What is a wave function in quantum physics?
Such a question generates long and loud debates among philosophers of physics (and more limited debate among most physicists, who tend to prefer to make predictions using wave functions rather than wondering what they are.) I have a foot in both camps, even though I have no real credentials among the former set. But no matter; today I won’t try to answer my own question in any profound way. We can debate the deeper meaning of wave functions another time.
Instead I just want to address the question practically: what is this function for, in what sense does it wave, and how does it sit in the wider context of physics?
Schrödinger’s Picture of the World
Quantum physics was born slowly, in stages, beginning around 1900. The most famous stage is that of 1925, when Heisenberg, along with Born and Jordan, developed one approach, using matrices, and Schrödinger developed another, using a “wave function”. Both methods could predict details of atomic physics and other systems, and Schrödinger soon showed the two approaches were equivalent mathematically. Nevertheless, he (and many others) felt his approach was more intuitive. This is why the wave function approach is emphasized — probably over-emphasized — in many books on quantum physics.
Suppose we want to investigate a physical system, such as a set of interacting subatomic objects that together make up a water molecule. We start by imagining the system as being in some kind of initial physical state, and then we ask how that state changes over time. In standard first-year undergraduate physics, using the methods of the 17th-19th century, we would view the initial physical state as consisting of the locations and motions of all the objects at some initial time. Armed with that information, we could then calculate precisely what the system would do in the future.
But experimental data on atomic physics revealed that this older method simply doesn’t agree with nature. Some other approach was needed.
In 1920s quantum physics in the style of Schrödinger, the state of the system is (under-)specified by an unfamiliar object: a function on the space of possibilities for the system. This function gives us a complex number for each possibility, whose absoulte-value-squared tells us the probability for that particular possibility. More precisely, if we measure the system carefully, Schrödinger’s function at the time of the measurement tells us the probability of our measurements giving one outcome versus another.
For instance, suppose the system consists of two particles, and let’s call the possible position of the first particle x1 and that of the second x2. Then Schrödinger’s function will take the form Ψ(x1,x2) — a function giving us a complex number for each of the possible locations of the two particles. (As I’ve emphasized repeatedly, even though we have a system of two particles, there is only one wave function; I’ve given you a couple of examples of what such functions are like here and here.)
If we want to know the probability of finding the first particle at some definite position X1 and the second at a definite position X2 — assuming we do the measurements right now — that probability is proportional to the quantity |Ψ(X1,X2)|2, i.e. the absolute-value-squared of the function when the first particle is at X1 and the second is at X2.
If we choose not to make a measurement right away, Schrödinger’s equation tells us how the function changes with time; if the function was initially Ψ(x1,x2; t=0) = Ψ(x1,x2), then after a time T it will have a new form Ψ(x1,x2; t=T) which we can calculate from that equation. If we then measure the positions of the particles, the probabilities for various measurement outcomes will be given by the absolute-value-squared of the updated function, |Ψ(x1,x2; t=T)|2.
Schrödinger’s function is usually called a “wave function”. But this comes with a caveat: it’s not always actually a wave…see below. So it is more accurate to call it a “state function.”
Wave Functions Are Not Things
Probably thanks to advanced chemistry classes, in which pictures of atoms are often drawn that suggest that each electron has its own wave function, it is a common error to think that every particle has a wave function, and that wave functions are physical objects that travel through ordinary space and carry energy and momentum from one place to another, much like sound waves and ocean waves do. But this is wrong, in a profound, crucial sense.
If the electrons and atomic nuclei that make up atoms are like characters in a 19th century novel, the wave function is like an omniscient narrator. No matter how many characters appear in the plot, there is only one such narrator. That narrator is not a character in the story. Instead the narrator plays the role of storyteller, with insight into all the characters’ minds and motivations, able to give us many perspectives on what is going on — but with absolutely no ability to change the story by, say, personally entering into a scene and interposing itself between two characters to prevent them from fighting. The narrator exists outside and beyond the story, all-knowing yet powerless.
A wave function describes the objects in a system, giving us information about all the locations, speeds, energies and other properties that they might have, as well as about how they influence one another as they move around in our familiar three-dimensional space. The system’s objects, of which there can be as many as we like, can do interesting things, such as clumping together to form more complex objects such as atoms. As they move around, they can do damage to these clumps; for instance, they can ionize atoms and break apart biological DNA molecules. The system’s wave function, by contrast, does not travel in three-dimensional space and has neither momentum nor energy nor location. It cannot form clumps of objects, nor can it damage them. It is not an object in the way that electrons , photons and neutrinos are objects. Nor is it a field like the electric field, the Higgs field, and the electron field, which exist in three dimensions and whose waves do have momentum, energy, speed, etc. Most important, each system has one, and only one, wave function, no matter how many objects are in the system.
