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Unformatted text preview: 37 Quantum Behavior 371 Atomic mechanics In the last few chapters we have treated the essential ideas necessary for an
understanding of most of the important phenomena of light—or electromagnetic
radiation in general. (We have left a few special topics for next year. Speciﬁcally,
the theory of the index of dense materialsand total internal reﬂection.) What we
have dealt with is called the “classical theory” of electric waves, which turns out
to be a completely adequate description of nature for a large number of effects.
We have not had to worry yet about the fact that light energy comes in lumps or
“photons.” We would like to take up as our next subject the problem of the behavior of
relatively large pieces of matter—their mechanical and thermal properties, for
instance. In discussing these, we will ﬁnd that the “classical” (or older) theory
fails almost immediately, because matter is really made up of atomicsized par
ticles. Still, we will deal only with the classical part, because that is the only part
that we can understand using the classical mechanics we have been learning. But
we shall not be very successful. We shall ﬁnd that in the case of matter, unlike the
case of light, we shall be in diﬁiculty relatively soon. We could, of course, con
tinuously skirt away from the atomic effects, but we shall instead interpose here a
short excursion in which we will describe the basic ideas of the quantum properties
of matter, i.e., the quantum ideas of atomic physics, so that you will have some
feeling for what it is we are leaving out. For we will have to leave out some im
portant subjects that we cannot avoid coming close to. So we will give now the introduction to the subject of quantum mechanics,
but will not be able actually to get into the subject until much later. “Quantum mechanics” is the description of the behavior of matter in all its
details and, in particular, of the happenings on an atomic scale. Things on a very
small scale behave like nothing that you have any direct experience about. They
do not behave like waves, they do not behave like particles, they do not behave
like clouds, or billiard balls, or weights on springs, or like anything that you
have ever seen. Newton thought that light was made up of particles, but then it was discovered,
as we have seen here, that it behaves like a wave. Later, however (in the beginning
of the twentieth century) it was found that light did indeed sometimes behave like
a particle. Historically, the electron, for example, was thought to behave like a
particle, and then it was found that in many respects it behaved like a wave. So it
really behaves like neither. Now we have given up. We say: “It is like neither.” There is one lucky break, however—electrons behave just like light. The
quantum behavior of atomic objects (electrons, protons, neutrons, photons, and
so on) is the same for all, they are all “particle waves,” or whatever you want to
call them. So what we learn about the properties of electrons (which we shall use
for our examples) will apply also to all “particles,” including photons of light. The gradual accumulation of information about atomic and smallscale be
havior during the ﬁrst quarter of this century, which gave some indications about
how small things do behave, produced an increasing confusion which was ﬁnally
resolved in 1926 and 1927 by Schrodinger, Heisenberg, and Born. They ﬁnally
obtained a consistent description of the behavior of matter on a small scale. We
take up the main features of that description in this chapter. Because atomic behavior is so unlike ordinary experience, it is very difﬁcult
to get used to and it appears peculiar and mysterious to everyone, both to the 37—1 37—1 Atomic mechanics 37—2 An experiment with bullets
37—3 An experiment with waves
37—4 An experiment with electrons 375 The interference of electron
waves 37—6 Watching the electrons 37—7 First principles of quantum
mechanics 37—8 The uncertainty principle novice and to the experienced physicist. Even the experts do not understand it
the way they would like to, and it is perfectly reasonable that they should not,
because all of direct, human experience and of human intuition applies to large
objects. We know how large objects will act, but things on a small scale just do
not act that way. So we have to learn about them in a sort of abstract or imagi
native fashion and not by connection with our direct experience. In this chapter we shall tackle immediately the basic element of the mysterious
behavior in its most strange form. We choose to examine a phenomenon which is
impossible, absolutely impossible, to explain in any classical way, and which has
in it the heart of quantum mechanics. In reality, it contains the only mystery.
