# Chapter 1: Superposition and Measurement

Welcome to Chapter 1 of our Quantum Explainers Series! If you are inspired dive deeper into this topic, check out the Quantum Shorts Contest. You can take inspiration from this article (we will even hint at potential topics that you can make your video about) and provide your own take on quantum superposition for the contest!

By Yi Zhu

April 14, 2024

Suppose you are a researcher and your principal investigator (i.e. your boss) has just tasked you to bake two identical cakes. You mix up two batters that are exactly — down to the atom — the same. After a set time in the oven, you pull both cake pans out and find that one cake is undercooked (raw) and the other overcooked (burnt). How is this possible?

In a panic to figure out what’s going on, you whip up 100 more identical cake batters, bake them all in one giant oven, and again find that approximately half of the cakes are raw and the other half burnt. This is precisely the scenario that physicists Stern and Gerlach encountered in 1922, albeit while experimenting with silver atoms rather than cake. They found that having prepared many identical silver atoms and measuring their spin angular momentum, that approximately half the time the atoms had spin +½ and the remaining half had spin -½.

In the language of quantum mechanics, right before we open the oven to check if the cakes are properly baked, we describe the condition of a cake as being in a superposition of raw and burnt. The term “superposition” means obtaining a new state by adding together multiple states. If we use the symbol |raw⟩ to represent an undercooked cake and the symbol |burnt⟩ to represent an overcooked cake, then the superposition state that we have prepared is:

The factor 1/√2 indicates that we have an equal chance of measuring the cake to be raw or burnt (why? see Chapter 3: Wave Function). We define the state of the cake as literally the sum of the raw and burnt states, or colloquially, “simultaneously raw and burnt.”

But why bother defining the state of the cake as a “superposition of raw and burnt” when we could just as easily say that “half the time it’s in raw state and the other half it’s in the burnt state”? The subtle distinction between these two descriptions is that the former is a single well-defined state while the latter is a statistical mixture of two different states. [Editor’s note: in more technical lingo, the former is a pure state while the latter is a mixed state]. We postulated that all the cakes are prepared identically, so all the cakes must be in the exact same state—the superposition state.

How is it then that we measure the cake to be in different states (raw/burnt) with some probability? Another subtly lies in the word “measure.” In quantum mechanics, measuring a state (e.g. checking if the cake is raw or burnt with a toothpick) is a disruptive process. Measuring a state changes, i.e. forces, the state to be in one of the measurement outcomes: either |raw⟩ or |burnt⟩. In other words, we say that measurement “collapses” the superposition.

For example, if we measure the cake to be |raw⟩, then our measurement process has (probabilistically) changed the state of the cake from

to |raw⟩. Subsequent measurements will still find the cake to be raw because the first measurement has collapsed the state to be in |raw⟩.

This model, though bizarre to our classical intuition, completely explains the result of real-life experiments: when measuring quantum objects prepared in identical superposition states, we find different outcomes with various probabilities and subsequent measurements yield the same outcome.

Can you come up with another model — another description of what is happening — that is consistent with the experimental results?

[Editor’s note: An appealing description is a “hidden variable theory”; however, this description is not compatible with our understanding of the nature of physics as demonstrated by John Bell via the eponymous “Bell’s theorem.”]