Quantum Networks Reading Group #2

Last time we discussed some of the mathematical basics for understanding quantum mechanics. Today, we will discuss the last parts of the mathematics needed including how operators in Hilbert spaces are constructed and how they work. We can then sashay into a little discussion about tensor products of quantum systems, and then understand what quantum entanglement is.

These are all really cool things, and it’s pretty neat to see how these physically realizable phenomena directly follow from the simple structure of Hilbert spaces and the algebraic idea of a tensor product. After that, I will talk more about two-level systems, where we will fully understand what the Bloch sphere is and how spin works. Time permitting, I will work out the canonical example of the hydrogen atom and then derive the structure of the periodic table, which in my opinion is one of the most amazing results of introductory quantum physics.

Recall from last time that the outer product of two vectors in a Hilbert space $\mathcal{H}$ in terms of bras and kets is given by

$|\psi\rangle\langle\phi |:\mathcal{H}\to\mathcal{H}.$

If we think of a simple vector space, the outer product of vectors gives a matrix. Likewise, the outer product of bras and kets gives a matrix in $\mathcal{H}$ , or more precisely it gives the matrix representation of an operator in $\mathcal{H}$ . The operator $|\psi\rangle\langle\phi |$ maps

$|x\rangle\to|\psi\rangle\langle\phi |x\rangle$ .

This is of course the vector $|\psi\rangle$ multiplied by the scalar $\langle\phi |x\rangle$ . Since operators are constructed from elements in $\mathcal{H}$ , we can derive the following properties of operators $X,Y$ on $\mathcal{H}$ :

$(X+Y)|\psi\rangle = X|\psi\rangle + Y|\psi\rangle$
$(\alpha X|\psi\rangle) = \alpha (X|\psi\rangle)$
$XY|\psi\rangle = X(Y|\psi\rangle)$

Recall that we assume our Hilbert space has a basis $\{|x_i\rangle\}$ (apologies to anyone who prefers ZF over ZFC) and that this basis is orthonormal: $\langle x_i| x_j\rangle = \delta_{ij}$ . We can construct the identity in our Hilbert space by considering the following expression:

$\sum_{i} |x_i\rangle \langle x_i|=I$ .

It is easy to show that this operating on any vector returns the same vector back. Let us write a ket in its basis expansion:

$|\psi\rangle =\sum_i \alpha_i|x_i\rangle$ .

Applying the operator above gives

$\sum_j\sum_{i} |x_i\rangle \langle x_i|\alpha_j|x_j\rangle=\sum_j\sum_{i} \alpha_j |x_i\rangle \delta_{ij}=\sum_i \alpha_i|x_i\rangle=|\psi\rangle$ .

This is pretty neat since now we have all the things we care about in a Hilbert space. We have elements in that space, we have things that map elements to other elements, and we have ways of representing the identity in terms of elements in the Hilbert space. Our Hilbert space is pretty much all grown up, and now it can face the world and do cool things like explain reality reasonably well.

Lets show this by considering a set $\Omega$ that has an integral measure $dx$ , and suppose that we have a basis set $|x\rangle$ . This is an uncountably infinite set, and so the Hilbert space has uncountably infinite dimension. We want to define the inner product so

$\langle x | x_1 \rangle = \delta(x-x_1)$ .

Here, the left-hand side is the Dirac delta function which isn’t really a function, but a functional. It maps functions to vectors in the complex Hilbert space via some integral. First, we need to understand that in the uncountably infinite Hilbert space, the basis expansion is no longer countable (obviously) and so instead of the basis expansion

$|\psi\rangle =\sum_i \alpha_i|x_i\rangle$ ,

where we have a set of “weights” $\{\alpha_i\}$ that tell us how much of each basis vector the state $|\psi\rangle$ occupies, we have some continuous function on $\Omega$ that for each infinitesimal increment $dx$ , the function maps “how much” of the basis vector $|x+dx\rangle$ the state takes up. We call this continuous function the “wave function” and denote it by $\psi(x)$ . It is a beautiful thing.

To summarize, when going from the countable Hilbert space to the continuous Hilbert space, the basis expansion looks like

$|\psi\rangle = \int_\Omega \psi(x)|x\rangle dx$

We can now see that using the Dirac delta function as the inner product, we get

$\langle x_1|\psi\rangle = \langle x_1|\int_\Omega \psi(x)|x\rangle dx = \int_\Omega \psi(x)\langle x_1|x\rangle dx=\int_\Omega \psi(x)\delta(x-x_1)dx = \psi(x_1).$

This is exactly what we want! We know that $\langle x_1|$ corresponds to (the dual of) a single point in $\Omega$ , and so the inner product with $\psi$ should give us the part of the vector $\psi$ that corresponds to the part of the basis that is $\langle x_1|$ . This is exactly $\psi(x_1)$ .

This gives us our first interpretation of the wavefunction. It is the continuous analogue of a basis expansion of a countable Hilbert space. We have the following “mappings” (if you want to call them that) from when we go from countable to uncountable:

$\sum_i \to \int_{\Omega}$
$\{\alpha\}\to\psi (x)$
$\{|x_i\rangle\}\to |x\rangle$
$\sum_{i} |x_i\rangle \langle x_i|\to \int_\Omega |x\rangle\langle x|dx$

where the last “mapping” is the definition of the identity in each setting.

Now that we have seen the first mathematical motivation of what the wavefunction is, let’s switch gears and discuss how to combine quantum systems to form new ones. Let’s consider two systems in Hilbert spaces $\mathcal{H}_1$ and $\mathcal{H}_2$ respectively. The state of the composite system, where we look at both systems as a single one, is given by the tensor product $\mathcal{H}_1\otimes\mathcal{H}_2$ . The basis of this space is given by all possible tensor products of basis vectors $\{|x_i\rangle\}$ of $\mathcal{H}_1$ and $\{|y_i\rangle\}$ of $\mathcal{H}_2$ . The set of basis vectors explicitly is