|HPS 0410||Einstein for Everyone|
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Department of History and Philosophy of Science
University of Pittsburgh
|The amplitude of a water wave is just the height
of the water level above or below the mean water level. The
amplitude of a light wave is just the strength of the electric and
magnetic fields that comprise it.
But what is the amplitude of a wave in quantum theory that represents a particle?
In the simple cases dealt with in these chapters, the amplitude of quantum wave is a complex number. Such a number is a sum of two parts: an ordinary real number and an "imaginary number." An imaginary number is some multiple of i, the square root of minus one.
So the amplitude of the wave can be things like 1, i, -1 -i and their multiples and sums, such as 1+i, 1-i, 37+23i, and so on.
If you have not seen it before, it will seem perverse to
take the square root of minus one seriously. Don't we all learn that
"(plus) x (plus) = (plus)" and
"(minus) x (minus) = (plus)" ?
So how can there be a number that, when multiplied by itself, gives us -1? If the number is negative, it will give us a positive number when multiplied by itself. If it is positive, the same will happen.
The simple answer is that all these worries are well placed for ordinary "real" numbers like 2, 7, -4.3, and so on. But i is a different sort of number. It is not a real number. It is an extension to the real number system.
Can we do this? Add to the number system? Yes--we do it all the time. It is fine as long as we are clear about the properties of the extension.
Negative numbers are familiar example of how we extend our number systems. If we think of natural numbers as just a way of counting things like apples in a basket, then we can make sense of the number 3: there are just 3 apples in the basket.
|But what about -3? How can we make sense of -3
apples in the basket? We can and do manage by thinking of "-3
apples" as an intermediate in calculation, such as in the
sample problem opposite.
All we need to know that "-3" apples has the key property that -3+7 = 4. That is, -3 when added to another number, cancels out 3 positive units.
-3 + 3 = 0
|How many apples are in the basket if we start with
1 apple, put 7 in ("+7") and take four out ("-4")? We have 1, 7 and
-4, all to be added together. It would be nice if we could add them
up in any order. But if we start with 1 - 4 and then add 7, our
intermediate quantity is 1-4 = -3.
If we allow -3 as an intermediate, then we free up our calculation. We can group the additions any way we please, but still end up with the right final result. That we never actually see -3 applies in the basket is no cause of concern.
The situation is the same with the imaginary number i. We will never have an imaginary distance in space or an imaginary time elapsed. However i will be very useful to us as an intermediate. What we need to know is that its defining formal property is just this:
i x i = -1
|We can mostly treat i as we would any other
number. We can add up i's. i + i + i = 3i. We can multiply them by
real numbers. 5 x 3i = 15i.
We can create complex numbers by adding real and imaginary numbers: 3 + 7i.
However there are some manipulations that do not go as you'd expect and these are used to generate recreational paradoxes.
|Writing sqrt for "square root," here's a standard
i = i
sqrt(-1) = sqrt(-1)
sqrt(-1/1) = sqrt(1/-1)
sqrt(-1)/sqrt(1) = sqrt(1)/sqrt(-1)
sqrt(-1)/1 = 1/sqrt(-1)
sqrt(-1) x sqrt(-1) =1
-1 = 1
Of course, 1 does not equal -1. The trick in the paradox is that these operations are all licit for real numbers, but one of them fails for complex numbers.
Which one fails?
Hint: Does sqrt(1/-1) = sqrt(1)/sqrt(-1) ?
It will turn out that i is very useful for us when we want
to represent how waves propagate in space according quantum theory.
To see how that comes about, we will first look at a very simple classical system. Imagine a classical particle moving at unit speed in some fixed direction. Say it starts at some position we label x=0. We can keep track of its position at subsequent times merely by adding one unit of distance to the position for each unit of time elapsed.
The rule of evolution in time is a simple "add one for each unit of time."
(one for each
unit of time)
When a wave of constant wavelength propagates, it maintains its shape but just shifts its location. Would it be possible to have a comparably simple rule for the time evolution of this wave?
Simply adding to the wave's amplitude does not work. It merely shifts the wave amplitude up. It doesn't move the wave in the direction of its motion. The wave peaks and troughs stay where they are.
