Copyright © 2013 Krishna Myneni

31 December 2013

It is often heard by the layperson that the quantum world is a strange place.
There is something mysterious about the behavior of particles of matter and light
in the quantum world, although that mystery itself is only in the sense of how we can
picture such behavior. In fact, there is nothing mysterious about the behavior of the
quantum world, as described by the mathematics which follow from a few rather simple
but seemingly absurd assumptions, or *postulates*. However, to the
layperson, the mathematics is often an insurmountable obstacle to gaining further insight
into the difference between the ordinary world of daily experience and the quantum
world.

When I speak of the *quantum world*, I'm imagining the world at a length
scale below about 10 nanometers. I want to illustrate the difference between
behavior of objects in the quantum world, and the behavior of objects in the world of our experience, the world at length scales of micrometers and larger, the
*classical world*. Let us consider the toss of a penny. We will use one ordinary
penny, at its classical world diameter of about 19 mm, which will be called the
*classical penny*. We will also imagine the existence of one *quantum penny*,
an object the size of maybe a few nanometers. Both the classical and quantum pennies
may be observed to determine if they are heads or tails. We know what it means
to observe the classical penny — we simply look at it, and unless we are in a
totally dark room, we can see the heads or tails pattern on its face.

What does it mean to *observe* the quantum penny? We imagine being able to do an
experiment on the quantum penny, where a very weak pulse of light is bounced
off the quantum penny. If the light pulse bounces (scatters) off the penny
so that some of the light appears at right angles, 90 degrees from the forward
direction, then we will say that the quantum penny is *heads*. If the light pulse
does not scatter at 90 degrees but continues on in the forward direction without
any loss, then we will say that the quantum penny is *tails*. We make a
*measurement* of the *state* of the penny, with the result being
heads or tails. A little reflection (pun intended) reveals that
we are also *making a measurement* on the classical penny in order to reveal
whether its state is heads or tails, although the measurement on the classical
penny is vastly more complex than a measurement on the quantum penny, since it
involves analysis of an image from the scattered light.

Now, when we toss our classical penny and *measure* its state, heads or
tails, we expect to find after a large number of tosses that the penny will be
in the heads state half the time, and in the tails state half the time, assuming
that we have a fair penny. And, even with a fair penny, the equal
distribution of heads and tails will only be true if our tosses are completely
*random* tosses. By a completely random toss, we mean that the state of the
classical penny after the toss cannot be predicted with better than 50% accuracy
from the state of the penny before the toss. It is not hard to image
a *deterministic* toss, say a robotic arm that can toss the coin with
sufficient precision so that it exactly reverses state on every toss. But
outcomes from such tosses will be *fully correlated*, meaning that after we
study the results from a number of tosses, we will be able to predict with
certainty what will be the outcome of the next toss.

What about our quantum penny? How do we *toss* our quantum penny? Somehow,
we must disturb our quantum penny in such a way that its state can possibly change.
Inevitably our toss will involve scattering something from our quantum penny,
perhaps using light, or maybe using a sufficiently energetic quantum particle.
For now, we will *assume* that we can do something to alter the state of
our quantum penny, without going into the details of how the toss is performed
on the quantum object. Furthermore, let us *assume* that we know how to
toss our quantum penny in such a way that we obtain heads half the time, and
tails half the time, and that we cannot predict with certainty whether the
quantum penny will be in the state, *heads*, or in the state, *tails*,
by measurement after each toss.

At this point you might say, well those are a lot of assumptions to make about
something we can't see directly. For the moment, my response would be that, even
though I can't "see" the quantum penny directly with my eyes, I can specify
in detail how to setup and perform an *experiment* which will give me such
results. Then, your next point may be, "Ok, so what? All you are suggesting so far
is that a penny in the so-called quantum world behaves like a penny in the classical
world." And, you would be correct, except for a crucial distinction between
the toss of the quantum penny and the toss of the classical penny, which we have not
mentioned up to this point: we can obtain the same kind of results with the quantum
penny, half heads and half tails and with no correlations, as in the *randomly
tossed* classical penny, by using a *deterministic toss* of the quantum penny.

