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Unformatted text preview: The Dependence of Amplitudes on Time 7—1 Atoms at rest; stationary states We want now to talk a little bit about the behavior of probability amplitudes
in time. We say a “little bit,” because the actual behavior in time necessarily
involves the behavior in space as well. Thus, we get immediately into the most
complicated possible situation if we are to do it correctly and in detail. We are
always in the difﬁculty that we can either treat something in a logically rigorous
but quite abstract way, or we can do something which is not at all rigorous but
which gives us some idea of a real situationipostponing until later a more careful
treatment. With regard to energy dependence, we are going to take the second
course. We will make a number of statements. We will not try to be rigorous—but
will just be telling you things that have been found out, to give you some feeling
for the behavior of amplitudes as a function of time. As we go along, the precision
of the description will increase, so don’t get nervous that we seem to be picking
things out of the air. It is, of course, all out of the air—the air of experiment and
of the imagination of people. But it would take us too long to go over the historical
development, so we have to plunge in somewhere. We could plunge into the ab
stract and deduce everything—which you would not understand—or we could
go through a large number of experiments to justify each statement. We choose
to do something in between. An electron alone in empty space can, under certain circumstances, have a
certain deﬁnite energy. For example, if it is standing still (so it has no translational
motion, no momentum, or kinetic energy), it has its rest energy. A more compli
cated object like an atom can also have a deﬁnite energy when standing still, but
it could also be internally excited to another energy level. (We will describe later
the machinery of this.) We can often think of an atom in an excited state as having
a deﬁnite energy, but this is really only approximately true. An atom doesn’t
stay excited forever because it manages to discharge its energy by its interaction
with the electromagnetic ﬁeld. So there is some amplitude that a new state is
generated—with the atom in a lower state, and the electromagnetic ﬁeld in a higher
state, of excitation. The total energy of the system is the same before and after,
but the energy of the atom is reduced. So it is not precise to say an excited atom
has a deﬁnite energy; but it will often be convenient and not too wrong to say that
it does. [Incidentally, why does it go one way instead of the other way? Why does an
atom radiate light? The answer has to do with entropy. When the energy is in the
electromagnetic ﬁeld, there are so many different ways it can be—so many diﬁerent
places where it can wander—that if we look for the equilibrium condition, we
ﬁnd that in the most probable situation the ﬁeld is excited with a photon, and the
atom is deexcited. It takes a very long time for the photon to come back and ﬁnd
that it can knock the atom back up again. It’s quite analogous to the classical
problem: Why does an accelerating charge radiate? It isn’t that it “wants” to lose
energy, because, in fact, when it radiates, the energy of the world is the same as it
was before. Radiation or absorption goes in the direction of increasing entropy.] Nuclei can also exist in different energy levels, and in an approximation which
disregards the electromagnetic effects, we can say that a nucleus in an excited state
stays there. Although we know that it doesn’t stay there forever, it is often useful
to start out with an approximation which is somewhat idealized and easier to
think about. Also it is often a legitimate approximation under certain circum
stances. (When we ﬁrst introduced the classical laws of a falling body, we did not
include friction, but there is almost never a case in which there isn’t some friction.) 7—1 7—1 Atoms at rest; stationary states
7—2 Uniform motion 7—3 Potential energy; energy
conservation 74 Forces; the classical limit 7—5 The “precession” of a spin
onehalf particle Review: Chapter 17, Vol. I, SpaceTime
Chapter 48, Vol. 1, Beats Then there are the subnuclear “strange particles,” which have various masses.