We cannot explain the mystery in the sense of “explaining” how it works. We will
tell you how it works. In telling you how it works we will have told you about the
basic peculiarities of all quantum mechanics. 37—2 An experiment with bullets To try to understand the quantum behavior of electrons, we shall compare
and contrast their behavior, in a particular experimental setup, with the more
familiar behavior of particles like bullets, and with the behavior of waves like
water waves. We consider ﬁrst the behavior of bullets in the experimental setup
shown diagrammatically in Fig. 37—1. We have a machine gun that shoots a stream
of bullets. It is not a very good gun, in that it sprays the bullets (randomly) over a
fairly large angular spread, as indicated in the ﬁgure. In front of the gun we have
a wall (made of armor plate) that has in it two holes just about big enough to let a
bullet through. Beyond the wall is a backstop (say a thick wall of wood) which will
“absorb” the bullets when they hit it. In front of the wall we have an object which
we shall call a “detector” of bullets. It might be a box containing sand. Any bullet
that enters the detector will be stopped and accumulated. When we wish, we can
empty the box and count the number of bullets that have been caught. The
detector can be moved back and forth (in what we will call the xdirection). With
this apparatus, we can ﬁnd out experimentally the answer to the question: “What
is the probability that a bullet which passes through the holes in the wall will
arrive at the backstop at the distance x from the center?” First, you should
realize that we should talk about probability, because we cannot say deﬁnitely
where any particular bullet will go. A bullet which happens to hit one of the holes
may bounce off the edges of the hole, and may end up anywhere at all. By “prob
ability” we mean the chance that the bullet will arrive at the detector, which we can
measure by counting the number which arrive at the detector in a certain time and
then taking the ratio of this number to the total number that hit the backstop during
that time. Or, if we assume that the gun always shoots at the same rate during the
measurements, the probability we want is just proportional to the number that
reach the detector in some standard time interval. For our present purposes we would like to imagine a somewhat idealized
experiment in which the bullets are not real bullets, but are indestructible bullets—
they cannot break in half. In our experiment we ﬁnd that bullets always arrive in
lumps, and when we ﬁnd something in the detector, it is always one whole bullet.
If the rate at which the machine gun ﬁres is made very low, we ﬁnd that at any given
moment either nothing arrives, or one and only one—exactly one—bullet arrives
at the backstop. Also, the size of the lump certainly does not depend on the rate
of ﬁring of the gun. We shall say: “Bullets always arrive in identical lumps.” What
we measure with our detector is the probability of arrival of a lump. And we meas
ure the probability as a function of x. The result of such measurements with this
apparatus (we have not yet done the experiment, so we are really imagining the
result) are plotted in the graph drawn in part (c) of Fig. 37—1. In the graph we plot
the probability to the right and x vertically, so that the xscale ﬁts the diagram of
the apparatus. We call the probability P12 because the bullets may have come
either through hole 1 or through hole 2. You will not be surprised that P12 is
large near the middle of the graph but gets small if x is very large. You may
wonder, however, why P12 has its maximum value at x = 0. We can understand 372 / // I r— ' ~
er; :1' 1 _ _ _\_‘_‘
GUN \‘3 2
. . STOP
Fig. 37—1 . Interference experiment WALL BACK
with bullets. (a) this fact if we do our experiment again after covering up hole 2, and once more
while covering up hole 1. When hole 2 is covered, bullets can pass only through
hole 1, and we get the curve marked P1 in part (b) of the ﬁgure. As you would
expect, the maximum of P1 occurs at the value of x which is on a straight line with
the gun and hole 1. When hole 1 is closed, we get the symmetric curve P2 drawn
in the ﬁgure. P2 is the probability distribution for bullets that pass through hole
2. Comparing parts (b) and (c) of Fig. 37—1, we ﬁnd the important result that P12=P1+P2. The probabilities just add together. The elfect with both holes open is the sum of
the effects with each hole open alone. We shall call this result an observation of
“no interference,” for a reason that you will see later. So much for bullets. They
come in lumps, and their probability of arrival shows no interference. at
e are. 3) Fig. 37—2. Interference experiment WALL
with water waves. (a)
37—3 An experiment with waves Now We wish to consider an experiment with water waves. The apparatus is
shown diagrammatically in Fig. 37—2. We have a shallow trough of water. A small
object labeled the “wave source” is jiggled up and down by a motor and makes
circular waves. To the right of the source we have again a wall with two holes,
and beyond that is a second wall, which, to keep things simple, is an “absorber,”
so that there is no reﬂection of the waves that arrive there. This can be done by
building a gradual sand “beach.” In front of the beach we place a detector which
can be moved back and forth in the xdirection, as before. The detector is now a
device which measures the “intensity” of the wave motion. You can imagine a
gadget which measures the height of the wave motion, but whose scale is calibrated
in proportion to the square of the actual height, so that the reading is proportional
to the intensity of the wave. Our detector reads, then, in proportion to the energy
being carried by the wave—or rather, the rate at which energy is carried to the
detector. With our wave apparatus, the ﬁrst thing to notice is that the intensity can
have any size. If the source just moves a very small amount, then there is just a
little bit of wave motion at the detector. When there is more motion at the source, 37—3 (b) II ABSORBER I, = nl2 Izlhf (b) Fiz'li *5 (d 2
It? "‘3th (C) there is more intensity at the detector. The intensity of the wave can have any
value at all. We would not say that there was any “lumpiness” in the wave intensity. Now let us measure the wave intensity for various values of x (keeping the
wave source operating always in the same way). We get the interestinglooking
:urve marked [12 in part (c) of the ﬁgure. We have already worked out how such patterns can come about when we
studied the interference of electric waves. In this case we would observe that the
original wave is diffracted at the holes, and new circular waves spread out from each
hole. If we cover one hole at a time and measure the intensity distribution at the
absorber we ﬁnd the rather simple intensity curves shown in part (b) of the ﬁgure.