In order to explain my last statement, let me first tell you about a
weird *postulate* about the quantum world. Objects in the quantum
world don't always have to be in *states* for which the measurement
gives only one answer, although they can be. What do I mean by such a
statement? In the case of our quantum penny, the penny *may* be
in a *heads* state, so that when we measure it we are guaranteed to
obtain a result of heads, or it *may* be in a *tails* state, so
that a measurement is guaranteed to give tails. But, the quantum penny is
*not required* to be in just either of these two states.
The *heads* state and the *tails* state happen to be just two
particular states out of an infinite possible set of states available to
the quantum penny. The notion of the state of an object is vastly enlarged
in the quantum world, and when we make our deterministic toss of
the quantum penny, its final state *can be predicted* from knowing
its initial state, but, when measured we may obtain *heads* or
*tails*.

Before we discuss this further, we should also be familiar with another
postulate: the state of the quantum penny will be *collapsed* by the
measurement. The final state of the quantum penny after the toss will be
known, and it will not necessarily be the *heads* state or the
*tails* state, but one of the other possible states from an infinite
set. However, after making a measurement of the penny's state, we measure
*either* heads *or* tails. We find light scattered either at
right angles or not at all, according to our prescription for measuring
the state of the quantum penny. The state of our quantum penny is
now collapsed to either the *heads* state or to the *tails*
state. This means that if we measured heads, then any subsequent measurement
made on the penny without performing another toss will also give heads, for as many
times as we care to measure the penny's state. Similarly for tails. The state of
the quantum penny is, in general, altered by the measurement so that the
penny is in one of two special states following the measurement. Physicists refer
to these special states as *eigenstates*. The two particular states of the
quantum penny, *heads* and *tails*, are special states.

The toss of the quantum penny may leave the penny in a state which is not exactly
a *heads* state or exactly a *tails* state, but a very peculiar combination
of *both* heads and tails. At this point, our words will begin to fail us, and we need
to invent a precise language for talking about the possible states of a quantum
penny. Let's write the *heads* state as **H**, and the *tails*
state as **T**. And let us invent symbols for the *initial* and *final*
states of the quantum penny, which represent its state before and immediately after
the toss. These states will be written as **I** and **F** representing the
initial and final states. The states are summarized in the table below.

Label | State | Special? |

H | heads | Yes |

T | tails | Yes |

I | initial | Not in general |

F | final | Not in general |

Neither the initial state,

where

The expression above for the general state of a quantum object having two special
states may appear, at first, intimidating to the layperson for several reasons.
What does it mean to multiply a special state by a number? Even stranger, what
does it mean to add the two special states, **H** and **T**, together,
after multiplying them by numbers? The answer is that such an expression is
nothing more than a convenient way of keeping track of the numbers *a*,
*b*, and *φ*, and performing the proper arithmetic to determine
how the state changes under a toss. The numbers, *a*, *b*, and
*φ*, along with the special states, **H** and **T**, completely
characterize the general state of the quantum penny (students specializing in
physics should also be cognizant of *ω*).
We mentioned that there is an infinite set of states available to the quantum penny,
and this is reflected in the fact that the real numbers *a* and *φ*
have a continuous range of possible values. The number *a* may be any number
in the continuous range of values, [0,1], and the number *φ* may be any
number in the continuous range, [0, 2*π*]. The reason I didn't mention
*b* is that choosing the value of *a* determines the value of *b*,
uniquely. We will see this below.