But the heavier ones disintegrate into other light particles, so again it is not correct
to say that they have a precisely deﬁnite energy. That would be true only if they
lasted forever. So when we make the approximation that they have a deﬁnite
energy, we are forgetting the fact that they must blow up. For the moment, then,
we will intentionally forget about such processes and learn later how to take them
into account. Suppose we have an atom—~or an electron, or any particle—which at rest
would have a deﬁnite energy E 0. By the energy E 0 we mean the mass of the whole
thing times c2. This mass includes any internal energy; so an excited atom has a
mass which is diﬁerent from the mass of the same atom in the ground state. (The
ground state means the state of lowest energy.) We will call E 0 the “energy at rest.” For an atom at rest, the quantum mechanical amplitude to ﬁnd an atom at a
place is the same everywhere; it does not depend on position. This means, of course,
that the probability of ﬁnding the atom anywhere is the same. But it means even
more. The probability could be independent of position, and still the phase of the
amplitude could vary from point to point. But for a particle at rest, the complete
amplitude is identical everywhere. It does, however, depend on the time. For a
particle in a state of deﬁnite energy E o, the amplitude to ﬁnd the particle at (x, y, 2)
at the time t is ae—1(E0/ﬁ)t , (7.1) where a is some constant. The amplitude to be at any point in space is the same
for all points, but depends on time according to (7.1). We shall simply assume
this rule to be true. Of course, we could also write (7.1) as tie—1"”, (7.2)
with hw = E0 = Mcz, where M is the rest mass of the atomic state, or particle. There are three different
ways of specifying the energy: by the frequency of an amplitude, by the energy in
the classical sense, or by the inertia. They are all equivalent; they are just different
ways of saying the same thing. You may be thinking that it is strange to think of a “particle” which has
equal amplitudes to be found throughout all space. After all, we usually imagine
a “particle” as a small object located “somewhere.” But don’t forget the uncer
tainty principle. If a particle has a deﬁnite energy, it has also a deﬁnite momentum.
If the uncertainty in momentum is zero, the uncertainty relation, Ap Ax = it,
tells us that the uncertainty in the position must be inﬁnite, and that is just what
we are saying when we say that there is the same amplitude to ﬁnd the particle
at all points in space. If the internal parts of an atom are in a different state with a different total
energy, then the variation of the amplitude with time is diﬁerent. If you don’t
know in which state it is, there will be a certain amplitude to be in one state and a
certain amplitude to be in another—and each of these amplitudes will have a dif
ferent frequency. There will be an interference between these diﬁerent components
—like a beatnote—which can show up as a varying probability. Something will
be “going on” inside of the atomeven though it is “at rest” in the sense that its
center of mass is not drifting. However, if the atom has one deﬁnite energy, the
amplitude is given by (7.1), and the absolute square of this amplitude does not
depend on time. You see, then, that if a thing has a deﬁnite energy and if you ask
any probability question about it, the answer is independent of time. Although
the amplitudes vary with time, if the energy is deﬁnite they vary as an imaginary
exponential, and the absolute value doesn’t change. That’s why we often say that an atom in a deﬁnite energy level is in a stationary
state. If you make any measurements of the things inside, you’ll ﬁnd that nothing
(in probability) will change in time. In order to have the probabilities change in 7—2 time, we have to have the interference of two amplitudes at two different frequencies,
and that means that we cannot know what the energy is. The object will have one
amplitude to be in a state of one energy and another amplitude to be in a state of
another energy. That’s the quantum mechanical description of something when
its behavior depends on time. If we have a “condition” which is a mixture of two different states with differ
ent energies, then the amplitude for each of the two states varies with time according
to Eq. (7.2), for instance, as —i(E1/ﬁ)t e—i(E2/if)t e and (7.3) And if we have some combination of the two, we will have an interference. But
notice that if we added a constant to both energies, it wouldn’t make any difference.
If somebody else were to use a different scale of energy in which all the energies
were increased (or decreased) by a constant amount—say, by the amount A—then
the amplitudes in the two states would, from his point of view, be Ai(E1+A)I/ﬁ e—i(E2+A)t/ﬁ e and (74) All of his amplitudes would be multiplied by the same factor e"“’“””, and all
linear combinations, or interferences, would have the same factor. When we take
the absolute squares to ﬁnd the probabilities, all the answers would be the same.
The choice of an origin for our energy scale makes no difference; we can measure
energy from any zero we want. For relativistic purposes it is nice to measure the
energy so that the rest mass is included, but for many purposes that aren’t rela
tivistic it is often nice to subtract some standard amount from all energies that
appear. For instance, in the case of an atom, it is usually convenient to subtract
the energy Mscz, where M, is the mass of all the separate pieces—the nucleus and
the electrons—which is, of course, different from the mass of the atom. For other
problems it may be useful to subtract from all energies the amount Macz, where
M9 is the mass of the whole atom in the ground state; then the energy that appears
is just the excitation energy of the atom. So, sometimes we may shift our zero of
energy by some very large constant, but it doesn’t make any difference, provided
we shift all the energies in a particular calculation by the same constant. So much
for a particle standing still. 72 Uniform motion If we suppose that the relativity theory is right, a particle at rest in one inertial
system can be in uniform motion in another inertial system. In the rest frame of
the particle, the probability amplitude is the same for all x, y, and 2 but varies with
t. The magnitude of the amplitude is the same for all 1, but the phase depends on t.
