The Wavefunction of a Tossed Coin

This is actually my Science Essay Prize submission – here it is for you to read / scoff at…

The title is a reference to a famous article by Hawking and Hartle: ‘Wave Function of the Universe’ (2)

Coin Toss. Taken at 21:34 on 3 Aug 2009 at home using an ad-hoc strobe and a 1p coin. Camera: Fujifilm Finepix S5800 (DSLR); 1” exposure; F/3.5 aperture

'Coin Toss'. Taken at 21:34 on 3 Aug 2009 at home using an ad-hoc strobe and a 1p coin. Camera: Fujifilm Finepix S5800 (DSLR); 1” exposure; F/3.5 aperture

The Quantum world, the world of the very small, is very different from the world we live in and are familiar with. In this world, the law of conservation of energy can be cheated for a short period of time owing to the time-energy relation to produce short-lived but vital ‘virtual photons’, and even the concept of probability has a quantum twist. This rather bizarre framework has however been enormously successful and has allowed physicists to make very accurate predictions of experiments’ results; it has never been shown to be incorrect or inaccurate. This seems to imply that, however different it may be from classical physics, quantum mechanics is correct. Yet it seems utterly inconceivable that such apparently opposing ideas are in fact both largely correct and thus identical on a macroscopic level, as evidence suggests, and that the correspondence principle, which states that Quantum Mechanics (QM) reduces to classical mechanics at high masses, is true. Classical mechanics provides a full description of a coin toss. Can QM make the same claim while keeping its predictions consistent with those of classical physics?

The Correspondence Principle

The most important relationship between the quantum mechanical description of this coin and its classical counterpart is embodied in the correspondence principle.

The Schrödinger equation is used principally to work with the energy of a system and is displayed here:

Where m corresponds to the mass of the system, E is a number whose possible values are the possible energies the system may take on, and Ψ(x) is a function of the position of the system (the wavefunction) whose square modulus is the probability density of the particle being at position x. In the example of the one-dimensional infinite square well in which a particle is trapped in a one-dimensional universe (such that V(x)≡0) with impassable ‘walls’ placed at 0 and l, it turns out that:

Ψ(x)=sin⁡Ax (In fact Ψ(x)=cos⁡Ax also works)

At both walls (at l and 0):

A=nπ/l (n=1,2,3…)

When Ψ(x)=sin⁡(nπ/l x) is plugged into the Schrödinger equation, one obtains:

This tells us that the greater the value of n, the more nodes the wavefunction has on the interval [0,l], so the more energy the system has.

Contrasting the shape of the graph of Ψ(x)²=sin²⁡Ax=sin²⁡(nπx/l) (a probability density plot) when n=1 (below, left) with that when n=50 (below, centre), one surmises the one corresponding more closely with what classical physics would predict (i.e. a constant probability density) is the case when n is high; when n is sufficiently high, the graph of Ψ(x)² can be faithfully approximated to a classical constant probability density (below, right) in which the particle can be anywhere along the line with equal probability.

Click to embiggenatrify

Thus as the energy of the system increases, the value of n rises and thus the correspondence between classical and quantum mechanics becomes closer.

Although this is a very specific example, wavefunctions tend to turn out to be trigonometric functions (such as sin⁡x, cos⁡x and eix) which are often the only eigenfunctions of the second differential operator in the Schrödinger equation, leading to the occurrence of a positive power of this naturally arising quantum number (n=1,2,3…) which increases with the number of nodes. One can use the periodicity of Ψ(x) to support the argument that this positive integer quantum number n exists for almost every wavefunction Ψ(x) (i.e. for almost every quantum system). So long as the system’s energy is over a certain limit known as the classical limit, it will have a sufficient number of nodes that it will behave effectively identically to a classical system; as its energy tends to infinity, its behaviour will tend to classical behaviour. Extending this concept, since the energy of the coin toss situation is so large relative to atomic energies (the kinetic energy of the entire coin while spinning is many orders of magnitude greater than that of a stream of particles about to collide at 99.9999991% the speed of light in the LHC), n is extremely large so the quantum description of the coin toss approximates extremely well, and in most respects identically, to its classical counterpart.

