Schrödinger's equation — what does it mean?

In the first article of this series we introduced Schrödinger's equation and in the second we saw it in action using a simple example. But how should we interpret its solution, the wave function? What does it tell us about the physical world? We went to speak to Tony Short and Nazim Bouatta, both theoretical physicists at the University of Cambridge, to find out.

Solvay conference

This picture was taken at the 5th Solvay conference in 1927. Wolfgang Pauli is fifth from the left in the back row, Werner Heisenberg 6th from the left in the back row, Louis de Broglie is 7th from the left in the middle row, Max born 8th from the left in the middle row, Niels Bohr 9th from the left in the middle row, Max Planck is second from the left in the front row and Albert Einstein is fifth from the left in the front row.

Schrödinger’s equation is to quantum mechanics what Newton’s second law of motion is to classical mechanics: it describes how a physical system, say a bunch of particles subject to certain forces, will change over time. In classical mechanics what you’re after are the positions and momenta of all particles at every time $t$: that gives you a full description of the system.

In quantum mechanics the information about the system is contained in the solution to Schrödinger’s equation, a wave function $\Psi .$ The square of the absolute value of the wave function, $|\Psi |^2,$ is interpreted as a probability density. For example, with our particle in a box $|\Psi (x)|^2$ gives the probability density for finding the particle at position $x.$ But it is also possible to solve Schrödinger’s equation for many particle systems and to find wave functions for other observable quantities, for example the momenta of the particles.

Where are all the waves?

So what, exactly, does quantum mechanics tell us about physical reality? Schrödinger's equation grew out of the idea that particles such as electrons behave like particles in some situations and like waves in others: that's the so-called wave-particle duality (see the first article of this series). One question that comes up immediately is why we never see big objects like tables, chairs, or ourselves behave like waves.


As a heuristic argument, recall de Broglie’s relationship between the wavelength $\lambda $ and the momentum $p$ of a "matter wave":

  \[ \lambda =h/p, \]    

where $h$ is Planck’s constant.

(Again see the first article of this series.) The momentum $p$ of an object is its mass times its velocity. One consequence of quantum mechanics is that no object is ever completely at rest, so $p$ will never be zero. But Planck's constant $h$ is so incredibly small, $h=6.626068 \times 10^{-34} m^2 kg/s,$ that even the tiniest bit of mass and velocity make the wavelength $\lambda $ vanishingly small as well. So small that we usually don't perceive the waviness of macroscopic objects. (We say "usually" here because this general argument does not always apply: there are situations in which quantum effects do become perceptible at a macroscopic scale.)

A wave is not a wave

The next question is how to interpret the wave function. Unlike a classical description of a physical system, the wave function does not give us definite information about the location of a particle at a given time t — it only gives us the probability of finding the particle in a given location at time t. We normally use probabilities to quantify our ignorance: if I say a coin has a fifty-fifty chance of coming up heads or tails, then this simply reflects the fact that I don't yet know how it will come up after the next throw. So perhaps the probabilities given by the wave function measure our ignorance in a similar way: as the particle is moving around in a box, say, it's somewhere definite all the time, only we just can't tell where until we make a measurement. De Broglie did indeed pioneer such a deterministic approach. It was later developed by David Bohm and has become known as the pilotwave or causal interpretation of quantum mechanics, or as Bohmian mechanics. But it's a minority view. Most physicists believe that experiments like the double slit experiment suggest that a particle really can become delocalised in space. "There is a sense in which the particle is in all these different places at once, but the worry about saying this is that there still is only one particle," says Short.

Solution

The particle in a box: the position of the particle is shown on the x-axis and the energy on the y-axis. The permitted energy levels for the first four quantum numbers are shown as horizontal dotted lines. The wavefunctions are shown superimposed on the diagram at the corresponding energies. Image: Papa November.

So perhaps the wave function describes a physical wave in space along which the particle is spread out like goo — we never see this particle goo because somehow it contracts to one point when we make a measurement. That’s an idea to banish from your head straight away! In our particle in a box example, the solutions to Schrödinger’s equation,

  \[ \psi _ n(x) = \sqrt {\frac{2}{L}} \sin {\left(\sqrt {\frac{npx}{L}\right),}} \]    

for $n = 1,2,3,4, ..$ do indeed describe waves along which the particle "goo" might conceivably be spread out. But this picture collapses once you have several particles. Suppose, for example, there are three of them. In this case the wave function is a function of many variables (three coordinates for the possible positions of each of the three particles and time) and in general it is not possible to break it up into components corresponding to each particle. Now you can’t even plot this "wave" because you’d need more than three dimensions to do it. In general the wave function doesn’t describe a physical wave because it isn’t a function defined on physical space. Rather, it’s defined on configuration space: it takes as input all the possible configurations of locations the particles could be in and it returns a value related to the probability that you will find the particles in the given configuration at the given time.