We can get a kind of a picture of the behavior of the amplitude if we plot lines of
equal phase~say, lines of zero phase—as a function of x and I. For a particle at
rest, these equalphase lines are parallel to the xaxis and are equally spaced in
the t—coordinate, as shown by the dashed lines in Fig. 7—1. In a different frame—x’, y’, z’, t’—that is moving with respect to the particle
in, say, the xdirection, the x’ and t’ coordinates of any particular point in space
are related to ac and t by the Lorentz transformation. This transformation can be
represented graphically by drawing x’ and t’ axes, as is done in Fig. 741. (See
Chapter 17, Vol. I, Fig. 17—2.) You can see that in the x’t’ system, points of equal
phaseT have a different spacing along the t’axis, so the frequency of the time
variation is different. Also there is a variation of the phase with x’, so the prob
ability amplitude must be a function of x’. 1‘ We are assuming that the phase should have the same value at corresponding points
in the two systems. This is a subtle point, however, since the phase of a quantum me
chanical amplitude is, to a large extent, arbitrary. A complete justiﬁcation of this assump
tion requires a more detailed discussion involving interferences of two or more amplitudes. 7—3 Fig. 7].
of the amplitude of a particle at rest in
the xf systems. Relativistic transformation Under a Lorentz transformation for the velocity 1.», say along the negative
xdirection, the time t is related to the time I’ by t = t’ — x’v/c2
V1 —— v2/c2 ’ so our amplitude now varies as e—u/mb'ot _ e—(i/VLMEOt’/\/1—1)2/‘oil—Eo‘uac'tc2 ‘/l—712/(22) In the prime system it varies in space as well as in time. If we write the amplitude as —(i’ﬁ)(E' t’—p'r’)
e ‘ P , we see that E,’, = EU/x/l —— 112/02 is the energy computed classically for a
particle of rest energy E0 travelling at the velocity Z), and p’ = E,§,v/c2 is the
corresponding particle momentum. You know that x, = (t, x, y, z) and 1),, = (E, pr, p1), p,) are fourvectors, and
that pﬂxﬂ = Et — p  x is a scalar invariant. In the rest frame of the particle,
pm, is just El; so if we transform to another frame, E1 will be replaced by Elt! __ pl . x1.
Thus, the probability amplitude of a particle which has the momentum p will be proportional to
e~(i/h)(Ept—px) , (7.5) where E, is the energy of the particle whose momentum is p, that is, E12 2 \/(pc)2 + E5, (7.6)
where E 0 is, as before, the rest energy. For nonrelativistic problems, we can write
E, = M,c'~’ + W,,, (7.7) where W1, is the energy over and above the rest energy M,c2 of the parts of the
atom. In general, W1, would include both the kinetic energy of the atom as well
as its binding or excitation energy, which we can call the “internal” energy. We would write
2 p»;
2M Wp = Wint —l_ and the amplitudes would be [WW—W). (7.9) Because we will generally be doing nonrelativistic calculations, we will use this
form for the probability amplitudes. Note that our relativistic transformation has given us the variation of the
amplitude of an atom which moves in space without any additional assumptions.
The wave number of the space variations is, from (7.9), _ 12.
k _ h, (7.10) so the wavelength is =2I=£ A k p (7.11) This is the same wavelength we have used before for particles with the momentum
p. This formula was ﬁrst arrived at by de Broglie in just this way. For a moving
particle, the frequency of the amplitude variations is still given by hw = W... (7.12)
74 The absolute square of (7.9) is just 1, so for a particle in motion with a
deﬁnite energy, the probability of ﬁnding it is the same everywhere and does not
change with time. (It is important to notice that the amplitude is a complex wave.
If we used a real sine wave, the square would vary from point to point, which
would not be right.) We know, of course, that there are situations in which particles move from
place to place so that the probability depends on position and changes with time.
How do we describe such situations? We can do that by considering amplitudes
which are a superposition of two or more amplitudes for states of deﬁnite energy.