What about multiplying the wave by some factor? That seems to work no better. Each time we multiply the wave amplitude by a number, we just increase the height of the peaks and the depth of the troughs without moving them in space.
It turns out that multiplying by a factor is very close to what we need to move the wave forward in time. Where it goes wrong is that we are (tacitly) using real numbers as amplitudes. Keep multiplying a positive real number by the same factor and the positive numbers keep getting bigger and the negative ones keep getting smaller. The wave doesn't move.
If however, the wave has a complex amplitude, then multiplication can have exactly the right effect. This comes from the peculiar properties of i. Let's start with 1 and keep multiplying it by i. We get
1 x i = i
i x i = -1
-1 x i = i x i x i = -i
-i x i = i x i x i x i = 1
That is, repeatedly multiplying by i
leads us to cycle
from 1 to i to -1 to -i and then finally back to 1.
We now have cyclic behavior that would not be possible just by multiplying a real number amplitude by the same real number over and over. This cyclic behavior is exactly what we need for waves that cycle periodically through their states.
If the wave amplitude is multiplied by i repeatedly, then the wave amplitude will cycle through these values 1, i, -1, -i and 1. After four multiplications by i, the amplitude returns to its original amplitude. Here is an amplitude, cycling through the space of complex numbers under these repeated multiplications.
We form a wave by distributing these wave amplitudes
through space. In distributing the amplitudes, we can adjust two
properties of the amplitudes:
• Magnitude: this is, informally speaking, the length of the arrow in the pictures given here. It is computed formally as the "norm" of the complex number amplitude. For amplitude 1, the magnitude is one; for amplitude i, it is one; for amplitude 3i, it is 3; and so on.
• Phase: this is, informally speaking, the direction in complex number space of the amplitude. Differences of phase produce interference phenomena. If a wave of amplitude i meets another of i, they interfere constructively to produce an increased amplitude of 2i: i + i = 2i. If, however, the second wave is of amplitude -i, then they interfere destructively and produce a zero amplitude: i - i = 0.
Now consider a wave of everywhere constant wavelength that, in quantum theory, represents a particle of definite momentum. We consider just a one dimension of space in the figure below: left to right represents the spatial extension of the wave. We use the remaining dimensions for the representation of the complex amplitudes. Up-down represents the imaginary +i and -i directions. Out-in represents the real +1 and -1 directions.
For this special case, the magnitude is constant. In the figure this is represented by the constant length of the amplitude arrow. The phase, however, advances uniformly with distance. That is, as we advance along the wave, the amplitude cycles through the values 1, i, -1, -i, 1 and, of course, all the infinitely many values in between. What results is a wave that is represented by the helical ("corkscrew") shape shown.
We can now solve the puzzle in the last chapter of the apparent inhomogeneities in the usual portrayal of a wave. Does the wave really vanish at the points indicated in red? That would mean that the wave is not really completely homogeneous:
|The illusion that the wave periodically vanishes in amplitude is merely an artefact of an incomplete representation of the wave. It arises by taking a slice through the "corkscrew" shape to show just values in one direction in the space of complex numbers used to represent amplitudes.The grey shape results when we take a vertical slice through the corkscrew.||The term "slice" here is not exactly correct. What we are doing is looking just at one component of the complex number amplitude. In this case, if the imaginary number direction is up-down, then we are considering just the imaginary part of the amplitude. If, for example, the amplitude is somewhere (1/2+i/2), this "slice" would have an amplitude of i/2. "Projection" might be a better term to use, but slice is more visceral.|
If we separate out the grey shape, we recover something like the usual representation of a plane wave. It gives the illusion of the wave repeatedly vanishing in amplitude and different places in space.
Recall that the attraction of a complex number amplitude was that it might enable a very simple multiplication rule for the time evolution of the wave. We can now see how that simple rule will evolve the wave in time. Multiplying the whole curve by i repeatedly will relocate the amplitudes in a way that gives us propagation of the wave. This is the basic rule of time evolution of quantum theory.
How this multiplication results in a wave propagation is shown in the figure below, for a small section of the wave. At each point along the wave, the amplitude is multiplied by i. The effect is to rotate each amplitude by one quarter turn around the space of values employed. Even though each multiplication affects only the amplitude at one point, the combination of these effects results in the wave advancing as shown. For example, if the wave had amplitude i at some position in space, that amplitude i will be relocated to a new position that turns out to be displaced by one quarter of the wavelength of the wave in the direction of its propagation.
