quantum wave funtion: real or probabilistic

The wave function of quantum theory has always been accepted as an abstract mathematical device – but could this cipher actually be real?

A BOAT trip on Lake Zurich seemed like the perfect way to unwind after intense discussions about the hot topic of the moment, quantum theory. Appropriate, too, given that all the physicists present were talking about an idea Erwin Schrödinger had put forward a few months earlier. He had suggested that all quantum particles, from atoms to electrons, could be described by intangible entities that spread out through space much like ripples on a lake's surface. He called them wave functions.

When Schrödinger had first published his insights in March 1926, theorists had been thrilled. Symbolised by the Greek letter psi, the wave function gave them a way to apply their much-loved mathematics of waves to the quantum world. And it worked, neatly explaining why electrons in atoms have the energies they do. Yet there was a problem, summed up by a cheeky verse penned on the boat trip:

Erwin with his psi can do
Calculations quite a few
But one thing has not been seen:
Just what does psi really mean?

Almost 90 years on, and quantum theory is still our very best description of the microscopic world of atoms and their constituents. It has given us lasers, computers and nuclear reactors, and even tells us how the sun shines and why the ground beneath our feet is solid. Yet the wave function remains an enigma, a ghost at the heart of the atom. To most physicists, it is little more than a convenient way of calculating how quantum systems such as atoms should behave. Few have considered it in any sense real.

It might now be time to rethink those attitudes. A group of physicists based in the UK has looked again at the wave function, and think they have the proof that there is more to it than meets the eye. The ghost in the atom may yet prove to be more real than we ever imagined.

The emergence of the wave function as a central element of quantum theory was by no means an overnight thing. The theory was born in 1900 when German physicist Max Planck found that he could explain the baffling spectrum of light emerging from a hot furnace only if the vibrational energy of the atoms giving out the light came in discrete chunks, or quanta. Planck himself thought this was merely a convenient mathematical device. It was left to one Albert Einstein to recognise that quanta were real five years later. He showed that the light streaming out of the atoms that make up matter consists of untold trillions of these tiny energy packets, now better known as photons.

Einstein's idea flew in the face of hundreds of experiments showing that light was like a ripple spreading on a lake or pond. The only plausible explanation was that somehow light must be both a wave and a stream of particles. In 1923, French physicist Louis de Broglie took this idea and ran with it, proposing that just as light waves could behave like particles, the fundamental particles of matter could behave like waves.

De Broglie imagined an electron as a "matter wave" rippling through space. But as the idea was fleshed out mathematically, that interpretation was quietly dropped. The waves associated with quantum particles were waves alright, but they appeared to be totally abstract things, unlike any waves anyone had ever imagined.

All waves can be described mathematically - a ripple across a pond, for example, is a disturbance in water; its wave function describes its shape at any point and time, while something called the wave equation predicts how the ripple moves. Schrödinger realised from de Broglie's work that actually every quantum system has a wave function associated with it, though he struggled to explain what the disturbance would be in the case of an atom or an electron. Even so, Schrödinger's work led to a radical new picture of the quantum world as a place in which certainties give way to probabilities.

Schrödinger's wave function is central because it encodes all the possible behaviours for a quantum system. Picture the simple case of an atom flying through space. It is a quantum particle, so you cannot say for sure where it will go. If you know its wave function, however, you can use that to work out the probability of finding the atom at any location you please. That is good enough for most physicists, who believe the wave function to be merely a probability distribution - a statistical summary of what large numbers of measurements would tell you about the whereabouts of the particle. But is that the full story?

An example helps to highlight the subtle difference between a purely mathematical and a physically real wave function. Say you arrive at a lake into which a large number of plastic bottles have been thrown. You notice that there are places where the bottles bunch up and places where there are very few bottles. By counting the number of bottles at different locations, you could create a probability distribution, which allows you to estimate the chance of finding a bottle at each point.

But suppose you notice that the bottles are most common where the amplitude of the real waves in the lake peak. Now you realise that the probability distribution is not the last word - there is a mechanism behind it. Real, physical waves have driven the bottles to their particular locations.

Back in quantum theory, we can similarly ask if the quantum wave function is merely a probability distribution or the manifestation of a real, underlying wave. The key to answering that question is to come up with a thought experiment in which the two possibilities produce different outcomes. That isn't so easy, and theorists have struggled for years to formulate the question in a tractable way.

Now Matthew Pusey and Terry Rudolph of Imperial College London, with Jonathan Barrett of Royal Holloway University of London, seem to have struck gold. They imagined a hypothetical theory that completely describes a single quantum system such as an atom but, crucially, without an underlying wave telling the particle what to do.

New reality

Next they concocted a thought experiment to test their theory, which involved bringing two independent atoms together and making a particular measurement on them. What they found is that the hypothetical wave-less theory predicts an outcome that is different from standard quantum theory. "Since quantum theory is known to be correct, it follows that nothing like our hypothetical theory can be correct," says Rudolph (Nature Physics, vol 8, p 476).

