bra-ket notation

Bra-ket notation is Paul Dirac's compact way of writing quantum states and the quantities built from them. A state is written as a 'ket', |ψ⟩, and its partner, called a 'bra', is written ⟨ψ|. Put a bra and a ket together and you get a 'bra-ket' or bracket, ⟨φ|ψ⟩, a single number that measures how much the two states overlap. The playful name comes from splitting the word 'bracket' into 'bra' and 'ket'.

The notation is more than a cute pun: it quietly keeps track of what kind of object you are holding. A ket on its own is a state; a bra on its own is the recipe that turns a state into a number; placing a ket then a bra the other way round, |ψ⟩⟨φ|, builds an operator that acts on states. Because the symbols carry this bookkeeping, you can manipulate equations almost mechanically and the answers come out with the right character.

What makes it powerful is that it never commits to a particular way of writing the state down. The same expression ⟨φ|ψ⟩ stands for an overlap whether you later choose to spell the states out as wavefunctions, as columns of numbers, or as spins pointing up and down. Dirac's notation lets you reason about quantum mechanics at the level of states and operators themselves, only descending to explicit components when you actually need to calculate.