11 is the intensity of the wave from hole 1 (which we ﬁnd by measuring when hole
2 is blocked off) and 12 is the intensity of the wave from hole 2 (seen when hole
1 is blocked). The intensity 112 observed when both holes are open is certainly not the sum
of 11 and 12. We say that there is “interference” of the two waves. At some
places (where the curve 11 2 has its maxima) the waves are “in phase” and the wave
peaks add together to give a large amplitude and, therefore, a large intensity. We
say that the two waves are “interfering constructively” at such places. There will
be such constructive interference wherever the distance from the detector to one
hole is a whole number of wavelengths larger (or shorter) than the distance from
the detector to the other hole. At those places where the two waves arrive at the detector with a phase differ
ence of 71' (where they are “out of phase”) the resulting wave motion at the detector
will be the difference of the two amplitudes. The waves “interfere destructively,”
and we get a low value for the wave intensity. We expect such low values wherever
the distance between hole 1 and the detector is different from the distance between
hole 2 and the detector by an odd number of halfwavelengths. The low values of
11 2 in Fig. 37—2 correspond to the places where the two waves interfere destructively. You will remember that the quantitative relationship between 11, 12, and 112
can be expressed in the following way: The instantaneous height of the water wave
at the detector for the wave from hole 1 can be written as (the real part of) Elem,
where the “amplitude” I31 is, in general, a complex number. The intensity is
proportional to the mean squared height or, when we use the complex numbers,
to lizllZ. Similarly, for hole 2 the height is ﬁgeiw‘ and the intensity is proportional
to lizzl2. When both holes are open, the wave heights add to give the height
(le + ﬁg)?“ and the intensity lizl + 132R Omitting the constant of proportion
ality for our present purposes, the proper relations for interfering waves are 11 = W2, 12 = lfzzlz, 112 = lle + 2242. (37.2)
You will notice that the result is quite different from that obtained with bullets
(Eq. 37.1). If we expand f11 + 5242 we see that [/31 + flzl2 = lizrl2 + lilzl2 + ZIﬁrllftleOS 5, (373) where 6 is the phase difference between izl and 22. In terms of the intensities, we
could write 112 = 11 + 12 + Nil—1; cos a. (37.4) The last term in (37.4) is the “interference term.” So much for water waves. The
intensity can have any value, and it shows inteference. 37—4 An experiment with electrons Now we imagine a similar experiment with electrons. It is shown diagram
matically in Fig. 3743. We make an electron gun which consists of a tungsten wire
heated by an electric current and surrounded by a metal box with a hole in it. If
the wire is at a negative voltage with respect to the box, electrons emitted by the
wire will be accelerated toward the walls and some will pass through the hole.
All the electrons which come out of the gun will have (nearly) the same energy.
In front of the gun is again a wall (just a thin metal plate) with two holes in it. 37—4 DETECTOR
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WALL msroe Fig. 373.
with electrons. Interference experiment (0) Beyond the wall is another plate which will serve as a “backstop.” In front of the
backstop we place a movable detector. The detector might be a geiger counter or,
perhaps better, an electron multiplier, which is connected to a loudspeaker. We should say right away that you should not try to set up this experiment
(as you could have done with the two we have already described). This experiment
has never been done in just this way. The trouble is that the apparatus would have
to be made on an impossibly small scale to show the effects we are interested in.
