Transcription
So the wavefunction can be thought of as the assignment of a special complex number to every point in three-dimensional physical space. You do this operation, it's called mod-squaring, and then it tells you the probability, roughly speaking, with which a measurement will find a particle at that location.
So it's very easy to think that a wavefunction is like a field, like the electric field, in three-dimensional space. But if you have two particles instead of one, then the possibility space is much more complicated. We call this possibility space configuration space because each possible configuration of a two-particle system requires six numbers, not three. You need to know x, y, and z for the first particle, and you need to know x, y, and z for the second particle.
So the wavefunction assigns a complex number to a point in a six-dimensional space. Six dimensions because you need six numbers to specify its points. And that means a two-particle wavefunction is a function, we say, whose domain is a six-dimensional space. Six-dimensional space is not physical three-dimensional space. If you've got three particles, it's a nine-dimensional space. If you've got ten particles, it's a thirty-dimensional space.
And actually, our universe does not have a well-defined number of particles at all. Our leading physical model for the universe, at least for the non-gravitational parts of the universe, is the standard model, which is based on a set of models known as quantum field theories. In quantum field theories, particles are emergent excitations of these sort of delocalized entities called quantum fields. And the number of particles can change from moment to moment. It's not always well-defined.
So it's not even clear how to think about wavefunctions that live in anything like physical space in the universe as we know it. I don't mean to say that we're teaching our students wrong. We're not. I mean, students will learn this as they go on in their physics trajectories. But I think a lot of people on the outside or new students who haven't yet started their own journey in physics have a certain idea about what wavefunctions are like that's actually quite different from the way we use them in practice.
I don't know if that clarifies what you were asking about. Okay. So you have a lecture about how Hilbert spaces are also not real. I'd like you to talk about that. Now, it's my understanding that you more precisely said Hilbert spaces are redundant in the same way that gauge symmetries are redundant, or the gauge transformations are redundant.
So this point, if it's all right, let me do a little bit of history, if it's okay. So where does quantum theory begin? Lightning grand tour of quantum theory. There are these actual physical objects called black bodies. They are chambers that are heated, and a little hole is poked in them. I mean, you don't really poke a hole in them, it's more complicated.
But this is, I'm very much a theorist, and my picture of experimental physics is it should be better. And I apologize in advance to anyone who's an experimentalist listening to me. But these were actual systems. People actually built these things. They were these chambers, they would heat them up. And then they would have a little hole in the chamber through which they could see radiation coming out of them.
And they could put that radiation through experimental devices that could show them how intense the radiation was as a function of wavelength. We all know that different wavelengths of light correspond to different colors in the visible range. And if you go outside the visible range, you have very long wavelengths that are infrared, or microwave radiation, or radio waves.
And at the very short wavelengths, you've got x-rays and so forth, gamma rays going all the way out, ultraviolet x-rays, gamma rays going all the way out to very short wavelengths. And you could just ask yourself, if I plot how strong the radiation is as a function of wavelength, what kind of pattern would I expect to see?
And the pattern that was revealed in experiments was difficult to explain on first principles theoretical grounds. A very important physicist, Max Planck, in 1900, found a way to generate a theoretical prediction of what this black body radiation curve should look like, and it agreed with experiments. But in order to get there, he had to kind of hack his formulas.
He had to introduce a fudge factor. He had to assume that radiation in this chamber could only occur in quantized amounts. That is, there were different wavelengths you could produce, but each wavelength could only be excited in discrete steps. This led to the quantum hypothesis. This was his quantum hypothesis.
And there was a sort of parameter that told you how big you wanted to make these steps. Today, we would call it a regulator. And this parameter, we now call it h, little h. We call it Planck's constant. Planck introduced it. And then I think at one point, his idea was he wanted to introduce an intermediate step in the calculation.
And once he calculated everything, send this clearly, in his mind, unphysical parameter to zero. But whenever he tried to send it to zero, he would get the wrong results. He realized this parameter was not zero. It was very small. It was, today, it has a value, you know, that's of order 10 to the negative 34, 10 to the negative 33 in conventional units.
