Almost every book or article about quantum mechanics seems to start with a passage like this:
Quantum mechanics is arguably the most successful physical theory in the history of science but strangely, no one really seems to agree about how it works.
And now I’ve done it to you too. One of the main reasons people write this sentence over and over again is because of what is called the measurement problem. Here is a way to state the measurement problem, which I will then try to explain to you.
The measurement problem refers to the following facts, which seem to contradict each other:
On the one hand, when we measure quantum systems we always see one answer.
On the other hand, if you want to use the regular rules of time evolution in quantum mechanics to describe measurements, then there are states for which measurements should not give you one answer.
In particular, measuring states that describe a superposition (see below) can cause a lot of confusion.
In part 1 of this series I gave you a bit of the history and motivation behind the development of quantum mechanics. It followed the development of the theory the way a lot of physics text books do, with lots of differential equations and other scary math. We will now leave all that behind us.
My plan here is to describe enough of the mathematical formalism of quantum mechanics in enough detail to express the measurement problem in a way that is relatively rigorous. This mostly boils down to a lot of tedious and basic facts about linear algebra, instead of all the scary differential equations from part 1. Personally I find the algebraic material a lot easier to understand than the more difficult differential equation solving. But, it will still be an abstract slog, but I’ll try to leave out enough of the really boring details to keep it light.
As with my other technical expositions on subjects that are not about computers, I am the furthest thing from an expert on this subject, I’m just organizing what I think are the most interesting ideas about what is going on here, and hoping that I’m not too wrong. I’ll provide a list of more better sources at the end.
Quantum States and Hilbert Space
The rules of quantum mechanics are about states and observables. These are both described by objects from a fancy sort of linear algebra. This involves a lot of axioms that are interesting (not really) but not needed for our purposes. To try and keep this section a bit shorter and less tedious I link out to Wikipedia for many of the mathematical details, and just provide the highlights that we need here.
Quantum states live in a thing called a Hilbert space, which is a special kind of vector space. Observables are a particular kind of linear function, or operator on a Hilbert space.
The ingredients that make up a Hilbert space are:
A set of scalars. In this case it’s always the complex numbers (\mathbb C).
A set of vectors. Here the vectors are the wave functions.
A long list of rules about how we can combine vectors and scalars together. In particular vector spaces define a notion of addition (+) for vectors that obeys some nice rules (commutativity, associativity, blah blah blah), and a notion of multiplying vectors by scalars that also obeys some nice rules. For reference, you can find the rules here.
We denote Hilbert spaces with a script “H”, like this: \cal H, and we use greek letters, most popularly \psi, to denote vectors in \cal H. For a reason named Paul Dirac, we will dress up vectors using a strange bracket notation like this: | \psi \rangle, or sometimes this way \langle \psi |. This is also how we wrote down the wave functions in part 1.
The most important thing about Hilbert spaces is that they are linear. What this means is that any given any two vectors | \psi \rangle and | \phi \rangle and two scalars a and b, any expression like
a | \psi \rangle + b | \phi \rangle
is also a vector in \cal H.
This rule, it turns out, is the most important rule in Quantum Mechanics and is famously called the superposition principle. You will also see states that are written down this way called superposition states. But, this terminology is more magic sounding than it needs to be. This is just a linear combination of two states, and the fact that you always get another state is also a straightforward consequence of the form of the Schrödinger equation (it is what we call a first order, or linear differential equation). Linearity plays a big role in the eventual measurement puzzle, so store that away in our memory for later.
Inner Products
The second most important thing about Hilbert spaces is that they define an inner product operation that allows us to define things like length and angle. We write this product this way:
\langle \psi | \phi \rangle
and its value is either a real or complex number.
Now we see a bit of the utility of this strange bracket notation. In Dirac’s terminology the | \psi \rangle is a “ket” or “ket vector” and the \langle \psi | is a “bra”. So you put them together and you get a “bra ket” or “braket”. So all of this silliness is in service of a bad pun. Those wacky physicists thought this joke was so funny that we’ve been stuck with this notation for almost a hundred years now.
