Examples of Quantum Superposition for the layman?

Question

I'm trying to introduce Quantum Superposition to high-schoolers, and I feel like it would be nice to start with some real-world examples of where it comes up. I'm hoping for non-contrived examples that are relevant for their lives (like I don't think the two slit experiment would be ideal because they've probably never heard of it, and particle beams being fired in a lab is very distant from daily life; I'm also shying away from schrodinger's cat because it seems like a bit of a contrived mathematical example).

I've got two examples in mind:

1. An atom. They probably know that an atom consists of a nucleus with electrons around it, and they may have also heard that the electrons don't actually "rotate". This is quantum superposition- the electrons are everywhere around the atom at once in a sort of "probability cloud".
2. Transistors. Quantum superposition will soon mess with how we make computers. A computer is made up of a bunch of tiny machines called transistors that can be 0 or 1, and these transistors interact with each other to do all the calculations and logic we use computers for. 1 means the transistor has more electrons on one side, and 0 means it doesn't. But if the transistor is small enough, we can't be sure the electrons are only on one side - they could be on both sides, and that possibility messes with our calculations. So the fact that electrons exist in quantum superposition means there's a limit to how small we can make our computers.

So example 1 is kind of based on stuff they already know, if not directly relevant to their lives, and example 2 is directly relevant to the technology they use on a daily basis, if a little harder to understand.

It was surprisingly hard to find examples on google, so I've turned here. Any suggestions? (examples of quantum entanglement also appreciated)

Answer 1

You could use the polarization of light to demonstrate superposition. It's conceptually and experimentally easily accessible.

Take a polarizing beamsplitter which transmits horizontally polarized light $|H\rangle$ and reflects vertically polarized light $|V\rangle$. This is the measurement device. If you send either $|H\rangle$ or $|V\rangle$ light it will come out completely transmitted or completely reflected. Then you prepare diagonally $|D\rangle$ or antidiagonally polarized light $|A\rangle$, which is a superposition of $|H\rangle$ and $|V\rangle$: $$ \begin{align} |D\rangle &= \frac{|H\rangle + |V\rangle}{\sqrt{2}} \\ |A\rangle &= \frac{|H\rangle - |V\rangle}{\sqrt{2}} \end{align} $$ Therefore it this light hits the detection device $50\,\%$ of the light is transmitted and the other $50\,\%$ reflected.

This might look the same as if you sent unpolarized light $\rho = \left( |H\rangle \langle H| + |V\rangle \langle V| \right) / 2$ into the detector, but here comes the twist: By using a λ/2 plate you can transform the diagonally polarized light such that it either comes out fully transmitted or fully reflected from the polarizing beamsplitter. In contrast, the unpolarized light can't be transformed to yield different measurement results.
This highlights an important feature of superpositions: Superpositions don't describe lack of knowledge, but everything we know about the system. Unpolarized light is therefore equivalent to rolling a dice coming to rest somewhere under the sofa. It's light which has a definite, but unknown polarization at each instant. If you instead know the wavefunction of light in a coherent superpostion of polarizations you have full control over its state.

Here's another advantage of this example: Superpositions are simply linear combinations of basis vectors of Hilbert spaces. Unfortunately many superpositions live in infinite-dimensional Hilbert spaces, one example being atomic orbitals. In contrast the Hilbert space of polarizations has only two basis vectors, which in the $\left\{ |H\rangle, |V\rangle \right\}$-basis coincide with two basis vectors of Euclidean space.

Answer 2

Your examples are good, but they may not help your pupils towards an intuitive grasp of superposition. To give your pupils a feel for how superposition works, put a dice in a closed box. When you open the box the dice has one number on top - it is in a definite state. But if you close the box and shake it, you lose the information about the state - all you can now say is that if you open the box and observe the dice, it is equally likely to be in any one of its six possible states.

You should emphasise that this is not actual quantum superposition - it is simply an illustration. There are some important differences which it may be worthwhile discussing with your pupils:

1. With the dice in a box, our knowledge about the state of the dice after shaking the box can be expressed by six real numbers - for a fair dice these are all $\frac 1 6$, but for a loaded dice they could be different. In a quantum system the parameters that determine how a superposition is related to observable states are complex numbers, not real numbers. They are called amplitudes. What we have constructed with the dice in a box is closer to what is called a mixed state in quantum mechanics.
2. The dice in the box is a hidden variable - it has a definite state whenever the box is still, even if we do not open the box and observe that state. In quantum mechanics we know there are no (local) hidden variables. This is a consequence of Bell’s theorem which was demonstrated experimentally in 1972.
3. The dice in the box needs to be explicitly shaken between observations, otherwise we will always observe the same state. The equivalent of shaking in a quantum system is called decoherence, and it occurs every time the quantum system interacts with anything outside it. It is very hard to prevent a quantum system from losing coherence - this is one of the big challenges in quantum computing.