When the quantum penny is in the state represented by **F**, the numbers
*a* and *b* are related to the probabilities, upon measurement, of finding
the penny in the **H** state and in the **T** state. The actual probabilities
are *a*^{2} and *b*^{2}, which is another *postulate*
of quantum mechanics. Thus, the numbers *a* and *b* are related to each
other through,

The special states

where

and these values of

since

Now, we will specify a *deterministic toss* of the quantum penny. The toss
may be specified by mapping each special state, **H** and **T**, to their
final states. Such a map can be used to represent the disturbance of the penny.
For example, we will choose our toss to transform the **H** state to a particular
final state, and to transform the **T** state to another final state, given
below,

In the above expressions, the right arrow, ⇒ , represents a toss of the quantum penny from the initial known state on the left. The reader may reasonably wonder, from looking at such a strange transformation of states as I have written above, whether or not we are dealing simply with mathematics and not with the physical world. Once again, my response is that I can specify an experiment which, when performed, will give results fully consistent with the above mathematics, and with our postulates about how to interpret such states. Indeed, in the field of

When we start with the quantum penny in an initial *heads* state, the
particular state **F**_{H} is the final state of the penny
after the toss, with

Note that

Both

The state, **F**_{H}, does not tell us whether we will find
**H** or **T**, only the probabilities of finding them. *It is
the act of measurement which gives us a random variation of heads or tails from toss
to toss, and not the toss itself*. For both final states,
**F**_{H} and **F**_{T}, the outcome of a measurement
of the state of the quantum penny is unpredictable, and both heads and tails
occur with equal probability upon measurement. It is now fair for the reader
to ask, why do we need all of these seemingly absurd assumptions about states
of an object in the quantum world, and the complex expressions for its state,
when the outcome of the toss of a quantum penny appears to be the same as that
of the classical penny! The answer is that very different behavior can occur,
the origin of which lies in the strange multiplication term involving
the phase, *φ*, of the final states. We will demonstrate this.

The final state, **F**, of our quantum penny is fully predictable when its
initial state is known. Let us now track the state of the quantum penny from
toss to toss, assuming that the penny is in an initial state of **I** = **H**.
We toss the penny, and we know that its final state will be **F**_{H}. Now,
suppose we do not measure the state of the penny, but toss it once again. What
will be the final state of the penny immediately after the second toss? The
rules of quantum mechanics say that we should apply the same transformation to
the individual parts of the state. Our initial state for the second toss is
neither **H** or **T**, but rather the final state from the first toss,
**F**_{1} = **F**_{H}, since we did not collapse the
state of the quantum penny to **H** or **T** by measuring it. This is
similar to keeping our hand over the coin for the classical penny so that
we cannot see whether it is heads or tails after the first toss.

The state of the quantum penny after the first toss is,

Tossing the penny a second time gives,

The final state after the second toss,

Using the rules of arithmetic, we can simplify the expression for state

which is, of course, just simply,

This means that for a measurement of the state of our quantum penny after the second toss,

Let us dissect what has happened in the second toss. What caused the **T** component
of the state to vanish? The origin of this effect goes back to how we specified the
toss. A toss transforms the **T** component of a state into a state with a
negative **T** component. Remember that the minus sign comes from the
`exp(`*iφ*`)` factor, with *φ = π*.
Therefore, in the second toss, the **H** component of **F**_{1}
transforms into a **T**, while the **T** component of **F**_{1}
transforms into a −**T**. The resulting state from the second toss,
**F**_{2}, contains both a **T** and a −**T**, and these
two parts of **F**_{2} add together as numbers would, and therefore
cancel each other. Of course, we have just said in a lot of words, what the
arithmetic above shows explicitly. This phenomena is known as
*quantum interference*. In the laboratory, physicists can make various kinds
of particles, such as neutrons, atoms, or even collections of thousands of atoms,
interfere with each other. The quantum penny is a simple example demonstrating
the difference between the quantum world and the world of our daily experience.

We have introduced a number of peculiar assumptions about the behavior of objects in the quantum world. It is reasonable to ask for the justification for these assumptions. Ultimately, the justification comes from the mountain of experimental evidence which shows that when we apply these assumptions, we can not only predict the behavior of quantum objects, but do so with extraordinary accuracy. The rules have been tested and no single violation of the rules has been found after nearly a century of careful experimentation.

Comments on this paper may be addressed to
*krishna.myneni@ccreweb.org*.