There is also some subtle math that you have to do to make sure that the “bra” \langle \psi | is a thing that makes sense in this context, but let’s assume we have done that and it has all worked out.
As always, I refer you to wikipedia for the comprehensive list of important inner product facts.
We can use the inner product to define a notion of distance in a Hilbert space that is similar to the familiar “Euclidean” distance that they teach you in high school. For a given vector \psi the norm of \psi is written \lVert \psi \rVert and is defined as
\lVert \psi \rVert = \sqrt{\langle \psi | \psi \rangle}
Since \langle \psi | \psi \rangle is always positive this is well-defined. You can also define the distance between two vectors in a Hilbert space as \lVert \psi - \phi \rVert.
The inner product and the norm will form the basis for how we compute probabilities using the Born rule, which we saw in part 1.
A Short Digression
All of this nonsense with Hilbert spaces and inner products is motivated by wanting to do calculus and mathematical analysis on objects that are functions rather than plain numbers (or vectors of numbers). This comes up because the big conceptual shift in quantum mechanics was moving from properties that had values which were real numbers to properties described by complex valued functions or wave functions. The issue was that we know how to do calculus over the reals, but calculus with function valued objects is a stranger thing. Functional analysis is the area of mathematics that studies this, and Hilbert spaces come from functional analysis. In the 30s von Neumann realized that functional analysis, Hilbert spaces, and operators were the right tools to use to build a unified basis for quantum mechanics. And that’s what he did in his famous book.
If we wanted to actually prove some of the things that I will later claim to be true about Hilbert spaces and operators we would need some of the more technical results from functional analysis. Doing such proofs is way above my pay grade so I’m mostly ignoring such things for now. But at the end of this whole story I’ll make a list of things that I left out.
After working out the mathematical basis for quantum theory Von Neumann went on to invent the dominant model that we still use to describe computers. So think about that next time you are feeling yourself after having written some clever piece of code.
Basis Vectors
The third important fact about Hilbert spaces that we will need is the idea of a basis. In a Hilbert space (really any vector space) a basis is a set of vectors that one can use to represent any other vector in the space using linear combinations. If this set is finite, meaning that you can count up the number of basis vectors you need with your fingers, then we say that the vector space is “finite dimensional”.
The most familiar example of a finite dimensional Hilbert space is \mathbb C^n, where the vectors are made up of n complex numbers, or \mathbb R^n, which is the same, but with real numbers. Here the basis that we all know about is the one made up of the unit vectors for each possible axis direction in the space. So, for n=3 the unit vectors are
\begin{pmatrix} 1 \\ 0 \\ 0 \\ \end{pmatrix}, \quad \begin{pmatrix} 0 \\ 1 \\ 0 \\ \end{pmatrix} \quad {\rm and} \quad \begin{pmatrix} 0 \\ 0 \\ 1 \\ \end{pmatrix}
To write down any vector v in the space all we need is three numbers, one to multiply each unit vector:
v = \begin{pmatrix} a \\ b \\ c \\ \end{pmatrix} = a\begin{pmatrix} 1 \\ 0 \\ 0 \\ \end{pmatrix} + b\begin{pmatrix} 0 \\ 1 \\ 0 \\ \end{pmatrix} + c \begin{pmatrix} 0 \\ 0 \\ 1 \\ \end{pmatrix}
By convention we write vectors in columns, which will make more sense in the next section.
And thus we have built the standard sort of coordinate system that we all know and love from 10th grade math.
This sort of basis for \mathbb C^n also has the property that it is orthonormal, meaning that with the standard inner product all of the unit vectors are orthogonal to each other (their mutual inner products are always zero).
In the rest of this piece we will assume that all of our Hilbert spaces have an orthonormal basis and that they are finite dimensional. Of course, the more famous state spaces in quantum mechanics (for position and momentum) are infinite dimensional, which is the other reason Hilbert spaces became a thing. But we will not deal with any of that complication here.