Some colleagues are impressed. "It's a fabulous piece of work," says Antony Valentini of Clemson University in South Carolina. "It shows that the wave function cannot be a mere abstract mathematical device. It must be real - as real as the magnetic field in the space around a bar magnet."

The UK trio's work has also received support from Lucien Hardy at the Perimeter Institute for Theoretical Physics in Waterloo, Ontario, Canada. Using slightly different assumptions, he has obtained a similar result indicating the reality of the wave function (arxiv.org/abs/1205.1439).

The conclusion appears inescapable. Quantum theory makes no sense if the wave function is merely a probability distribution. Instead, the wave function has to be a real thing associated with a single quantum system, informing it how to behave. "Under seemingly benign assumptions, you cannot escape the wave function being 'real'," says Rudolph.

Valentini agrees with this interpretation. "There is a definite reality underpinning quantum theory," he says. Not everyone shares his conviction, however. Some point to problems with the team's "benign" assumptions. One is the notion that a quantum system has true properties even before any measurement has been made on it. The pioneers of quantum theory, Niels Bohr and Werner Heisenberg, famously maintained that there was no real world out there and that quantum properties were brought into existence by the very act of measurement. Taken to its extreme view, the Bohr-Heisenberg view implies that the universe did not exist until we came on the scene to observe it. "But that was then and this is now," says Valentini. These days the consensus among most quantum theorists is that the universe, electrons and atoms exist even when they are not being measured.

Pusey and his colleagues also suppose that the two atoms in their thought experiment are truly independent of each other, so that a measurement made on one does not affect the other. They also take for granted that the laws of cause and effect hold, so that a measurement made on an atom at, say, 3 pm does not affect its state earlier at 1 pm.

Valentini argues that these are reasonable assumptions to make. To violate them would require a kind of universal web of interactions in which almost every system in the universe influences every other system at different times as well as different places. "It seems simpler to accept the quantum wave as real, just as classically it is simpler to accept the electromagnetic field as real," he says.

So what does it mean if Pusey's team and Hardy are right and the wave function is real? On the one hand, they have overturned almost 90 years of thinking by showing that the wave function is part and parcel of a single quantum system. On the other, if the wave function is real, it is a very weird kind of real. According to the mathematics, Schrödinger's wave function encodes everything there is to know about a single particle in three dimensions. But things get more complicated very quickly. The wave function for two particles exists in an abstract six-dimensional space and for three particles, it exists in nine dimensions, and so on. "We need to expand our imaginations, widen our view of what constitutes fundamental reality," says Valentini.

He draws parallels with the time when no one believed in an electric field, which permeates all of space and tells charged particles such as electrons how to behave. Without the intermediary of an electric field, it was tremendously difficult to predict how charged particles would react with each other. "Similarly, people might not like to believe in a wave function that lives in an abstract space and tells quantum systems how to behave," says Valentini. The new work, however, proves that it is difficult to find a theory which predicts the behaviour of quantum systems without that property.

That could have implications for how we interpret the seemingly counter-intuitive predictions of quantum theory, a major source of debate ever since the theory's inception (see "The way they tell it"). Valentini believes it could be the most important shift in the foundations of quantum theory for almost 50 years, when physicist John Bell argued that quantum theory allows something then thought scarcely believable - that making a measurement on a particle could instantaneously influence another particle on the other side of the globe. It was another 18 years before Alain Aspect at the University of Paris South in France proved experimentally that Bell's theorem was correct.

In the interim, the abstract notion had surprisingly practical applications, such as protocols for transmitting cryptographic signals securely using quantum mechanics. It may take just as long before the ideas of Pusey and his colleagues are vindicated - or not, as the case may be. Pusey is confident that initial experiments are "fairly well within reach of current technology", but devising an experiment that could silence all the criticisms may take a little longer.

Valentini remains enthusiastic. "I certainly hope that the new work will lead to the discovery of new phenomena," he says. Most importantly, however, he thinks it will open up new avenues in research into the foundations of quantum theory. "You've seen nothing yet," he says. "Mark my words, this will run and run."

The way they tell it

Historically, there have been more than a dozen different interpretations of what quantum theory means (New Scientist, 22 January 2011, p 30). The chief problem is reconciling the many possibilities for a quantum particle's state that are encoded in its wave function with the fact that we can only ever measure one of them. If the wave function is indeed associated with something physically real, that would favour certain ways of achieving that trick.

One interpretation given a fillip by the latest work (see main story) would be the "many worlds" interpretation put forward by US physicist Hugh Everett in 1957. According to this view, all those possibilities encoded in the wave function are actualities that play out in parallel realities. We observe only one of these possibilities because we are confined to one reality - one branch of the wave function. That wave function, with all its parallel strands, is ultimate reality.

Another interpretation in which the wave function is real is the "collapse interpretation". Here, the wave function contains all possibilities until some random event inherent to the wave function collapses it down to one possibility.