We are doing a “thought experiment,” which we have chosen because it is easy to
think about. We know the results that would be obtained because there are many
experiments that have been done, in which the scale and the proportions have
been chosen to show the effects we shall describe. The ﬁrst thing we notice with our electron experiment is that we hear sharp
“clicks” from the detector (that is, from the loudspeaker). And all “clicks” are
the same. There are no “halfclicks.” We would also notice that the “clicks” come very erratically. Something like:
click . . . . . clickclick . . . click . . . . . . . . Click . . . . clickclick . . . . . . click . . . ,
etc., just as you have, no doubt, heard a geiger counter operating. If we count
the clicks which arrive in a sufficiently long time—say for many minutes—and
then count again for another equal period, we ﬁnd that the two numbers are very
nearly the same. So we can speak of the average rate at which the clicks are heard
(soandsomany clicks per minute on the average). As we move the detector around, the rate at which the clicks appear is faster
or slower, but the size (loudness) of each click is always the same. If we lower the
temperature of the wire in the gun the rate of clicking slows down, but still each
click sounds the same. We would notice also that if we put two separate detectors
at the backstop, one or the other would click, but never both at once. (Except that
once in a while, if there were two clicks very close together in time, our ear might
not sense the separation.) We conclude, therefore, that whatever arrives at the
backstop arrives in “lumps.” All the “lumps” are the same size: only whole
“lumps” arrive, and they arrive one at a time at the backstop. We shall say:
“Electrons always arrive in identical lumps.” Just as for our experiment with bullets, we can now proceed to ﬁnd experi
mentally the answer to the question: “What is the relative probability that an
electron ‘lump’ will arrive at the backstop at various distances x from the center?”
As before, we obtain the relative probability by observing the rate of clicks, holding
the operation of the gun constant. The probability that lumps will arrive at a
particular x is proportional to the average rate of clicks at that x. The result of our experiment is the interesting curve marked P12 in part (c)
of Fig. 37—3. Yes! That is the way electrons go. 37—5 The interference of electron waves Now let us try to analyze the curve of Fig. 37—3 to see whether we can under
stand the behavior of the electrons. The ﬁrst thing we would say is that since they
come in lumps, each lump, which we may as well call an electron, has come either
through hole 1 or through hole 2. Let us write this in the form of a “Proposition”: 37—5 I2 Fi2"'¢+¢zl2 (C) Proposition A: Each electron either goes through hole 1 or it goes through
hole 2. Assuming Proposition A, all electrons that arrive at the backstop can be di
vided into two classes: (1) those that come through hole 1, and (2) those that come
through hole 2. So our observed curve must be the sum of the effects of the elec
trons which come through hole 1 and the electrons which come through hole 2.
Let us check this idea by experiment. First, we will make a measurement for those
electrons that come through hole 1. We block off hole 2 and make our counts of
the clicks from the detector. From the clicking rate, we get P1. The result of the
measurement is shown by the curve marked P1 in part (b) of Fig. 37—3. The result
seems quite reasonable. In a similar way, we measure P2, the probability distribu
tion for the electrons that come through hole 2. The result of this measurement
is also drawn in the ﬁgure. The result P1 2 obtained with both holes open is clearly not the sum of P1 and
P2, the probabilities for each hole alone. In analogy with our waterwave experi
ment, we say: “There is interference.” For electrons: P12 75 P1 + P2. (37.5) How can such an interference come about? Perhaps we should say: “Well,
that means, presumably, that it is not true that the lumps go either through hole
1 or hole 2, because if they did, the probabilities should add. Perhaps they go in a
more complicated way. They split in half and . . .” But no! They cannot, they
always arrive in lumps . . . “Well, perhaps some of them go through 1, and then
they go around through 2, and then around a few more times, or by some other
complicated path . . . then by closing hole 2, we changed the chance that an elec
tron that started out through hole 1 would ﬁnally get to the backstop . . .” But
notice! There are some points at which very few electrons arrive when both holes
are open, but which receive many electrons if we close one hole, so closing one
hole increased the number from the other. Notice, however, that at the center
of the pattern, P1 2 is more than twice as large as P1 + P2. It is as though closing
one hole decreased the number of electrons which come through the other hole.
It seems hard to explain both effects by proposing that the electrons travel in
complicated paths. It is all quite mysterious. And the more you look at it the more mysterious
it seems. Many ideas have been concocted to try to explain the curve for P12 in
terms of individual electrons going around in complicated ways through the holes.
