The wave function and superposition

Describing a particle differently

In classical physics, we describe a particle by its position and its velocity. Quantum mechanics describes it using a mathematical object, the wave function (often written ψ, "psi"). This function does not say where the particle is, but with what probability it would be found at each location or in each state.

The central idea: superposition

The most baffling fact: as long as it is not measured, a particle can be in a superposition of several states at once. It is not that we are ignorant of its state — it is that it really is in all these states simultaneously.

Let's take a system with two possible states, written |0⟩ and |1⟩ (for example the spin of an electron: "up" or "down"). A superposition is written:

   |ψ⟩ = a |0⟩ + b |1⟩

where a and b are numbers encoding the "share" of each state. The system is neither in |0⟩ nor in |1⟩: it is in a combination of the two.

   Classical :  either 0 , or 1        (a coin : heads OR tails)

   Quantum :    a|0⟩ + b|1⟩            (both at once, like a coin
                                         still spinning in the air)

The image of the coin spinning in the air is useful: as long as it is spinning, talking about "heads" or "tails" makes no sense; it is in an in-between. Only when it is caught (the measurement) does a result become fixed.

The probabilities

The coefficients a and b determine the probabilities of each outcome upon measurement:

   probability of getting 0  =  |a|^2
   probability of getting 1  =  |b|^2

   always with :  |a|^2 + |b|^2 = 1   (there must be a result)

For example, if a = b, there is a one-in-two chance of getting each state: a "balanced" superposition.

This is not ignorance

A crucial point: superposition is not our ignorance of a value that would already be fixed. A coin hidden under your hand is already heads or tails, we simply don't know which. A particle in superposition, on the other hand, has no definite value before measurement — and this is experimentally demonstrable (Young's double-slit experiment, or the Bell inequalities, prove it). The difference is real, not philosophical.

In summary

A particle is described by its wave function, which encodes probabilities. Before measurement, it can be in a superposition of several states, |ψ⟩ = a|0⟩ + b|1⟩: really in both at once, with probabilities |a|^2 and |b|^2. This is not ignorance about a hidden value: the value does not yet exist.