Operators and Observables
In classical mechanics we did not think about observables too much. They were just simple numbers or lists of numbers that in principle you can just read off of the mathematical model that you are working with.
But, in quantum mechanics, observables, like the states before them, become a more abstract thing, and that thing is what we call a self-adjoint linear operator on the Hilbert space \cal H. All this means is that for everything we want to observe we have to find a function from \cal H to \cal H that is linear and also obeys some more technical rules that I will sort of define below.
Linearity we have seen before. This just means that if you have a operator O that takes a vector \psi and maps it to another vector, then you can move O in and out of linear combinations of vectors. In particular
O(\alpha \psi) = \alpha O (\psi)
and
O(\psi + \phi) = O(\psi) + O(\phi)
The “self-adjoint” (or Hermitian) part of the definition of observables is more technical to explain.
As we all know from basic linear algebra, in finite dimensional vector spaces you can, once you fix a basis, write linear operators down as a matrix of numbers. Then the action of the operator on any given vector is a new vector where each component of the new vector is the dot product of the original vector with the appropriate row of the matrix.
So the easiest operator to write down is the identity (\bf 1)… which just looks like the unit vector basis vectors written next to one another
{\bf 1} = \begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{pmatrix}
We can check that the application rule I outlined above works … here we write the vector we are acting on vertically for emphasis:
{\bf 1} \begin{pmatrix} a \\ b \\ c \\ \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{pmatrix} \begin{pmatrix} a \\ b \\ c \\ \end{pmatrix} = a\begin{pmatrix} 1 \\ 0 \\ 0 \\ \end{pmatrix} + b\begin{pmatrix} 0 \\ 1 \\ 0 \\ \end{pmatrix} + c \begin{pmatrix} 0 \\ 0 \\ 1 \\ \end{pmatrix}
So it works!
With this background in hand, we can define the adjoint of an operator A, which we write as A^* (math) or A^\dagger (physics). Anyway, the adjoint of A is an operator that obeys this rule:
\langle A \psi | \phi \rangle = \langle \psi | A^* \phi \rangle
for any two vectors \psi and \phi in \cal H.
In finite dimensional complex vector spaces (e.g. \mathbb C^n), where operators can be written down as matrices, you can visualize what the adjoint is by transposing the matrix representation and taking some complex conjugates. This is not the cleanest way to define this object since the matrix representation is dependent on a basis, and we can (and did!) define the notion of an adjoint without referencing a basis at all. But it’s not the end of the world.
In infinite dimensional spaces and other more complicated situations finding the adjoint is more complicated. I’ll leave it at that.
A self-adjoint operator is just one whose adjoint is equal to itself. So it obeys the rule:
\langle A\, \psi | \phi \rangle = \langle \psi | A\, \phi \rangle .
We can remove the ^* because A = A^*.
In a lot of physics books you will also see self-adjoint operators referred to as Hermitian operators. In the finite dimensional complex case the two terms are equivalent.
Self-adjoint operators have some nice properties for physics. The reason why has to do with eigen-things.
Eigen-things
Linear operators map vectors to vectors in a fairly constrained way. You have some freedom in how you transform the vector, but you don’t have total freedom since whatever you do has to preserve linear combinations.
But, for every operator there might be a special set of vectors that map to some scalar multiplied by themselves. That is, for some operator A and vector \psi you will have
A \psi = \alpha \psi
where \alpha is just a scalar. What this means, in some sense, is that the operator transforms the original vector to itself. The only thing that changes is its length, or magnitude.
Vectors with this property are called eigenvectors, and the constants are called eigenvalues. Both words are derived from the German word “eigen” meaning “proper” or “characteristic”, but that doesn’t really matter. This just one of those weird words that stuck around by habit.