Since multiplying by i takes the amplitude just one quarter the way through its space of values, four multiplications by i takes it through one full period. That is, we multiply by i x i x i x i = i4 = 1. So the rule in its simplest form is just
"multiply by i for each quarter period"
Multiplication by different numbers advances the wave through a time that is any desired fraction of its full period. We just need to scale the factor by that fraction.
The scaling is a little complicated, however, since the
factor must have unit magnitude, so that the magnitude of the wave
amplitude is unchanged. Multiplying by i advances the wave by a quarter
period. Multiplying by 1 leaves the wave unchanged. So, following the
scaling rule above, to advance it by 1/8th
period, we expect to use a complex number that is midway between 1 and i.
You might expect this number just to be their average: (1+i)/2. It is
close, but not quite the right one, since this number does not have unit
magnitude. Instead we need to use (1+i)/sqrt(2), where "sqrt(2)" is the
square root of two.
The factor by which we multiply affects only the phase of
the wave amplitudes. Hence it is known as a "phase factor." We can now
state the simple multiplication rule for time evolution of a wave of
constant wavelength. This is what results when Schroedinger's wave
equation is applied to the wave and it yields the rule
of Schroedinger evolution for a wave of constant wavelength:
(scaled to i for each
quarter wave period)
Here's an animation of the propagation. As the helix propagates to the right, the red spot traces out the amplitude of the wave in the vertical direction at one fixed location of space. It traces over the gray outline, which represents the amplitude in the vertical direction.
This depiction of the time evolution of a quantum wave allows us to solve another little puzzle in quantum theory. Recall that the Schroedinger evolution of the wave is deterministic. That means that the present state of the wave fixes its future state.
Now consider a wave of constant wavelength. We usually draw it something like this:
The arrows indicate the direction of propagation expected for the wave. Selecting the correct direction of propagation is the problem. The waveform as drawn here is symmetric as far as the left and right are concerned. There seems to be no property of the wave itself that would distinguish right over left. We have to add in the arrows to indicate the direction of propagation. But the determinism of Schroedinger evolution requires that some property of the wave picks out its direction of propagation. Otherwise its present state does not fix its future state. So how does the wave "know" that it should propagate to the right?
Why can it not do this?
Here is another way to state the problem. Schroedinger evolution is time reversible. Any propagation that goes one way in time can also go the other way in time. If there's a wave that propagates to the right, then its reversal in time--a wave propagating to the left--is also possible. But if both these waves have the same state at one moment, as these figures suggest, then they must belong both to a wave propagating to the right and to one propagating to the left. That contradicts the determinism of Schroedinger evolution. It looks like we cannot have both time reversibility and deterministic time evolution.
The solution to the "which way?" puzzle is the same as the solution to the vanishing amplitude problem. The pictures above are incomplete. They just show a single slice through the complex space of amplitude values. The fuller picture is of complex-valued wave amplitudes that wind in a helix around a spatial axis in the direction of propagation. To recall, the picture from above is:
We saw above that this wave will propagate to the right. It "knows" that it should propagate to the right because this direction is encoded in the handedness of the helix. To see this handedness, consider a helix with a reversed handedness:
This reversed helix encodes a direction of propagation to the left.
|The difference of handedness is similar to the handedness
of ordinary hardware screws.
The universal standard is a right-handed thread that is tightened by a clockwise turn, looking from the screw head.
public domain image from https://pixabay.com/en/screw-phillips-roundhead-2066580/
|A left-handed screw, however, tightens by an anti-clockwise turn, looking from the screw head.|
|A caution: these real-world screws inhabit an ordinary three dimensional space. The helices pictured above for quantum waves, however, are in a different sort of space. The left-right dimension is an ordinary spatial dimension. The two remaining dimensions, however, belong to the complex space of wave amplitudes. They are not ordinary spatial dimensions.|
The paradox arises when we consider just a slice of the helix. That gives a flatten figure to represent the wave. We then have the same figure for both helices:
The distinguishing property of the handedness of the helix is lost when we use these flattened picture. With it we lose the property that will determine the direction of propagation of the wave. This distinguishing property is encoded in what are described as "phase differences" in the next section.