None of them has succeeded. None of them can get the right curve for P12 in
terms ofP1 and P2. Yet, surprisingly enough, the mathematics for relating P1 and P2 to P12 is
extremely simple. For P12 is just like the curve 112 of Fig. 37—2, and that was
simple. What is going on at the backstop can be described by two complex numbers
that we can call $1 and $2 (they are functions of x, of course). The absolute square
of 651 gives the effect with only hole 1 open. That is, P1 = [$1]? The effect with
only hole 2 open is given by $2 in the same way. That is, P2 = <i>22. And the
combined effect of the two holes is just P12 = ¢i>1 + $22. The mathematics
is the same as that we had for the water waves! (It is hard to see how one could
get such a simple result from a complicated game of electrons going back and forth
through the plate on some strange trajectory.) We conclude the following: The electrons arrive in lumps, like particles, and
the probability of arrival of these lumps is distributed like the distribution of
intensity of a wave. It is in this sense that an electron behaves “sometimes like a
particle and sometimes like a wave.” Incidentally, when we were dealing with classical waves we deﬁned the in
tensity as the mean over time of the square of the wave amplitude, and we used
complex numbers as a mathematical trick to simplify the analysis. But in quantum
mechanics it turns out that the amplitudes must be represented by complex num
bers. The real parts alone will not do. That is a technical point, for the moment,
because the formulas look just the same. 376 Since the probability of arrival through both holes is given so simply, although
it is not equal to (P1 + P 2), that is really all there is to say. But there are a large
number of subtleties involved in the fact that nature does work this way. We
would like to illustrate some of these subtleties for you now. First, since the num
ber that arrives at a particular point is not equal to the number that arrives through
1 plus the number that arrives through 2, as we would have concluded from
Proposition A, undoubtedly we should conclude that Proposition A is false. It is
not true that the electrons go either through hole 1 or hole 2. But that conclusion
can be tested by another experiment. 37—6 Watching the electrons We shall now try the following experiment. To our electron apparatus we
add a very strong light source, placed behind the wall and between the two holes,
as shown in Fig. 37—4. We know that electric charges scatter light. So when an
electron passes, however it does pass, on its way to the detector, it will scatter some
light to our eye, and we can see where the electron goes. If, for instance, an electron
were to take the path via hole 2 that is sketched in Fig. 37—4, we should see a ﬂash
of light coming from the vicinity of the place marked A in the ﬁgure. If an electron
passes through hole 1 we would expect to see a ﬂash from the Vicinity of the upper
hole. If it should happen that we get light from both places at the same time, because the electron divides in half . . . Let us just do the experiment!
f—"l E ',..,,§l>°u'!a'ce , I
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ELECTRON ‘ ‘ ’2 '7‘
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(0) Here is what we see: every time that we hear a “click” from our electron de
tector (at the backstop), we also see a ﬂash of light either near hole 1 or near hole
2, but never both at once! And we observe the same result no matter where we put
the detector. From this observation we conclude that when we look at the electrons
we ﬁnd that the electrons go either through one hole or the other. Experimentally,
Proposition A is necessarily true. What, then, is wrong with our argument against Proposition A? Why isn’t
P1 2 just equal to P1 + P2? Back to experiment! Let us keep track of the electrons
and ﬁnd out what they are doing. For each position (xlocation) of the detector
we will count the electrons that arrive and also keep track of which hole they went
through, by watching for the ﬂashes. We can keep track of things this way:
whenever we hear a “click” we will put a count in Column 1 if we see the ﬂash near
hole 1, and if we see the ﬂash near hole 2, we will record a count in Column 2.
Every electron which arrives is recorded in one of two classes: those which come
through 1 and those which come through 2. From the number recorded in Column
1 we get the probability Pl' that an electron will arrive at the detector via hole 1;
and from the number recorded in Column 2 we get P;, the probability that an
electron will arrive at the detector via hole 2. If we now repeat such a measurement
for many values of x, we get the curves for Pi and P; shown in part (b) of Fig. 37—4. Well, that is not too surprising! We get for P1 something quite similar to
what we got before for P 1 by blocking oﬂ hole 2; and P5 is similar to what we got
by blocking hole 1. So there is not any complicated business like going through
both holes. When we watch them, the electrons come through just as we would 377 (b) I I I
PIEPl +Pz (c) expect them to come through. Whether the holes are closed or open, those which
we see come through hole 1 are distributed in the same way whether hole 2 is open
or closed. But wait! What do we have now for the total probability, the probability that
