## 3.6 Some quantum dynamics

We will finish this chapter with a short aside on some more fundamental quantum theory. Although this isn’t our main focus — we will happily black box away this stuff, happy in the knowledge that some scientists in a lab somewhere have already packaged everything up into nice quantum logic gates that we can use — it is a nice opportunity to talk about other aspects of the subject that might be of interest.

The time evolution of a quantum state is a unitary process which is generated by a Hermitian operator called the **Hamiltonian**, which we denote by **Schrödinger equation**:

The first approach towards classical mechanics that you might meet is the **Newtonian** framework, where we talk about the equations that are satisfied by **forces**.
It is Newton’s second law that we usually apply the most in order to describe the behaviour of classical systems, and it is usually stated as **momentum**: the product of mass (a scalar) with velocity (a vector).

Instead of talking about *forces* within a system, we can instead describe things entirely in terms of either *position and velocity* (where the latter is just the time derivative of the former) — using coordinates *position and momentum* — using coordinates

If we take either of these two approaches, then we have a suitable replacement for Newton’s second law:

- The first approach results in
**Lagrangian mechanics**, where we have some function\mathcal{L}(t,\mathbf{q}(t),\dot{\mathbf{q}}(t)) called the**Lagrangian**, and study the**Euler–Lagrange equations**\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial\mathcal{L}}{\partial \dot{\mathbf{q}}}\right) = \frac{\partial\mathcal{L}}{\partial\mathbf{q}} which is a second-order differential equation. - The second approach results in [
**Hamiltonian mechanics**], where we have some function\mathcal{H}(t,\mathbf{p}(t),\mathbf{q}(t)) called the**Hamiltonian**, and study the**Hamilton equations**\begin{aligned} \frac{\mathrm{d}\mathbf{q}}{\mathrm{d}t} &= \frac{\partial\mathcal{H}}{\partial\mathbf{p}} \\\frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t} &= \frac{\partial\mathcal{H}}{\partial\mathbf{q}} \end{aligned} which is a pair of first-order differential equations.

These two important functions, the Lagrangian and the Hamiltonian, are given by the total energies of the system: the former is the *difference* of the kinetic and potential energies; the latter is the *sum* of the kinetic and potential energies.

There are many situations where one framework is more useful than the other, but in quantum physics we normally find the Hamiltonian approach more easier than the Lagrangian, since momentum is a conserved quantity, whereas velocity is not.
In fact, the Hamiltonian approach is hidden all over the place: the position and momentum operators in quantum physics are truly fundamental, and will show up again when we talk about **uncertainty principles** in Section 4.6.

Here *very*) small number known as the **Planck constant**.
Physicists often pick a unit system such that *exactly*^{81} equal to

As a historical note, Planck’s constant *any* particle would be **quanta** (whence the name quantum physics!), which we now call electrons.^{82}
We will see the Planck constant turn up again when we talk about **uncertainty principles** in Section 4.6.

Back to quantum dynamics.
For **time-independent** Hamiltonians **separable**: it is written as a product of two functions **stationary**, or refer to it as a **standing wave**.

We will *not* delve into a proper study of the Schrödinger equation — this is the subject of entire books already, and deserves a lengthy treatment — but it is nice to mention at least one worked example (although we will skip almost all of the details!), since its applications are commonplace in day-to-day life.

In the time-independent case, the Schrödinger equation can simply be written as

One particularly instructive situation to consider is that of a particle in a box: we have some

What is utterly unique and important to quantum physics is not really this specific fraction, but the fact that *the possible energy levels of the system are purely discrete* — energy cannot be any real value, as is the case in the classical world, but it can only take values within some discrete set

But what are the applications of this particle in a box?
Well, this phenomena of a system having a *discrete* energy spectrum when restrained to small enough spaces is known as **quantum confinement**, and **quantum well lasers** are laser diodes which have a small enough active region for this confinement to occur.
Such lasers are arguably the most important component of fiber optic communications, which form the underlying foundations of the internet itself.

Before moving on to understand the relevance of this to what we have already been discussing, let us take a moment to see why we might have expected to stumble across such a solution as *continuous*, whatever that might formally mean.

For a start, this means that we want not only to be able to multiply the matrices that represent these transitions, but also to do the inverse: take any transition and “chop it up” into smaller time chunks, viewing any evolution *roots* (square roots, cube roots, and so on) of our matrices, which means that they must at the very least have complex entries.

But let us try to take this continuity requirement a bit more seriously: say that any transition

Next, as we’ve already mentioned, complex matrices have a polar form — analogous to how any *unitary* matrices.
And if we want

In summary, from just asking for our evolutions to be continuous and not have any convergence issues, we end up with the conclusion that we are interested in evolutions described by exponentials of anti-Hermitian matrices, i.e.

This correspondence between so-called **one-parameter unitary groups** — families **Stone’s theorem (on one-parameter unitary groups)**.

For example, if we consider the **translation** operators **momentum operator**.
In fancy words, this says that * 1-dimensional motion is infinitesimally generated by momentum*.

Now, to relate this to the earlier parts of this chapter, we note that the Hamiltonian of a qubit can always be written in the form

The kilogram is now

*defined*in SI in terms of the Planck constant, the speed of light, and the atomic transition frequency of of caesium-133.↩︎The whole history of quantum physics, arguably starting with the black-body problem, accounting for the Rayleigh–Jeans law, and leading on to the discovery of the photoelectric effect, is a wonderful story, but one that we do not have the space to tell here.↩︎