The conclusion above is that a constant wavelength quantum wave is completely homogeneous in space after all. We have seen that the wave amplitude never vanishes, contrary to the pictures usually drawn. However there still remains an awkwardness. The wave amplitude is, at different places, 1, i, -1, -i and any of the many values mediating between them. 1 and i are different. So is this not an inhomogeneity? Is there not a physical difference between these amplitudes?
In principle, there might be a difference. Perhaps some other physical theory might exploit it. However quantum theory does not exploit it. Rather the theory treats the difference as a "gauge freedom." That is, it is a difference in the mathematical description that has no physical consequences.
A good enough analog is the way clocks read in different time zones in geography. It may be one o'clock in California, by the clocks set to its Pacific time zone; and four o'clock in Pennsylvania, by the clocks set to its Eastern time. However, physically, they represent the same time.
If we want to talk unambiguously about the time of some event, merely saying that it happened at one o'clock is not enough. We have to specify the time zone. Analogously, we "fix a gauge" to communicate our descriptions of the physical properties of a quantum wave unambigiously.
This lack of absolute meaning applies to the times, in the sense of the numbers assigned to the various moments ("one o'clock"). By using a clock in a suitably selected time zone, we can assign any time number we please to some chosen event. This lack of physical meaning does not apply to time differences. Take for example, times elapsed for some process, such as a one hour TV broadcast that is viewed in all the time zones. No matter which time zone's clock we use to measure the broadcast duration, we will always recover the same one hour length from differences in the clock times.
Correspondingly, differences of amplitude in a quantum
wave are physically significant. They matter
physically in two ways:
Magnitude ratios: We saw in an earlier chapter that, through the Born rule, the magnitude of the wave amplitude at some position in space determines the probability that measurement will reveal the particle to be located at that position. It is actually not the magnitude at each point in isolation. Rather, it is the magnitude at one point in relation to the magnitudes at other points. The ratio of these magnitudes squared give, by the Born rule, the ratio of probabilities.
Phase differences: When portions of a wave interfere, phase differences determine whether the superposition produces constructive inference (e.g. i + i = 2i) or destructive interference (e. g. i - i =0). Magnitudes also enter into these interference effects. We get different results if we change the magnitude of the waves. (e. g. i - i = 0, but 2i - i = i.)
Otherwise, the amplitude assigned to some particular point in the wave, by itself, considered in isolation, has no physical meaning. We can always rescale it to any amplitude we like merely by multiplying the wave overall by some suitably chosen complex number. To do that is the analog of choosing a clock from some other time zone in order to secure some desired time number. Multiplying the whole wave by our chosen factor alters the amplitudes of the entire wave. In the same way, choosing a clock from a different time zone, alters the time numbers assigned to all events in the process tracked. Neither changes the physically meaningful differences of amplitude or of time numbers.
You may wonder now how multiplication by a phase factor can yield the physically significant time evolution of the wave. This evolution is actually dependent on phase differences. We must start with some initial wave. When we evolve the wave forward in time by multiplication, the change in the evolved state is judged in relation to the initial state. Evolution manifests in the differences between the initial and evolved wave.
We saw in the example above that multiplying a wave's amplitude everywhere by i, four times, takes it through a complete cycle. We multiply it by i x i x i x i = i4 = 1. The time of this cycle is the period of the wave motion. During that time, the wave propagates in space through one wavelength. The two factors together, the period and wavelength, determine the speed of propagation. Since speed is just distance covered over time, that means that the speed of the wave is given by:
speed = wavelength / period
We have seen already that the wavelength is fixed by the particle's momentum. The period is fixed by the particle's energy. This last result is actually familiar to us. We have from Planck's original relationship that energy and frequency are related. However the period of a wave is just the inverse of the frequency: 1/period. Planck's relation tells us:
Energy = h / period
Thus we arrive at the important conclusion that the energy and momentum of a particle fix its speed of propagation, since these two quantities in turn fix its period and its wavelength.