And he couldn't get rid of it. And so he had, people had to accept that there was this basic discreteness in nature he couldn't explain. And then for the next 22, 23 years, the period from 1900 to 1922, 1923, physicists working on these questions were living in a time that today we call the old quantum theory, a paradigm we call the old quantum theory.
The old quantum theory was based on a mixture of physical pictures of particles moving around in space, in orbits, like in atoms, and then a collection of ad hoc formulas and rules that people didn't fully understand that seemed to capture some of the observations that were coming out of experiments. But it was a very murky time.
There wasn't anything like an underlying theory from which this whole picture emerged. And the formulas weren't perfect. They didn't work exactly right. In 1913, Bohr proposed, Niels Bohr proposed his famous model of the hydrogen atom. This is a model in which, yeah, you've got, you know, the nucleus, and for the hydrogen atom, it's just a proton, positively charged proton in the middle, and an electron going around.
And by making use of these sort of heuristic formulas that were characteristic of the time, Bohr argued that the electron could only be in certain definite orbits, and when it transitioned between them, its energy changed in discrete steps, and whenever its energy changed, it would either absorb or emit radiation in discrete amounts.
And with this, he was able to account for the particular colors of radiation that we would see emitted from excited hydrogen atoms or that were absorbed by hydrogen atoms. The so-called line spectra. And we also found that other atoms had line spectra as well. Famously, helium was discovered because we looked at the spectral lines from the sun, and after accounting for all the spectral lines we knew about, there were still some more that we didn't account for.
And so people conjectured there was a new element. They called it helium from Helios the sun, and eventually we found helium on Earth, but it was first discovered in the sun. So this was obviously a very important thing, and Bohr ended up winning the Nobel Prize in 1922 for this Bohr model.
But by around 1922, certainly at least by then, people were becoming very skeptical because people couldn't take these clear pictures of particles moving in definite ways, particles interacting with fields like the electric and magnetic field, which were well understood by this point. They couldn't take this world picture, this ontology, this picture of what was physically out there.
They couldn't find laws that when combined with this picture gave you the right predictions, that gave you an empirically adequate, empirical meaning subject to experiment, empirically adequate, a theory that was able to account for the things we were seeing and make predictions that were regularly confirmed.
Niels Bohr gave a bunch of lectures in 1922. Heisenberg, Werner Heisenberg, attends one of these lectures, at least one of them. And the story goes, and I'm not sure quite how much of the story is apocryphal or how much really happened, but the story goes that Heisenberg objected at one point to this lecture.
Now, Heisenberg at this point was 20 years old, maybe 21. Niels Bohr was a Nobel Prize winning physicist, widely regarded as the quantum whisperer, someone who just innately, intuitively understood quantum mechanics in a way that no one else did. And here was Heisenberg challenging Bohr. People, I think, probably could have been expected to have thought that Bohr would be upset at this.
But the story goes that after the talk, Heisenberg took a long walk with Bohr, a many hours walk, and it had a huge impact on how Heisenberg thought about nature. The story, the legend goes that Bohr revealed to Heisenberg that, as Bohr put it, he was not convinced there were particles or orbits after all.
You start seeing comments by people like Sommerfeld. Arnold Sommerfeld was actually Heisenberg's doctoral advisor in Munich where Heisenberg was. And then people like Wolfgang Pauli, many of the people sort of at the foundations of this developing theory were increasingly skeptical that this world picture could survive.
And then in 1925, Heisenberg, who was visiting the premier institution in mathematics and theoretical physics at the time, Göttingen, where people like David Hilbert were, Max Born was there, Felix Klein, top mathematicians and physicists. Heisenberg was there visiting. He was visiting Max Born's group, Max Born and his student, Max Born and his student, Pascal Jordan.
And he was thinking about all these questions, and he was very confused about all of it. There were all these new, more complicated formulas that people were writing down that were connecting energy levels and transitions and atoms to spectra. And he develops this very terrible case of hay fever. In the spring of 1925, he goes out to Heligoland, which is this island where the pollen levels are much lower.