Eigenvectors and eigenvalues come up in all kinds of contexts. They are important because they provide a way to characterize complicated transformations in a simpler way. If you have all the eigenvectors you can in principle switch to working in a basis where the transformation is a diagonal matrix, which is a usually simpler representation. The applications of this idea come up all over, from image processing to Google PageRank, to quantum mechanics.
The reason we wanted to have the operators that represent observables be self-adjoint above is that self-adjoint operators have two nice properties related to eigen-things.
All the eigenvalues of a self-adjoint operator are real-valued (even though our state space is over the complex numbers).
There is a famous theorem that says that every self-adjoint operator has a set of eigenvectors that form a orthonormal basis of the underlying Hilbert space. This theorem is called the spectral theorem and the eigenvectors/values of the operator are called its spectrum. This is a very important result for quantum mechanics.
Circling Back to the Atom
At this point you might be thinking to yourself, “I have seen this word spectrum before”. And you have. One of the earliest problems in quantum mechanics was to explain the spectral lines of the hydrogen atom. So you might be wondering, how do we get from these abstract quantum states and operators to energy? The answer is the next important rule of quantum mechanics, which we are already familiar with from part 1: there is a special observable for the energy of the system whose operator we call H, for the Hamiltonian. Time evolution of quantum states is then given by the Schrödinger equation:
i \hbar \frac{\partial}{\partial t} | \psi(t) \rangle = H | \psi(t) \rangle .
You will recall from part 1 that the wave functions, which we now know are the quantum states of a system, were all solutions to this equation.
Now, the trick to solving the hydrogen atom is first finding a Hamiltonian H that correctly describes the behavior of the electron in the atom. It turns out that when you do this H will be one of our coveted self-adjoint linear operators on the Hilbert space of wave functions. This means that there will be some set of states that obey this rule:
H | \psi \rangle = E | \psi \rangle
where E here is just a real number, rather than an operator. We use the letter E to stand for energy. These energies will be the energies that appear in the spectrum of the atom.
So here is why we were going on about eigen-things before (and linear operators before that, and vector spaces before that). The Hamiltonian for the hydrogen atom H is a self-adjoint operator whose the eigenvalues are the energies in the spectrum of the atom. The eigenvectors are the electron wave functions that define the fixed energy levels at which we see spectral lines. The two amazing facts about spectral lines are that they teach us almost everything we know about astrophysics (at least up until the 80s and 90s), and that quantum mechanics can explain them so well if you set things up the right way. The setup is somewhat technical and complicated, so I don’t cover that here. I’ll use a simpler system to describe the rest of what I want to talk about.
Speaking of which.
Break Time
At this point we have put together almost all of the formalism that we need. But this post has gone on too long, so I am going to make you read yet another part to get to the real point of this entire exercise. Meanwhile, here is a quick summary of what we have so far:
States are vectors in a Hilbert space, usually over \mathbb C.
Observables are self-adjoint linear operators on that space.
The possible values of observables are the eigenvalues of the corresponding operator, and the eigenvectors are the states that achieve those values. In addition, for the operators that represent observables, we can find eigenvectors that form an orthonormal basis of the underlying state space. Which is really convenient.
There is a special observable for the energy of the system whose operator we call H, for the Hamiltonian. Time evolution of states is then given by the Schrödinger equation.
Of course, I still have not said anything about measurement, and you should be furious with me. I promise I will in part 3.
References
Here are some things I like.
Isham’s Lectures on Quantum Theory, is a nice treatment of the subject that is more mathematically rigorous than most.
Peres and Ballentine are more “physics oriented” books that start from the algebraic point of view. Weinberg is also covers this material, but from a more traditional point of view, but it’s a nice illustration of how the physics view and the algebraic view are related.
Scott Aaronson’s Quantum Computing since Democritus is a nice computer nerd’s view of the world.
Brian C. Hall’s book on Quantum Theory for Mathematicians covers a lot of the more technical details about Hilbert spaces and their operators in more mathematically rigorous way.
Frederic Schuller’s lectures on quantum mechanics also gives you a rigorous mathematical view of this material.