The Quantum Penny: An Introduction to the Quantum World

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.

IinitialNot in general
FfinalNot in general

Neither the initial state, I, nor the final state, F, are required to be special states. However, if we want to set the initial state of our quantum penny to a special state before tossing it, we can do so by making a measurement to see whether we obtain heads or tails. Then, the initial state, I, will be H or T, by virtue of our measurement. It is important to note that the final state, F, of the quantum penny is its state immediately after the toss, but before measurement of the penny's state. How can we write this state, F? The conventional way of writing a general state G for the quantum penny, in quantum mechanics, is the following:

G = aH + bexp(iφ)T

where a and b and φ represent real numbers, i represents the imaginary number which is the square root of negative one, and exp(x) is a function representing raising the real irrational number, e = 2.71828..., to the power of x. The real number, φ, is called the phase of the quantum state, and the a and b are related to the probabilities for measuring heads and for measuring tails, respectively. A note for physics students: there should also be a factor of exp(−iωt) multiplying the second term on the right hand side, but for sufficiently small ωt we may ignore this term, which will not be relevant for the present discussion.

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, 2π]. 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 a2 and b2, which is another postulate of quantum mechanics. Thus, the numbers a and b are related to each other through,

a2 + b2 = 1

The special states H and T are particular cases for b=0 and a=0, respectively. For a state with equal probability of measuring heads or tails, we may take

a = b = 1/sqrt(2)

where sqrt() represents taking the square root of the number inside the parentheses. Then, the probabilities of measuring heads or tails will both be 1/2 for the quantum penny. Note that the particular assignment of a and b we used above is not the only assignment which will give equal probabilities for heads and tails. We might also have taken,

a = 1/sqrt(2)
b = −a

and these values of a and b will give the same probabilities for measuring heads and for measuring tails. However, the negative value of b is really an artifact of combining part of the phase in the value of b. More properly, the above set of values for a and b is equivalent to

a = 1/sqrt(2)
b = a
φ' = φ+π

since exp() = −1.

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,

H  ⇒  FH = (1/sqrt(2)) { H + T }

T  ⇒  FT = (1/sqrt(2)) { HT }

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 quantum computing, our prescription for the quantum penny toss is exactly a particular type of quantum logic gate known as a Hadamard gate.

When we start with the quantum penny in an initial heads state, the particular state FH is the final state of the penny after the toss, with

a = b = 1/sqrt(2)
φ = 0

Note that exp(i0) = 1. Similarly, when we start with the penny in an initial tails state, the particular state FT is the final state after the toss, with

a = b = 1/sqrt(2)
φ = π

Both FH and FT are possible states of the quantum penny before we make our measurement. After measurement, the state of the penny will become either H or T. Examine the expressions for the two final states, FH and FT, to see that the probability of obtaining heads or tails will both be 1/2, for either case.

The state, FH, 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, FH and FT, 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 FH. 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, F1 = FH, 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,

F1  =  (1/sqrt(2)) { H + T }

Tossing the penny a second time gives,

F1  ⇒  F2

The final state after the second toss, F2, is given by applying the rules of the toss to both component special states present in F1.

F2  =  (1/sqrt(2)){ 1/sqrt(2){H + T}  +  (1/sqrt(2)){HT} }

Using the rules of arithmetic, we can simplify the expression for state F2 to,

F2 = (1/2){   (1 + 1)H + (1 − 1)T  }

which is, of course, just simply,

F2 = H

This means that for a measurement of the state of our quantum penny after the second toss, without having measured it after the first toss, we are guaranteed to obtain heads! Measuring the quantum penny after every other toss will give heads every time, but measuring the quantum penny after every toss will give us completely unpredictable outcomes, with heads occuring half the time and tails occuring half the time. Try doing that with a classical penny!

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() factor, with φ = π. Therefore, in the second toss, the H component of F1 transforms into a T, while the T component of F1 transforms into a −T. The resulting state from the second toss, F2, contains both a T and a −T, and these two parts of F2 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.

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