The fact that you cannot always neatly separate the wave function of a many-particle system into individual components illustrates another weirdness of quantum mechanics: that two particles that have once interacted, so that the system they form is described by a single wave function, can remain mysteriously linked even when they have moved light years apart. This mysterious connection is called quantum entanglement. When something happens to one of the entangled particles, a corresponding thing happens to its distant partner, a phenomenon Einstein described as "spooky action at a distance".

But while the wave function generally doesn't represent a straightforward wave in three-dimensional space, the question remains whether there is some sort of physical wave associated to it. Several physicists, including de Broglie, Schrödinger and Bohm, believed that there should be, but although their efforts to find one still continue today, they have not resulted in theories that enjoy mainstream approval.

Others, including Wolfgang Pauli, Werner Heisenberg and Niels Bohr were against this realistic picture and regarded the wave function as a mere mathematical tool to provide probabilities. Indeed, they argued that questions such as "where is the particle when we are not looking" are meaningless: science cannot describe nature per se, but only our knowledge of it. So the only kind of questions we can answer are questions about possible outcomes of measurements. And that's precisely what the wave function gives us. This view is known as the Copenhagen interpretation of quantum mechanics. It's in stark contrast to the intuition classical physics is based on: that there exists an objective reality even when we're not looking and that science can describe that reality.

The measurement miracle

Whether there’s a physical wave or not, the big question that remains is what happens when we come along and make a measurement. We will only ever find the particle in one spot, yet Schrödinger’s equation tells us nothing about why this should be the case, or where that spot should be. Its solution only gives us probabilities. And there is another aspect of the equation which illustrates this tension with reality even more forcefully: it’s a linear equation. This means that if some wave function $\Psi _1$ is a solution and some other wave function $\Psi _2$ is also a solution, then the sum $\Psi = \Psi _1 + \Psi _2 $ is a solution too. But $\Psi _1$ and $\Psi _2 $ could correspond to vastly different situations, for example $\Psi _1$ might correspond to the particle being on the Moon (so it’s zero for all locations that are not on the Moon) and $\Psi _2$ might correspond to the particle being on Earth (so it’s zero everywhere apart from on Earth). Since the sum $\Psi _1 + \Psi _2$ is also a solution, there is a sense in which the particle is in both places at once. When this happens we say that the particle is in superposition of the two states $\Psi _1$ and $\Psi _2$.

"When we do an experiment we don't see all these superimposed solutions, we see only one of them," says Bouatta. "This is in tension with the equation. When you look only to the equation you don't understand what happens at the measurement." There is no consensus among physicists as to how this measurement miracle comes about. "Most people probably don't commit to an interpretation, they just say we're not sure," says Short. But there are several schools of thought that try and provide an answer.

Collapsing waves

One of them asserts that when a measurement is made, the particle somehow "decides" where it is going to be. The corresponding wave function then simply collapses. In our particle in a box example, the wave function was non-zero at many places in the box, reflecting the fact that there's a non-zero probability of finding the particle at these places. Once we have opened the box and found the particle in a definite location, it's definitely nowhere else, so the wave function now has a single non-zero value at that location and is zero everywhere else — if you plot it, which you can in this simple example, it will have a single spike at the non-zero location. When we make a measurement an instant later, the particle is very likely still in the immediate vicinity, so the wave function spreads out a bit, but still has a single peak. Over time it will spread out more and more.

One problem with the collapse approach is how distant parts of the wave function "know" that a measurement has been made and thus that they are supposed to collapse. For example, suppose a particle is in superposition between two locations, as described above, and that one of these locations is on the Earth and the other on the Moon. If an observer on Earth now detects the particle, the Moon wave function has to vanish instantaneously. But Einstein's theory of relativity says that no message and no signal can travel faster than light.

Roger Penrose

Sir Roger Penrose is one of the scientists who have toyed with the idea that consciousness might be necessary to collapse the wave function. Image: Festival de la Scienza.

The collapse idea also raises another question: what is a measurement? Some physicists, including Eugene Wigner and Roger Penrose, have toyed with the idea that a measurement requires an observer and that it is the consciousness of the observer that causes the collapse (which then begs the question of whether a snail, say, has enough consciousness to collapse a wave function). But this approach has largely fallen out of favour. Instead, a measurement is defined as an interaction between the system you're measuring and the measuring device. "For example, an interaction could be: if the particle is on the right, move the pointer on my device to the right and if it's on the left, move the pointer to the left," says Short. "You can model this easily in normal quantum theory."