[One might argue that a wave function narrator is less omniscient, thanks to quantum physics, than in a typical novel; but then again, that might depend on the author, no? I leave this to you to debate.]
I wrote the article “Why a Wave Function Can’t Hurt You” to emphasize these crucial points. If you’re still finding this confusing, I encourage you to read that article.
Some Facts About Wave Functions
Here are a few interesting facts about wave functions. I’ll state them mostly without explanation here, though I may go into more details sometime in the future.
- It is widely implied in books and articles that wave functions emerged for the first time in quantum physics — that they were completely absent from pre-quantum physics. But this is not true; wave functions first appeared in the 1830s.
In the “Hamilton-Jacobi” reformulation of Newton’s laws, the evolution of a non-quantum system is described by a wave function (“Hamilton’s characteristic function”) that is a function on the space of possibilities and satisfies a wave equation quite similar to Schrödinger’s equation. However, in contrast to Schrödinger’s function, Hamilton’s function is a real number, not a complex number, at each point in the space of possibilities, and it cannot be interpreted in terms of probabilities. In very simple situations, Hamilton’s function is the argument (or phase) of Schrödinger’s function, but more generally the two functions can be very different. - Wave functions are essential in Schrödinger’s approach to quantum physics. But in other approaches, including Heisenberg’s and the later method of Feynman, wave functions and wave equations do not directly appear. (The situation in pre-quantum physics is completely analogous; the wave function of Hamilton appears neither in Newton’s formulation of the laws of motion nor in the reformulation known as the “action principle” of Maupertuis.)
This is an indication that one should be cautious ascribing any fundamental reality to this function, although some serious scientists and philosophers still do so. - The relevant space of possibilities of which the wave function is a function is only half as big as you might guess. For instance, in our example of two particles above, even though the function specifies the probabilities for the various possible locations and motions of the objects in the system, it is actually only a function of either the possible locations or the possible motions (more specifically, the particles’ momenta.) If we write it as a function of the possible locations, then the probabilities for the objects’ motions are figured out through a nontrivial mathematical procedure, and vice versa.
The fact that the wave function can only give half the information explicitly, no matter how we write it down, is related to why it is impossible to know objects’ positions and motions precisely at the same time. - For objects moving around in a continuous physical space like the space of the room that you are sitting in, waves are a natural phenomenon, and Schrödinger’s function and the equation that governs it are typical of waves. But in many interesting systems, objects do not actually move, and there’s nothing wavy about the function, which is best referred to as a “state function”. As an example, suppose our system consists of two atoms trapped in a crystal, so that they cannot move, but each has a “spin” that can point up or down only. Then
- the space of possibilities is just the four possible arrangements of the spins: up-up, up-down, down-up, down-down;
- the
wavestate function doesn’t look like a wave, and is instead merely a discrete set of four complex numbers, one for each of the four arrangements; - the absolute-value-squared of the each of these four complex numbers gives us the probabilities for finding the two spins in each of the four possible arrangements;
- and Schrödinger’s equation for how the state function changes with time is not a wave equation but instead a 4 x 4 matrix equation.
- So although the term “wave function” suggests that waves are an intrinsic part of quantum physics, they actually are not. For the design and operation of quantum computers, one often just needs state functions made of a finite set of complex numbers, as in the example I’ve just given you.
- Another case where a state function isn’t a wave function in the sense you might imagine is in quantum field theory, widely used both in particle physics and in the study of many materials, such as metals and superconductors. In this context, the state function shows wavelike behavior but not for particle positions, in contrast to 1920s quantum physics. More on this soon.
- For particle physics, we need relativistic quantum field theory, which incorporates Einstein’s special relativity (with its cosmic speed limit and weird behavior of space and time). But in a theory with special relativity, there’s no unique or universal notion of time. Unfortunately, Schrödinger’s approach requires a wave function defined at an initial moment in time, and his equation tells us how the function changes from the initial time to any later time. This is problematic. Because my definition of time will differ from yours if you are moving relative to me, my form of the wave function will differ from yours, too. This makes the wave function a relative quantity (like speed), not an intrinsic one (like electric charge or rest mass). That means that, as for any relative quantitiy, if we ever want to change perspective from one observer to another, we may have to recalculate the wave function — an unpleasant task if it is complicated.