Yet another is the de-Broglie-Bohm "pilot wave" theory, proposed in 1927, the year after Schrödinger devised his equation. De Broglie maintained that every quantum particle possesses an invisible pilot wave, which runs alongside it and tells the system how to behave. He even used the interpretation to predict that particles of matter would display wave phenomena such as interference, something also observed in 1927.

Despite this success, pilot wave theory fell out of favour. It was supplanted by the view of Niels Bohr and Werner Heisenberg, who believed that the wave function encapsulates merely what we can know about reality rather than reality itself. If that turns out not to be the case, the pilot wave interpretation could be in with a shout again.

Marcus Chown is the author of Tweeting the Universe (Faber & Faber, 2011)

Issue 2875 of New Scientist magazine
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Have your say

A Wave Function Is Not A Probability Function

Fri Jul 27 17:25:41 BST 2012 by Eric Kvaalen

There's a huge difference between a wave function and a probability function. A probability function has values that are non-negative real numbers, whereas a wave function has values that are complex numbers -- that means negative numbers, imaginary numbers like i (the square root of −1), and combinations of real and imaginary numbers. The "inner product" of the wave function with itself gives the probability function. But that doesn't mean that it's just another way of expressing the probability function. The wave function contains information that the probability function does not contain. For example, the wave functions for an electron in an orbital can have opposite angular momentum quantum numbers (so opposite angular momenta), but the same probability function.

On the other hand, we can never say what the actual value of the wave function is (which we can do with a magnetic or electric field). Multiplying the wave function by a number like i or −1 gives an exactly equivalent wave function. So its phase, although important, is not determined!

"...John Bell argued that quantum theory allows something then thought scarcely believable - that making a measurement on a particle could instantaneously influence another particle on the other side of the globe."

This is another thing that New Scientist almost systematically gets wrong. One cannot say that there is an instantaneous influence. All one can say is that there are only certain possible combinations of measurements. For instance finding that both of two photons are polarized vertically, or both at some other angle, but not one at one angle and the other at an angle perpendicular to it. It doesn't matter in which order you make the measurements, and in fact the order of two distant events may depend on whom you ask (of observers moving at different velocities). So how can one say that "the first measurement influenced the other particle"?

A Wave Function Is Not A Probability Function

Mon Jul 30 23:03:59 BST 2012 by Julian Mann

Eric, you make a very good point regarding Wave Functions and Probability theory. Wave Theory is an example of Applied Maths, I suppose. The question that needs to be addressed is why QM is subject to Wave Mechanics in the first place and not beholden to Classical Physics? To my mind all the varied explanations usually given for this are faulty. I think the answer must be to do with an alternative time dimension. That might be the reason behind the use of Complex Numbers etc in the Wave Function. If this is correct, it follows that QM operates in that alternative time dimension, rather than in Classical time. (Hence the FIT/BIT in Villata's work) thus both Classical Mechanics and QM are real, but are subject to different time frames. This is an obvious paradox, as I am saying that a football for example is subject to Classical Mechanics operating in Classical Time, but at the sub-atomic level, the individual atoms follow BIT and not FIT.

With regard to Entanglement, I think you are wrong. You can say that there is an Instantaneous connection between the Entangled parts of a Photon for example. Although this appears to contradict SR, if we say that the connection is made at light speed, but in BIT and FIT, SR is not violated. We cannot perceive motion in BIT. To us who can only view such events in FIT, such motion appears to have happened instantly.

The same reasoning can be applied to Gravitation which is assumed to be subject to C(GR), however the force may propagate in BIT rather than FIT and appear to us to be instant, whereas it probaly took a finite amount of time of the BIT variety.

A Wave Function Is Not A Probability Function

Tue Jul 31 06:50:31 BST 2012 by Eric Kvaalen

Let's say you have two entangled particles 300 000 km apart, and you measure the spin of one, particle A. You say that this instantaneously influences the other, particle B. Half a second later we measure particle B, and find that the second measurement is in accordance with the first.

But now let's look at this situation from the point of view of an observer moving very fast in the direction going from A toward B. In the frame of reference of this observer the two measurements occur in reverse order. So we cannot say that the measurement of A instantaneously influenced B. The measurement of B took place first!

A Wave Function Is Not A Probability Function

Tue Jul 31 12:08:33 BST 2012 by Julian Mann

I don't think you are technically correct on this one Eric. These are not separate particles we are dealing with here, rather say a single photon which has been split at source. The laws of Classical Mechanics/Relativity do not apply to this scenario. The 2 halves of the split particle still behave for all intents and purposes as if they were joined, although they might be on opposite sides of the universe. Thus the sum total of the measurements carried out on the parts is the same as it would be if the original split had not occurred.

The idea behind Entanglement could equally be applied to for example the Double Split experiment, where only a single particle is used and yet it traverses both slits at the same time. I would think that "one half" of the original particle say a photon is travelling through one slit in BIT, whereas it's partner is going through in FIT

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