Ensemble Interpretation

At its heart, the interpretation states that the Quantum Mechanics is a statistical abstraction from reality: No single particle has a wavefunction; rather the wavefunction applies to a group, or ensemble, of classical particles and it merely describes their distribution. Although this interpretation is largely rejected on the grounds of the Young’s double slit experiment run using a light source that emits only one photon at a time in which an interference pattern still appears, it is an apt intuitive argument for the correspondence principle: the time evolution of the graph of position against probability for a single particle with a particular wavefunction is very similar to the that of position against an approximate measure of density of an ensemble of classical particles. Since in the example of a coin toss there are a large number of particles participating with similar wavefunctions, the distinction between a Quantum and a Classical approach to determining the behaviour of the experiment becomes merely academic and a matter of paradigm since both methods will yield very nearly identical results – whether every particle has a wavefunction or the entire system is considered collectively is no longer relevant to predicting the result of a coin toss.

Classical Probability: Coarse Graining

When a decision event occurs, there are several histories associated with it. For example in a simplified universe containing only one photon, a target, and a piece of card with two slits in it, when the photon ‘chooses’ whether to travel through the right or left slit in the Young’s double slit experiment before hitting the target, a history is created for each of the two possibilities: one for when the photon chooses the left slit and one for the right slit. In a more complicated universe containing several particles, every combination of ‘decisions’ the particles ‘make’ constitutes a different history: each possible history (containing complete information about every particle in the universe at every point in time) is a separate history according to quantum mechanics. These histories are known as fine-grained histories.

As I mentioned, the quantum version of probability is very different from its classical counterpart. Instead of assigning each fine-grained history a probability, Quantum Mechanics assigns pairs of fine-grained histories values. This value for a pair of histories A and B can be denoted by a function: D(A,B). The pair of histories can be a pair constituting of just one history, for example D(A,A), which would in fact be a number between zero and one and can be interpreted as the probability of history A occurring. However the function D follows the following rule:

D(A or B,A or B)=D(A,A)+D(B,B)+[D(A,B)+D(B,A)]

The last term, [D(A,B)+D(B,A)], is called the interference term between histories A and B and can have both positive, negative and zero values; if it is not zero, histories A and B interfere with each other, making it difficult, and sometimes impossible, to assign a probability to each separately.

The problem has now been implicitly stated: if interference makes it so difficult to assign probabilities to histories, what makes it possible to say that the probability of tossing ‘heads’ on an unbiased coin is ½; how can such a prohibitive concept of probability be reduced at large scales to classical probabilities?

The solution lies in an idea called coarse graining: histories are organised into sets. For example there might be the set of all histories in which Photon A travels through the left slit – all histories in which this event takes place are organised into a set. The purpose of this classification is to ignore all factors that do not matter to the critical situation by taking the set of all fine-grained histories which agree that the critical event occurs but may disagree on the goings-on in the rest of the universe. This allows the construction of a single coarse-grained history whose probability is:

D(A1 or A2 or A3…,A1 or A2 or A3…)

where A1, A2 etc. are all the different histories that make up this set. If all the fine-grained histories can be divided into sets such that each fine-grained history belongs to one and only one set, one obtains a set of mutually-exclusive coarse-grained histories. The result of all this is that, for two coarse-grained histories α and β consisting of fine-grained histories {Ai } and {Bi } respectively (where {Ai } stands for A1,A2,A3,…), D(α,β)+D(β,α)—the interference term between α and β—is the sum of all the interference terms between pairs of fine-grained histories that belong to those two coarse-grained histories: the net interference between α and β is the sum of all the smaller interferences between {Ai } and {Bi }. This summation often leads to cancellation of positive and negative interference terms, leading to a near-zero interference term between α and β: D(α,α) and D(β,β)—the probabilities of α and β respectively—are very well-defined numbers, no longer dependent on interference terms, leading to independent probabilities of mutually-exclusive events. This collapse of D values into classical probabilities via coarse-graining is called decoherence, and statistically, the more fine-grained histories that are summed over, the closer to zero the net interference term becomes, and the more definite and independent the probabilities of coarse-grained histories become. Returning to the coin-toss example, since a huge number of particles interact with the coin’s faces, an enormous number of histories have to be summed over to obtain the two coarse-grained histories of the coin: heads or tails, leading to a near-zero interference term between the heads-up and tails-up histories: it can be said with much confidence that the probability of tossing heads is ½.