And he comes back with this totally bizarre draft of a paper. And he gives this paper to Max Born and to Jordan, and then he goes on a vacation again. He goes on some vacation. He's exhausted from all of his intellectual exertions. And this paper begins with these incredibly, these remarkable statements, statements that a philosopher of science would immediately recognize as indicating some kind of paradigm shift.
Heisenberg says at the beginning, we shouldn't be thinking about orbits anymore or particles. We should rephrase our physics, our theories, entirely in terms of quantities that can in principle be measured. And so he formulates this different way to think about quantum mechanics in terms of abstract objects that Max Born identifies are matrices.
People who are listening to this may have heard of a matrix. Matrices and the subject that they belong to, linear algebra, that was not a standard part of the physics curriculum at this time. Heisenberg actually independently discovered matrices in this work and their mathematical operations. And Born recognized what Heisenberg was doing.
And then together, they wrote a couple of papers in which they introduced this thing called matrix mechanics. In matrix mechanics, there's no picture anymore. There's no picture of atoms, of electrons, of particles going around atoms. There's just this abstract mathematical apparatus for predicting energy levels of things.
Schrodinger comes along a few months later and through a series of theoretical arguments that I will not be able to do justice to here that connect with what's called Hamlet-Jacobi theory, which is a beautiful area of classical physics, Heisenberg proposes a different way to think about quantum mechanics. He introduces his wave function.
It's what shows up in these papers. Schrodinger introduces his wave function. This is in early 1926. In some papers, both in German and in English, he wrote in both. He called this an undulatory theory of quantum mechanics, which is a beautiful word, right? Undulatory theory. It's very, it's a lovely, lovely way to, lovely term.
And he introduces his wave function. And at the time he treats it, he regards it as a mechanical object. He knows right away that it lives in configuration space. He says this in these early papers, but he even says maybe this is what reality is. Maybe reality is some giant mechanical wave undulating in configuration space and some, and through some mechanism that we don't fully understand, this undulating wave in this high dimensional abstract space somehow projects facts down into three-dimensional reality.
Facts about where electrons are. He doesn't have the full picture here, but this is what he basically describes. And he's able to use this wave function as an indirect method for predicting energy levels. Max Born comes along very shortly thereafter and proposes that wave functions are not mechanical objects. They are mathematical tools.
The reason they're complex valued is because they're not really physical things. They're things that we operate on using these sort of mathematical operations to generate probabilities. That wave functions, when you take the complex numbers that they describe and you do this operation to them, what comes out is probabilities.
That a wave function should be understood as a mathematical mechanism for generating probabilities. By 1928, Schrodinger is already recanting his view. He gives a lecture in 1928, his fourth lecture wave mechanics in which he says he used to think that wave functions were physical objects.
He even presages in some ways the many worlds interpretation. He says they're an object where every single thing that the system could be doing is really playing out in this giant wave function, but he no longer thinks that. He's accepted what seems to be the new idea.
And then within the next couple of years, 1930, Paul Dirac writes a textbook in which he summarizes all of the quantum mechanics that was known at the time. And he sews together Schrodinger's wave functions with Heisenberg and Born and Jordan's matrix mechanics, and he realizes they're all part of this deeper reality.
There's this mathematical construction. It's a kind of a space called a Hilbert space, which is kind of a vector space, but it's a vector space with complex numbers in it. And when you look at the vector space in one way, you see wave functions. If you look at the vector space in another way, you see matrices, you see what Heisenberg was doing.
They're all parts of this mathematical structure called the Hilbert space. And then two years later, John von Neumann, the great mathematician, he writes a book, Mathematical Foundations of Quantum Mechanics, in which he formalizes this theory in mathematical terms. And that's basically what we've had ever since.
Today, when people refer to the Dirac-von Neumann axioms or the textbook axioms of quantum mechanics, they mean this prescription for generating predictions about what we'll see in experiments, empirical predictions. And this framework is based on Hilbert spaces.