What is common to all these cases is that the propagation of the wave can be represented fully merely by multiplying the wave repeatedly by i, pacing ourselves so that we multiply by i four times for each period. All this applies to waves of a single wavelength, that is waves of a definite momentum. It constitutes the Schroedinger time evolution of these waves, as described above and in the preceding chapter.
There are many more quantum waves, however, than this easy case of waves of constant wavelength. One might imagine that expressing Schroedinger evolution for these other cases might be difficult, or at least require mathematics much fancier than simple multiplication by i four times per period.
The amazing thing is that this simple multiplication is pretty much the whole story! The key fact is that any wave can be expressed as a sum (i.e. superposition) of waves of definite wavelength. Here for example is some arbitrarily shaped curve and the components of constant wavelength that are summed to produce it. (As usual, the figure only shows a single slice through the complicated corkscrew shapes that would more fully represent the waves.)
To find how some arbitrarily chosen wave like this evolves in time, we merely need carry out this simple three step procedure:
1. Decompose the
wave into component waves of constant wavelength.
2. Determine how each component wave evolves in time by multiplying it by a phase factor suitably scaled to its period.
3. Superpose (add up) the evolving component waves to recover the time evolution of the original wave.
That his procedure is possible results from an essential and characteristic property of the Schroedinger equation of quantum mechanics: it is linear. That linearity tells us that we can recover many of the properties of any wave by adding up the corresponding properties of its component waves.
|Here's an illustration of the procedure. The
figure shows how constant wavelength waves can be superposed to give
something closer to a spatially localized particle, which looks like
a spike. The wave shown is a superposition of
20 constant wavelength waves. Only the first four of
the waves superposed are shown individually. Even with 20 components
superposed, the localization of the spike is imperfect. Infinitely
many are needed to get perfect localization.
The waves are evolved in time by the multiplication rule and the evolved waves are shown as dotted lines. These time evolved waves are summed to give the time evolved spike, also shown as a dotted line. The localized particle's spike moves and, at the same time, spreads out. We see the spreading out in the increase in amplitude of the evolved wave away from the main spike.
|The computation represented is for a classical
particle, whose energy is proportional to the square of its
momentum. As a result, the wave period is proportional to the square
of the wavelength. This means that, in some fixed time, the
different waves advance by different amounts. These differences lead
to the pulse moving correctly in time and spreading as it does.
The waves plotted are:
initial wave: cos(x) + cos(2x) + cos(3x) + cos(4x) + cos(5x) + cos(6x) + cos(7x) + cos(8x) + cos(9x) + cos(10x) + cos(11x) + cos(12x) + cos(13x) + cos(14x) + cos(15x) + cos(16x) + cos(17x) + cos(18x) + cos(19x) + cos(20x)
Final wave: cos(x-1/20) + cos(2x-2^2/20) + cos(3x-3^2/20) + cos(4x-4^2/20) + cos(5x-5^2/20) + cos(6x-6^2/20) + cos(7x-7^2/20) + cos(8x-8^2/20) + cos(9x-9^2/20) + cos(10x-10^2/20) + cos(11x-11^2/20) + cos(12x-12^2/20) + cos(13x-13^2/20) + cos(14x-14^2/20) + cos(15x-15^2/20) + cos(16x-16^2/20) + cos(17x-17^2/20) + cos(18x-18^2/20) + cos(19x-19^2/20) + cos(20x-20^2/20)
The wave components are cos(x), cos(2x), cos(3x), cos(4x), cos(x-1/20), cos(2x-2^2/20), cos(3x-3^2/20), cos(4x-4^2/20).
With this procedure, we can use our knowledge of how waves
of constant wavelength evolve in time to determine how the spikes,
representing localized particles, evolve in time.
This procedure embodies the more general case of the rule of Schroedinger evolution:
(scaled to i for each
quarter wave period)
Using complex valued amplitudes in quantum theory may seem perverse at first. Do we really need to stretch our thinking around the square root of minus one? The answer is that it is well worth the effort. For it is the simplest and easiest way to treat waves and their propagation. Simple multiplication lets us cycle through all the phases of a wave and then recover how any wave propagates in space. We are repaid handsomely for rather little effort.
Copyright John D. Norton. April 18, 2017. November 10, 2020.