Despite this, the wave function approach could still be used. But it is far more common for physicists to choose other approaches, such as Feynman’s, which are more directly compatible with Einstein’s relativity.
23 Responses
So, a system, any system, has ONE wave function, so it like a system can be characterized by one polynomial and every term could be viewed as the interaction between the particles (quantum states).
So, if you compare the electric force, Fe = k q1 q2/r2, and the gravitational force, Fg = G m1 m2/r2, they have similar form because the influence spacetime in a similar way.
So, shouldn’t the wave function of these two systems also be similar? I would also include that there are theories that electric charge, q, is a function of changing masses, m. Much like the Dirac fermion.
Is it possible that the key of finding a unified theory lies in understanding the relationship between mass and electric charge? To be more precise, like EMR which has both electric and magnetic fields, could the gravitational field, which I believe to be fundamental, have three components, three fields, mass, electric, and magnetic.
“Westart” -> “We start”
“Unfortunately, Schrödinger’s approach requires a wave function defined at an initial moment in time, and his equation tells us how the function changes from the initial time to any later time. This is problematic.”
This seems to be equivalent to what is required for fields as well — different observers get different slices of the field and swapping viewpoints requires some mathematical complications. Is this *so* problematic?
I have found that for some listeners … Conservatively, the wave function is a bookkeeping tool used to calculate what happens in systems where quantum effects are important. Its “reality” does not impact its utility. Maybe it’s a real thing in the Universe and maybe it only exists as much as any mathematical abstraction does. This is not the content of today’s discussion.
Well, as I said, you can do it. But it’s almost never useful in particle physics, and rarely in more general quantum field theory. The other problem, which I have not emphasized because it is too early, is that the Schrodinger wave function for a field is mathematically horrendous to write down, and unnecessarily so, simply because the ground state of a field theory is very complicated. We’ll see more of that quite soon.
Isn’t the square of a complex number generally another complex number? If so, isn’t the probability given by the complex number multiplied by its complex conjugate? This is what I remember from my modern physics class several decades ago.
Whereever I have said square I implicitly mean “absolute-value squared.” I will double check to see where I neglected to say this explicitly and fix it.
Dr. Strassler,
Is there a name for the problem of two confined “objects” with spin states by which I could search for an explicit solution?
No, other than “two-spin system”. The solution in any case depends on the magnetic fields impinging on the objects, which I haven’t specified, and on whether the two spins interact with each other, which I haven’t specified either.
Thanks. I actually found much (but not all) of what I had been seeking at the link,
https://phys.libretexts.org/Bookshelves/University_Physics/Radically_Modern_Introductory_Physics_Text_II_(Raymond)/19%3A_Atoms/19.01%3A_Fermions_and_Bosons
As you have observed, much more must be specified than what I had asked about.
I’ve always struggled to reconcile this: “Because my definition of time will differ from yours if you are moving relative to me, my form of the wave function will differ from yours, too. This makes the wave function a relative quantity …” with the idea that there is exactly one wavefunction that describes everything, even galaxies separated by 100s of Mpc. Are theoretical physicists generally ok with this?
I also struggle to reconcile it with “… ascribing any fundamental reality to this function, although some serious scientists and philosophers still do so”. The Everettian MW interpretation only seems to explain anything if one regards the wavefunction as being ontological, being the (one) thing that “exists” at the most fundamental level. Yet how can that be reconciled with relativistic notions of time?
[1] Yes and no; defining time across the whole universe, all the way back to the Big Bang, and across a complicated curved surface, is a non-trivial discussion even in classical gravity, much less a quantum system where gravity should somehow be included.
[2] I don’t know. I’m not sure if others know. It’s a good question for Sean Carroll, as he has thought about the many-worlds interpretation far more than I have, and I imagine he is far more clear about how this plays with general relativity (whose issues dwarf the problems with special relativity that I have mentioned here.)
Something I’ve been wondering about for a while is the extent to which it’s possible to talk about the wave function of a smaller system when presumably the larger system that encompasses it must have its own state function.
In the MWI, as I understand it, there is a universal wave function. But we use wave functions to describe much smaller systems within it (and obviously we get correct results). Can you post sometime about how we make sense of this?