We have already discussed this situation in Chapter 48 of Vol. I——even for prob
ability amplitudes! We found that the sum of two amplitudes with different wave
numbers k (that is, momenta) and frequencies w (that is, energies) gives inter
ference humps, or beats, so that the square of the amplitude varies with space
and time. We also found that these beats move with the socalled “group velocity”
given by where Ak and Aw are the differences between the wave numbers and frequencies
for the two waves. For more complicated waves—made up of the sum of many
amplitudes all near the same frequency—the group velocity is dw
v” = a}. (7.13)
Taking w = Ep/h and k = p/f’i, we see that
dB
1),, = (7.14)
Using Eq. (7.6), we have
dEp _ 2 i.
71; — c Ep (7.15)
But Ep : Mc2, so
dEp P
Up— A—l (7.16) which is just the classical velocity of the particle. Alternatively, if we use the non
relativistic expressions, we have W
w=7p and k=%,
and
dw_dW_ d p2 _ p
at — :1;— a; (in) — 17! (7'17) which is again the classical velocity. Our result, then, is that if we have several amplitudes for pure energy states
of nearly the same energy, their interference gives “lumps” in the probability that
move through space with a velocity equal to the velocity of a classical particle
of that energy. We should remark, however, that when we say we can add two
amplitudes of ditTerent wave number together to get a beatnote that will corre
spond to a moving particle, we have introduced something new—something that
we cannot deduce from the theory of relativity. We said what the amplitude did
for a particle standing still and then deduced what it would do if the particle were
moving. But we cannot deduce from these arguments what would happen when
there are two waves moving with different speeds. If we stop one, we cannot stop
the other. So we have added tacitly the extra hypothesis that not only is (7.9) a
possible solution, but that there can also be solutions with all kinds of p’s for the
same system, and that the different terms will interfere. 7—5 ; +
Fig. 7—2. A particle of mass M and momentum p in a region of constant
potential. Re (Am p) >
DlST a x. e were [FOR ¢2‘ 4") Fig. 7—3. The amplitude for a par
ticle in transit from one potential to
another. 7—3 Potential energy; energy conservation Now we would like to discuss what happens when the energy of a particle
can change. We begin by thinking of a particle which moves in a force ﬁeld de
scribed by a potential. We discuss ﬁrst the efl‘ect of a constant potential. Suppose
that we have a large metal can which we have raised to some electrostatic potential
(1). as in Fig. 7—2. If there are charged objects inside the can, their potential energy
will be qu, which we will call V, and will be absolutely independent of position.
Then there can be no change in the physics inside, because the constant potential
doesn’t make any difference so far as anything going on inside the can is concerned.
Now there is no way we can deduce/what the answer should be, so we must make
a guess. The guess which works is more or less what you might expect: For the
energy, we must use the sum of the potential energy V and the energy Ep—which
is itself the sum of the internal and kinetic energies. The amplitude is proportional to
e»ri/ﬁ>t<Ep+V) t—Pxr (7.18) The general principle is that the coefﬁcient of t, which we may call co, is always given by the total energy of the system: internal (or “mass”) energy, plus kinetic
energy, plus potential energy: hw=Ep+ V. Or, for nonrelativistic situations, (7.19) 0 m = Wint + + V. (7.20) Now what about physical phenomena inside the box? If there are several diﬁercnt energy states, what will we get? The amplitude for each state has the
same additional factor
ear/mm over what it would have with V = 0. That is just like a change in the zero of our
energy scale. It produces an equal phase change in all amplitudes, but as we have
seen before, this doesn’t change any of the probabilities. All the physical phenomena
are the same. (We have assumed that we are talking about different states of the
same charged object, so that qd; is the same for all. If an object could change its
charge in going from one state to another, we would have quite another result,
but conservation of charge prevents this.) So far, our assumption agrees with what we would expect for a change of
energy reference level. But if it is really right, it should hold for a potential energy
that is not just a constant. In general, V could vary in any arbitrary way with
both time and space, and the complete result for the amplitude must be given in
terms of a differential equation. We don’t want to get concerned with the general
case right now, but only want to get some idea about how some things happen,
so we will think only of a potential that is constant in time and varies very slowly
in space. Then we can make a comparison between the classical and quantum ideas. Suppose we think of the situation in Fig. 7—3, which has two boxes held at
the constant potentials ¢1 and ¢2 and a region in between where we will assume
tha...
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