Classic Mistakes: Misinterpretations

Common misinterpretations of the implications of the postulates tend to make QM sound more unbelievable and divergent from classical physics than it really is. For example a famous example often quoted when explaining QM is Schrödinger’s Cat. This is a thought experiment invented by Erwin Schrödinger in which a truly random (unpredictable) process produces a Boolean (yes/no) output which determines the fate of a cat contained in a sealed box. If the output is ‘no’, poison is released into the box and the poor animal perishes; if the output is ‘yes’, no poison is released and the cat remains alive. The concept Schrödinger was attempting to communicate by producing this analogy is that of superposition: in the story of the cat, after the cat’s fate has been determined and appropriate actions effected but before the box has been opened and the ‘aliveness’ of the cat observed, the cat is in a superposition of being dead and alive: while classical physics would state that the cat is either dead or alive, quantum physics would assert that the cat is in a curious state of being neither dead nor alive but in some half-way state, and only when the box is opened does the cat decide its ‘aliveness’. My personal opinion is that this is a brilliant analogy and can be a great help in explaining quantum superposition. Unfortunately the analogy is far too often taken literally, leading many to believe that in such a situation the cat really would be neither dead nor alive. As Gell-Mann pointed out in his book The Quark and the Jaguar (1), the cat frequently interacts with air particles inside the box which in turn interact with the box which in turn interacts with the rest of the universe; since a large number of particles are involved the two outcomes (living cat vs dead cat) decohere leading to a more or less classical situation in which the cat is either dead or alive: by the time the cat is observed, the wavefunction has already collapsed and the cat’s ‘aliveness’ is completely classical.

The idea that ‘anything can happen because of Quantum Mechanics’ is another myth. It is to some extent true, but often abominably misinterpreted by unthinking readers, perhaps a situation exacerbated by episodes of The Big Bang Theory and chapters of The Hitchhiker’s Guide to the Galaxy in which references to this idea are made through jokes and invented technology such as the improbability drive without making clear the true nature of the numbers behind these events happening. The probability of a familiar object visible to the naked eye such as a coin spontaneously appearing on Earth within a person’s lifetime is probably smaller than the probability of someone winning the lottery every time for the entirety of his/her lifespan (which itself is in the order of about 10(-105) ). It is in fact so unlikely that it is, in all practical sense, impossible (an argument often used, and bizarrely often rejected, to support the case against God).

So what about the coin?

In conclusion, the quantum mechanical description of a tossed coin is identical to its classical counterpart. The probability of obtaining heads is the same in both models; the coin spins and is affected by the air in the same way in both models; in neither model is it possible for the coin simply to disappear in mid-air: the coin never has strange quantum-mechanical properties such as superposition or ill-defined probabilities. By obeying the correspondence principle, QM completely describes classical mechanics and in theory, everything previously explainable and predictable by classical mechanics with regards to a coin toss can be fully explained and predicted by quantum mechanics.


1. Gell-Mann, Murray. The Quark and the Jaguar. s.l. : Abacus, 1995.
2. Wave function of the Universe. Hawking, Stephen and James, Hartle. s.l. : Physics Review, 1983.
3. Hawking, Stephen. A Brief History of Time. s.l. : BCA, 1996.
4. Gillespie, Daniel. A Quantum Mechanics Primer. s.l. : International Textbook Company Limited, 1973.
5. Ballentine, Leslie. Quantum mechanics: a modern development. s.l. : World Scientific Publishing Co Pte Ltd, 1998.
6. Gribbin, John. Q is for Quantum. s.l. : Simon & Schuster, 2000.


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