an electron will arrive at the detector by any route? We already have that informa
tion. We just pretend that we never looked at the light ﬂashes, and we lump to
gether the detector clicks which we have separated into the two columns. We
must just add the numbers. For the probability that an electron will arrive at the
backstop by passing through either hole, we do ﬁnd Pig = P1 + P2. That is,
although we succeeded in watching which hole our electrons come through, we
no longer get the old interference curve P12, but a new one, P12, showing no
interference! If we turn out the light P12 is restored. We must conclude that when we look at the electrons the distribution of them
on the screen is diﬂerent than when we do not look. Perhaps it is turning on our
light source that disturbs things? It must be that the electrons are very delicate,
and the light, when it scatters oﬂ the electrons, gives them a jolt that changes their
motion. We know that the electric ﬁeld of the light acting on a charge will exert
a force on it. So perhaps we should expect the motion to be changed. Anyway,
the light exerts a big inﬂuence on the electrons. By trying to “watch” the electrons
we have changed their motions. That is, the jolt given to the electron when the
photon is scattered by it is such as to change the electron’s motion enough so that
if it might have gone to where P12 was at a maximum it will instead land where
P12 was a minimum; that is why we no longer see the wavy interference effects. You may be thinking: “Don’t use such a bright source! Turn the brightness
down! The light waves will then be weaker and will not disturb the electrons so
much. Surely, by making the light dimmer and dimmer, eventually the wave
will be weak enough that it will have a negligible effect.” O.K. Let’s try it. The
ﬁrst thing we observe is that the ﬂashes of light scattered from the electrons as
they pass by does not get weaker. It is always the samesized ﬂash. The only .hing
that happens as the light is made dimmer is that sometimes we hear a “click”
from the detector but see no flash at all. The electron has gone by without being
“seen.” What we are observing is that light also acts like electrons, we knew that
it was “wavy,” but now we ﬁnd that it is also “lumpy.” It always arrives—or is
scattered—in lumps that we call “photons.” As we turn down the intensity of
the light source we do not change the size of the photons, only the rate at which
they are emitted. That explains why, when our source is dim, some electrons get
by without being seen. There did not happen to be a photon around at the time
the electron went through. This is all a little discouraging. If it is true that whenever we “see” the electron
we see the samesized ﬂash, then those electrons we see are always the disturbed
ones. Let us try the experiment with a dim light anyway. Now whenever we hear
a click in the detector we will keep a count in three columns: in Column (1) those
electrons seen by hole 1, in Column (2) those electrons seen by hole 2, and in
Column (3) those electrons not seen at all. When we work up our data (computing
the probabilities) we ﬁnd these results: Those “seen by hole 1” have a distribution
like Pi; those “seen by hole 2” have a distribution like P; (so that those “seen by
either hole 1 or 2” have a distribution like Hz); and those “not seen at all” have a
“wavy” distribution just like P12 of Fig. 37—3! If the electrons are not seen, we
have interference! That is understandable. When we do not see the electron, no photon disturbs
it, and when we do see it, a photon has disturbed it. There is always the same
amount of disturbance because the light photons all produce the samesized effects
and the eﬂect of the photons being scattered is enough to smear out any inter
ference eﬂect. Is there not some way we can see the electrons without disturbing them?
We learned in an earlier chapter that the momentum carried by a “photon”
is inversely proportional to its wavelength (p = h/A). Certainly the jolt given
to the electron when the photon is scattered toward our eye depends on the
momentum that photon carries. Aha! If we want to disturb the electrons only 37—8 slightly we should not have lowered the intensity of the light, we should have
lowered its frequency (the same as increasing its wavelength). Let us use light of
a redder color. We could even use infrared light, or radiowaves (like radar), and
“see” where the electron went with the help of some equipment that can “see”
light of these longer wavelengths. If we use “gentler” light perhaps we can avoid
disturbing the electrons so much. Let us try the experiment with longer waves. We shall keep repeating our ex
periment, each time with light of a longer wavelength. At ﬁrst, nothing seems to
change. The results are the same. Then a terrible thing happens. You remember
that when we discussed the microscope we pointed out that, due to the wave nature
of the light, there is a limitation on how close two spots can be and still be seen
as two separate spots. This distance is of the order of the wavelength of light. So
now, when we make the wavelength longer than the distance between our holes,
we see a big fuzzy ﬂash when the light is scattered by the electrons. We can no
longer tell which hole the electron went through! We just know it went somewhere!
