Some Implications of Quantum Superposition 
You have already seen an introduction
to quantum mechanics and the ideas of wave functions. This reading
will build upon that introduction and illustrate some of the consequences
of quantum superposition of states.
You should understand by now that all objects have both particle and wave properties. The wavelengths of macroscopic objects such as elephants, bullets, and computers are so small that they cannot be measured. Thus the wave nature of these macroscopic objects is never observed. Subatomic particles like the electron, however, can easily have wavelengths on the order of micrometers. Electrons traveling in a narrow beam through a double slit will produce an interference pattern of maxima and minima rather than the single spot expected if electrons always behaved as particles. This interference pattern is identical to the pattern produced by light passing through slits. The first figure below illustrates what would occur if electrons always acted like particles; the second if they always acted like waves; and the third what is actually observed to occur.
THESE DIAGRAMS ARE NOT AT ALL TO SCALE






As shown in the rightmost figure above, electrons travel through the slit like waves but do not strike the screen as waves. Instead, the same electron that exhibits wave properties traveling through the slits interacts with the screen as a particle, striking one and only one spot. If one were able to slow the electron emission and observe single electrons interacting with the screen, one would see the interference pattern build up one spot at a time. As the number of electrons having hit the screen increases, the pattern of where individual electrons strike takes the form of an interference pattern. Some of the "spots" produced by individual electrons striking the screen are located far away from the points directly in front of the slits that a particle would be expected to hit. Thus each electron passes through both slits like a wave, interfering with itself, before interacting with the screen in one point like a particle.
As if this dual behavior is not confusing enough, trying to detect the electron as it passes through the slits changes the entire outcome of the experiment. The electron is never detected simultaneously in both slits; instead, a detector will find it passes through only one opening. But when this detector is in place and making this measurement, the interference pattern disappears. We are left instead with the leftmost pattern on the screen above which was predicted if electrons are particles going through one opening or another: two bright spots in front of the each individual slits. Our measurement at the openings has forced the electron to behave like a classical particle.



(a) Light from a laser is incident on a beam splitter. The beam splitter will transmit half of the light while reflecting the other half  (b) The transmitted and reflected light is directed by completely reflecting mirrors back to the second beam splitter, where each beam is split once more.  (c) Light that has been reflected twice (RR) or transmitted twice (TT) ends up at the U detector, while light that has been reflected once and transmitted once (RT and TR) ends up at the D detector. 



(d) In a MachZhender interferometer, one of the mirror can be adjusted (amount it moves is exaggerated here) to make the top path different from the bottom path.  (e) Depending upon the location of the mirror, you can achieve complete destructive interference at the bottom detector and complete constructive interference at the top detector, or . . .  (f) . . . complete constructive interference at the bottom detector and complete destructive interference at the top detector, or anything in between. This occurs no matter how low the intensity of the input beam. 
The only explanation for the behavior of figures
(e) and (f) above is interference. So far, this experiment, like
the doubleslit experiment, seems to verify the wave theory of light.
Even if you decrease the intensity of the input to the point that the particle
theory predits only one photon would be in the detector at a time, you
still see the interference effect. But another interesting aspect
of the experiment is illustrated below.

