light, where >\ is much bigger
than the spacing, of course it does go through and there is no pattern of reﬂection
from the planes of the crystal. This fact also has an interesting consequence in the case of piles which make
neutrons (these are obviously particles, for anybody’s money!). If we take these
neutrons and let them into a long block of graphite, the neutrons diﬂuse and
work their way along (Fig. 2—7). They diffuse because they are bounced by the
atoms, but strictly, in the wave theory, they are bounced by the atoms because
of diﬂraction from the crystal planes. It turns out that if we take a very long piece
of graphite, the neutrons that come out the far end are all of long wavelength!
In fact, if one plots the intensity as a function of wavelength, we get nothing except
for wavelengths longer than a certain minimum (Fig. 2—8). In other words, we
can get very slow neutrons that way. Only the slowest neutrons come through;
they are not diffracted or scattered by the crystal planes of the graphite, but keep
going right through like light through glass, and are not scattered out the sides.
There are many other demonstrations of the reality of neutron waves and waves
of other particles. 24 The size of an atom We now consider another applicatio...
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