And it is just with light of this color that we ﬁnd that the jolts given to the electron
are small enough so that P; 2 begins to look like Ply—that we begin to get some
interference effect. And it is only for wavelengths much longer than the separation
of the two holes (when we have no chance at all of telling where the electron went)
that the disturbance due to the light gets sufﬁciently small that we again get the
curve P12 shown in Fig. 37—3. In our experiment we ﬁnd that it is impossible to arrange the light in such a
way that one can tell which hole the electron went through, and at the same time
not disturb the pattern. It was suggested by Heisenberg that the then new laws of
nature could only be consistent if there were some basic limitation on our experi
mental capabilities not previously recognized. He proposed, as a general principle,
his uncertainty principle, which we can state in terms of our experiment as follows:
“It is impossible to design an apparatus to determine which hole the electron passes
through, that will not at the same time disturb the electrons enough to destroy the
interference pattern.” If an apparatus is capable of determining which hole the elec
tron goes through, it cannot be so delicate that it does not disturb the pattern in
an essential way. No one has ever found (or even thought of) a way around the
uncertainty principle. So we must assume that it describes a basic characteristic
of nature. The complete theory of quantum mechanics which we now use to describe
atoms and, in fact, all matter depends on the correctness of the uncertainty prin
ciple. Since quantum mechanics is such a successful theory, our belief in the
uncertainty principle is reinforced. But if a way to “beat” the uncertainty principle
were ever discovered, quantum mechanics would give inconsistent results and
would have to be discarded as a valid theory of nature. “Well,” you say, “what about Proposition A? It is true, or is it not true,
that the electron either goes through hole 1 or it goes through hole 2?” The only
answer that can be given is that we have found from experiment that there is a
certain special way that we have to think in order that we do not get into incon
sistencies. What we must say (to avoid making wrong predictions) is the following.
If one looks at the holes or, more accurately, if one has a piece of apparatus which
is capable of determining whether the electrons go through hole 1 or hole 2, then
one can say that it goes either through hole 1 or hole 2. But, when one does not
try to tell which way the electron goes, when there is nothing in the experiment to
disturb the electrons, then one may not say that an electron goes either through
hole 1 or hole 2. If one does say that, and starts to make any deductions from the
statement, he will make errors in the analysis. This is the logical tightrope on
which we must walk if we wish to describe nature successfully. If the motion of all matter—as well as electrons—must be described in terms
of waves, what about the bullets in our ﬁrst experiment? Why didn’t we see an
interference pattern there? It turns out that for the bullets the wavelengths were so
tiny that the interference patterns became very ﬁne. So ﬁne, in fact, that with any 37—9 l 32 (smoothed) X § § (0) (b)
Fig. 37—5. Interference pattern with bullets: (a) actual (schematic), (b) ob served. detector of ﬁnite size one could not distinguish the separate maxima and minima.
What we saw was only a kind of average, which is the classical curve. In Fig. 37—5
we have tried to indicate schematically what happens with largescale objects.
Part (a) of the ﬁgure shows the probability distribution one might predict for
bullets, using quantum mechanics. The rapid wiggles are supposed to represent
the interference pattern one gets for waves of very short wavelength. Any physical
detector, however, straddles several wiggles of the probability curve, so that the
measurements show the smooth curve drawn in part (b) of the ﬁgure. 37—7 First principles of quantum mechanics We will now write a summary of the main conclusions of our experiments.
We will, however, put the results in a form which makes them true for a general
class of such experiments. We can write our summary more simply if we ﬁrst
deﬁne an “ideal experiment” as one in which there are no uncertain external
inﬂuences, i.e., no jiggling or other things going on that we cannot take into ac
count. We would be quite precise if we said: “An ideal experiment is one in which
all of the initial and ﬁnal conditions of the experiment are completely speciﬁed.”
What we will call “an event” is, in general, just a speciﬁc set of initial and ﬁnal
conditions. (For example: “an electron leaves the gun, arrives at the detector, and
nothing else happens”) Now for our summary. SUMMARY (1) The probability of an event in an ideal experiment is given by the square of
the absolute value of a complex number ¢ which is called the probability
amplitude. P = probability,
¢ = probability amplitude,
P = I442. (37.6) (2) When an event can occur in several alternative ways, the probability ampli
tude for the event is the sum of the probability amplitudes for each way
considered separately. There is interference. ¢ = ¢1 + (#2,
P = Im + ¢2I2 (3) If an experiment is performed which is capable of determining whether one or
another alternative is actually taken, the probability of the event is the sum
of the probabilities for each alternative. The interference is lost. (37.7) P = P1 + P2. (37.8) One might still like to ask: “How does it work? What is the machinery behind
the law?” No one has found any machinery behind the law. No one can “explain”
any more than we have just “explained.” No one will give you any deeper repre
sentation of the situation. We have no ideas about a more basic mechanism from
which these results can be deduced. We would like to emphasize a very important difference between classical and
quantum mechanics. We have been talking about the probability that an electron
will arrive in a given circumstance. We have implied that in our experimental
arrangement (or even in the best possible one) it would be impossible to predict
exactly what would happen. We can only predict the odds! This would mean, if
it were true, that physics has given up on the problem of trying to predict exactly
what will happen in a deﬁnite circumstance. Yes! physics has given up. We do
not know how to predict what would happen in a given circumstance, and we believe
now that it is impossible, that the only thing that can be predicted is the prob
ability of diﬁ'erent events. It must be recognized that this is a retrenchment in our
earlier ideal of understanding nature. It may be a backward step, but no one
has seen a way to avoid it. 3710 We make now a few remarks on a suggestion that has sometimes been made
to try to avoid the description we have given: “Perhaps the electron has some kind
of internal works—some inner variables—that we do not yet know about. Perhaps
that is why we cannot predict what will happen. If we could look more closely at
the electron we could be able to tell where it would end up.” So far as we know,
that is impossible. We would still be in difficulty. Suppose we were to assume that
inside the electron there is some kind of machinery that determines where it is
going to end up. That machine must also determine which hole it is going to go
through on its way. But we must not forget that what is inside the electron should
not be dependent on what we do, and in particular upon whether we open or close
one of the holes. So if an electron, before it starts, has already made up its mind
(a) which hole it is going to use, and (b) where it is going to land, we should ﬁnd
P1 for those electrons that have chosen hole 1, P 2 for those that have chosen hole
2, and necessarily the sum P1 + P2 for those that arrive through the two holes.