When a detector is placed to determine which path the light takes, the interference effects vanish! 
If a detector is placed in one of the paths, the
U and D detectors will once again each get half of the light. This
happens even if the detector is a "nondemolition" detector rather than
the highdemolition bomb we discussed in class. And if the intensity
of the input beam is decreased to send one photon at a time, a dector sensitive
enough to measure a single photon would only detect light half of the time.
This last behavior seems to indicate particle behavior, but neither the
wave nor the particle theory can explain why the presence of an interference
pattern depends on the type of measurements taken.
In our electron doubleslit experiment, the electron has two possible paths: take the top slit, or the bottom slit. Rather than try to figure out an exact way to describe those options, we will use a notation developed by Dirac: going through the top slit will be denoted top>, while going through the bottom slit will be denoted bot>. Some portion of our electron wave has passed through the top, and some portion through the bottom, so the wavefunction is
c(x) = a top> + b bot>
Unlike the amplitude y(x,t) describing a water wave, the wavefunction is NOT measurable. When the position of an electron is measured (such as when we put a detector at the slit), the electron will appear in one point, not as a wave spread out over the screen. The probability of finding the electron at a given point is found from the square of the wavefunction:
Prob = òc(x)^{2}dx = a^{2} <toptop> + b^{2} <botbot>
So our electron has probablility of a^{2} of going through the top slit and b^{2} of going through the bottom slit. Since the total probability must be one,
a^{2} + b^{2} =1
If the slits are equally likely, then a^{2} = b^{2} = 0.5. This equation may appear to give us a and b, and thereby tell us what the wavefunction of the electron is, but it does not. a and b can be (and generally are) imaginary and/or negative. We can measure the magnitude of the wavefunction amplitudes, but we cannot directly determine the entire amplitude.
If a measurement is made to determine which slit the electron goes through, it will turn up that a^{2} of measurements find the electron in the top slit, and the remaining b^{2} of measurements find it in the bottom slit. After this measurement, the electron is no longer a superposition of states, but it is entirely the state corresponding to the slit in which it was measured:
c(x) = top>, or c(x) = bot>.
We say that the measurement collapses the wavefunction into a single state. Since the wavefunction no longer contains both possibilities, the interference between different possible paths doesn't occur. The act of measurement alters the results of the experiment.
Let me restate that since it is a key concept in quantum computing applications:
measuring
a system changes the system in an irreversible manner.
This aspect of the quantum theory could be utilized to indicate whether
a message has been intercepted and read by someone on its way to a recipient,
as explained later in this reading.
c(x) = a RR> + b TT> + c TR> + d RT>.
Since each path is equally likely when the mirrors are truly halfmirrored and splite the beam 50/50, the probability of a photon taking a given path must be 1/4:
a^{2} = b^{2} = c^{2} = d^{2} = 1/4.
It is tempting to just make each amplitude equal to 1/2, since 1/2^{2} is 1/4. We must, however, not give way to this temptation. The wavefunction at the upper detector contains the RR> and the TT> paths, while the wavefunction at the lower detector contains the other two paths:
Prob (U detector) = a + b^{2}, and
Prob (D detector) = c + d^{2},
where we add the amplitudes of the two indistinguishable
paths before squaring to find the probability. If each amplitude
is 1/2, the probability at the U detector is 1, as is the probability of
the photon reaching the D detector. The total probability is therefore
2, which is impossible. If no interference occurs, the probability
at each detector is 1/2. The only way to meet this condition while
also requiring that each amplitude squared is 1/4 is to use complex amplitudes.
One (but certainly not the only) solution is
a = c =


b = d =


a + b^{2 }= c + d^{2}
=

1/2 
Again consider the simple double slit experiment. Let's say electrons
are sent one at a time toward the double slit. A counter is placed in the
top slit. As the electron approaches the slits, it is described by a wavefunction
that spreads out in space encompassing both slits. Once the wavefunction
hits the counter, however, it collapses. If the collapse of the wavefunction
leaves the electron in the top slit, the counter is triggered, and a "1"
bit is sent to the computer. If the collapse leaves the electron in the
bottom slit, the counter does not trigger and a "0" bit is sent. This procedure
could be repeated for each bit of an arbitrarilylong random number. Assuming
the electron beam is symmetric around the slits, half of the time an electron
will be found in the top slit. But you cannot determine which slit the
electron will be found in. Quantum behavior is truly unpredictable
and governed by probabilities of random events.
Einstein also wrestled with the implications of quantum theory, never
coming to accept it. He coauthored a paper putting forth an objection
now known as the EinsteinPodolskyRosen (EPR) paradox. While quantum
mechanics may involve uncertainty and superposition of states, it does
not involve breaking physics' conservation laws (such as energy, charge,
or momentum) in any experiment. And it does involve Heisenberg's
uncertainty principle (you can't know both position and momentum to arbitrary
accuracy). The EPR paradox considers two electrons emitted by a system
at rest. If the net spin is zero to start with, the two electrons
must have opposite spins. But until a measurement is taken, each electron
could have either spin, so the wavefunction of each must contain both possibilites.
Once you measure the spin of one elctron, however, the spin of the other
is immediately determined. But how could the electron that is not
measured "know" that its partner electron has been detected with a particular
spin? It would seem that some sort of information is sent from the
first electron to the second instantaneously. Such instantaneous
information transfer is at odds with relativity and violates causality.
Quantum cryptography provides the ability to determine whether a message
has been intercepted, since Eve's measurement would change the system.
It relies upon the very behavior that Einstein, Podolsky, and Rosen claimed
was impossible in their EPR paradox. This behavior, in which measuring
a property of a photon in one location immediately determines that property
for its partner photon that could be miles away, is commonly known as "entangled
states." Alice produces pairs of photons and sends one of each pair
to Bob while keeping the others for herself. They each take measurements,
and convey what they have measured to create a key. If Eve has intercepted
the signal, and thus discovered the key, Bob's photons will have inconsistencies
when compared to Alices' photons.
Like it or not, understand it or not, quantum mechanics works.
The properties that the paradoxes take issue with, the superposition of
states and the uncertainty principle, have been tested and retested, showing
that they are indeed the best way we have developed so far to describe
the world around us. You will see in class how this quantum behavior
can be used to perform parallel computation and for cryptography.
Copyright © 2000 Doris Jeanne Wagner. All Rights Reserved.