The challenge for advocates of the collapse approach is to come up with models that describe the workings of the collapse — how exactly does it happen and what causes it? There's a minority view that help may come from the one force that physicists haven't yet been able to reconcile with quantum mechanics: gravity. Finding a unified theory of quantum gravity is the holy grail of modern physics and some people believe that it might shed light on the collapsing behaviour. But this is a highly speculative approach.

Many worlds

As it stands, the idea that the wave function collapses after a measurement needs to be postulated as an extra rule of quantum mechanics. Pulling a new law of nature out of a hat like that isn't a very satisfactory solution for purists — but there is another approach: perhaps all the possible outcomes of a measurement are equally real. "The idea is that there are different worlds that are all real and in each of them the particle is in a different position," explains Short. The problem then becomes how to interpret the probabilities given by the wave function. "You can regard it as a kind of weight you ascribe to each world. If you pick a world at random you're most likely to pick [one with a bigger probability]."

Schrödinger's

Schrödinger devised a famous thought experiment in which a cat in a box is in superposition of two states: dead and alive. But when you open the box to observe the cat you only ever find it in one of these states. According to the Everett interpretation, when you make the observation the world splits into two branches: in one of them the cat is dead and in the other alive. Image: Dc987.

This is already pretty weird when you're only thinking of tiny little particles. But what about us, the observers? If you include them in this many worlds view, you get the so-called Everett interpretation of quantum mechanics (named after the physicist Hugh Everett). "Suppose you have a little microscopic particle that could be here or there, and then I look at it," says Short. "A collapse model would say that this really decides the issue, the particle makes up its mind and I see it over here or I see it over there, but there still is only one truth to what I see. But macroscopic objects are made up of particles, so there is no reason to believe they should obey different laws of physics. And if you apply [to me, as an observer,] the same laws of physics that you apply to microscopic objects, what you find is that you get this superposition of 'the particle is here and I saw it here' and 'the particle is there and I saw it there'. The Everett interpretation says, well maybe this is what happens. It seems pretty crazy because there are then two copies of me seeing different things. But each of these people, if you ask them if they saw something reasonable, will say yes."

The obvious question now is why we're never aware of these other copies of ourselves. But there's a straightforward answer. Two separate worlds could only interfere with each other if their wave functions were non-zero in the same region in configuration space — that's the possibility space containing all the different configurations of the particles that make up you, the particle you're observing, the measuring device, and so on. But this means that the two copies of you must have identical memories. "In order for two [copies of you] to get mapped into the same state in configuration space, you essentially have to unwind everything in their brain," says Short. "If they have any memory then they're not going to the same place in configuration space because that memory is part of them. If their brain is different, then they can't meet." So only complete amnesia would enable you to meet and merge with yourself. If you're suffering from that, you probably wouldn't notice the merger; and even if you did nobody would believe your incoherent ramblings anyway.

The Everett interpretation may lose out on common sense appeal, but you can argue that it gains in terms of mathematical simplicity. "People say that it seems completely unreasonable, parallel worlds and all this craziness, but actually it's very simple: all you do is take Schrödinger's equation and keep it all the way up to the macroscopic level. Everett simply drops the second law [which postulates the collapse of the wave function] completely," says Short. And Bouatta suggests that we shouldn't push our common sense perception of the world too far. "We have an intuition of our every day life — chairs, tables, birds, etc — and we try to apply this to all different regimes in the Universe: the largest scale but also the smallest scale. But why should the world at every scale be described by our common sense, everyday-life intuition? The consequences of Everett may seem a bit strange, but one may argue that this is the most conservative approach to quantum mechanics because you don't have to [introduce new] laws."

Collapse models and the Everett view are among the most prominent interpretations of quantum mechanics, but there are others too. The truth is that we simply don't know yet what really goes on in the physical world and how to interpret the mathematical formalism that describes it so well. What seems certain is that we have to radically expand our view of the world, but this wouldn't be the first time: who would have believed a thousand years ago that the Earth is just a tiny spherical speck in a vast expanding Universe?

But if you still can't help being weirded out by all this, take consolation from a famous quote that has been attributed to Richard Feynman: "Anyone who claims to understand quantum theory is either lying or crazy."

You might also want to listen to our podcast, Does quantum physics really describe reality, featuring Roger Penrose and John Polkinghorne.


About this article

Nazim Bouatta is a Postdoctoral Fellow in Foundations of Physics at the University of Cambridge.

Tony Short is a Royal Society Research Fellow in Foundations of Quantum Physics at the University of Cambridge.

Marianne Freiberger is Editor of Plus. She interviewed Bouatta and Short in Cambridge in May 2012. She would also like to thank Jeremy Butterfield, a philosopher of physics at the University of Cambridge, for his help in writing these articles.

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