There seems to be no way around this. But we have veriﬁed experimentally that
that is not the case. And no one has ﬁgured a way out of this puzzle. So at the
present time we must limit ourselves to computing probabilities. We say “at the
present time,” but we suspect very strongly that it is something that will be with us forever—that it is impossible to beat that puzzle—that this is the way nature
really is. 37—8 The uncertainty principle This is the way Heisenberg stated the uncertainty principle originally: If you
make the measurement on any object, and you can determine the xcomponent of
its momentum with an uncertainty Ap, you cannot, at the same time, know its
xposition more accurately than Ax = h/Ap. The uncertainties in the position
and momentum at any instant must have their product greater than Planck’s
constant. This is a special case of the uncertainty principle that was stated above
more generally. The more general statement was that one cannot design equipment
in any way to determine which of two alternatives is taken, without, at the same
time, destroying the pattern of interference. Let us show for one particular case that the kind of relation given by Heisen
berg must be true in order to keep from getting into trouble. We imagine a modiﬁ
cation of the experiment of Fig. 37—3, in which the wall with the holes consists of a
plate mounted on rollers so that it can move freely up and down (in the xdirection),
as shown in Fig. 37—6. By watching the motion of the plate carefully we can try to
tell which hole an electron goes through. Imagine what happens when the detector
is placed at x = 0. We would expect that an electron which passes through hole 1
must be deﬂected downward by the plate to reach the detector. Since the vertical
component of the electron momentum is changed, the plate must recoil with an
equal momentum in the opposite direction. The plate will get an upward kick.
If the electron goes through the lower hole, the plate should feel a downward kick.
It is clear that for every position of the detector, the momentum received by the
plate will have a different value for a traversal via hole 1 than for a traversal via
hole 2. So! Without disturbing the electrons at all, but just by watching the plate,
we can tell which path the electron used. Now in order to do this it is necessary to know what the momentum of the
screen is, before the electron goes through. So when we measure the momentum
after the electron goes by, we can ﬁgure out how much the plate’s momentum has
changed. But remember, according to the uncertainty principle we cannot at the
same time know the position of the plate with an arbitrary accuracy. But if we do
not know exactly where the plate is we cannot say precisely where the two holes are.
They will be in a different place for every electron that goes through. This means
that the center of our interference pattern will have a different location for each
electron. The wiggles of the interference pattern will be smeared out. We shall show
quantitatively in the next chapter that if we determine the momentum of the plate
sufﬁciently accurately to determine from the recoil measurement which hole was
used, then the uncertainty in the xposition of the plate will, according to the un 37—11 ROLLERS
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WALL aacxsmp Fig. 37—6. An experiment in which
the recoil of the wall is measured. certainty principle, be enough to shift the pattern observed at the detector up and
down in the xdirection about the distance from a maximum to its nearest minimum.
Such a random shift is just enough to smear out the pattern so that no interference
is observed. The uncertainty principle “protects” quantum mechanics. Heisenberg recog
nized that if it were possible to measure the momentum and the position simultane
ously with a greater accuracy, the quantum mechanics would collapse. So he
proposed that it must be impossible. Then people sat down and tried to ﬁgure out
ways of doing it, and nobody could ﬁgure out a way to measure the position and
the momentum of anything—a screen, an electron, a billiard ball, anything—with
any greater accuracy. Quantum mechanics maintains its perilous but accurate
existence. 37—12 ...
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 Spring '09
 LeeKinohara
 Physics, Uncertainty Principle, Electrons

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