When a subsystem within a larger system is isolated, so that there is no entanglement with the rest of the system and no interactions between it and the rest of the system, the wavefunction of the larger system can be written as a product Psi_S(X)*Psi_E(Y), where X are the degrees of freedom of the subsystem S, and Y are the remaining degrees of freedom in the larger system. Then we can treat Psi_S(X) as the wavefunction of the subsystem.
See my answer to Domenico cabras. The answer I’ve given is vague, but to be more precise would take us too far afield; it is too early to discuss the subtleties of making measurements. (Matthew Dickau’s answer is correct, but leaves out the question of how we arrange for this to happen, in some approximation.
“… it is a common error to think that … wave functions are physical objects that travel through ordinary space and carry energy and momentum from one place to another, much like sound waves and ocean waves do. But this is wrong, in a profound, crucial sense.”
When a cold neutron reflects coherently from a smooth surface on a pool of mercury, does it impart downward momentum to that pool?
Yes. Irrelevant to the question; you are apparently making the common error.
Think of it as a terminology question. Using a slow source, you have a neutron whose mathematical representation during reflection impact has the form of a wave, which in this case is best modeled as containing only one neutron. You agree that this wave-like entity imparts momentum to the mercury, making its reflection an experimentally recorded, historical event. (BTW: For clarity, would you agree that this momentum transfer should be distributed across the entire mercury surface, rather than localized to, say, one atom?)
Yet this decidedly wave-like version of a single neutron remains fully wave-like at all times, including the ability to reflect coherently again from another surface, again without reverting into a point-like neutron. Enough such reflections could even hurt you, since over time they would apply enough momentum (as you noted) to, say, push a button that then causes harm to you.
I understand the concept of defining some version of a wave that, by definition, cannot impart momentum any other body. My question is this: What do you call the wave-like entity that keeps imparting momentum to smooth surfaces without reverting to particle form?
Wavicle [or “particle”, with quotation marks.] It is not a question of the wave function of the system, which describes both the neutron and the mercury together.
Generally, we have a long way to go before we can understand the wave-like properties of a composite system like a neutron in quantum field theory. We have to first understand in what sense a simple spinless boson is a wavicle, and then why an electron or photon is a wavicle, and then have to think very carefully about something like positronium, and only then do we have a chance of understanding a neutron in this context. And finally, we have to understand what coherent reflection means, as opposed to local absorption, and the conditions under which we have one rather than the other. Remind me when we get to the summer to make sure I don’t forget to address this.
Some philosophers of physics seem quite keen on the pilot wave interpretation ( Bohm theory ).
Given the wave function is defined on configuration space and that it “can’t hurt you”, would you simply dismiss the pilot wave interpretation ?
You are getting way ahead of me. As a general rule, we would be well-advised to go through the facts themselves with extreme care before we try to interpret them.
That said, I would be very concerned about any intepretation of quantum physics that doesn’t work well for quantum field theory, as I believe is still the case for pilot-waves (which would be pushing fields around in the space of possible field shapes, not particles around in the space of possible particle positions.)
Your blog is the only place I have encountered the notion that there is one wave function per system, not one wave function per particle. Thank you for clarifying this! That leads to the question, how do you identify a system? If I have two pairs of entangled particles, one pair in New York and one pair in LA, do I have two systems, or is it one system? I am guessing that it is really one system, but for all practical purposes we can treat them as two separate systems. Intuitively, it would seem that distance is a good guide for delineating systems — the further apart two particles are, the less error we will introduce by considering them as belonging to two systems instead of one. Unless they are entangled with each other, then distance is irrelevant. Can you shed some light on this?
See my answer to Domenico cabras; this is a long story, quite subtle, and I don’t want to go into this yet.
Thank you, Dr. Strassler. Three questions. First, is there any logic behind the selection of the “system” described by the wave function beyond “dealer’s choice”? I assume the Cosmos, as a system, could be described by one wave function. Second, and related to the first, what are your views on QBism in this context? I understand that QBists deem the wave function to be the agent’s subjective probability assessment. Third, is there any relationship between the wave function and Bayesian approaches? For example, I think Bayesian cognitive scientists assign “probabilities” to sets whose members belong to an outcome space (described by the Greek letter omega).
[1] See my answer to Domeinco cabras below. [2] As a general rule, we would be well-advised to go through the facts themselves with extreme care before we try to interpret them. [3] To a degree, perhaps; but see [2].