# Can we describe quaternions using bra-ket in quantum mechanics?

For example, the rotation plus translation of a point using the language of quaternions is written as

$$Q(0,x,y,z)Q^* + T$$

where $$Q$$ is the unit quaternion, $$(x,y,z)$$ is the point, and $$T$$ is some translation vector

This looks remarkably similar to computing the expected state in quantum mechanics, namely

$$\langle Q^*|V|Q^*\rangle$$ where $$Q^*$$ is some quantum state and $$V$$ is some operator

Can an analogy be made between the quarternion and quantum mechanics in terms of the bra-ket formulation?

## 2 Answers

The only thing in common between unit quaternion rotation formula and bra-ket notation is that they rely involution. First of all, in the sense of s (both ℍ and linear operators on a Hilbert space are *-algebras) and, more generally, an idea that there is some operation that swaps left and right in algebraic compositions of (heterogeneous) objects. It is written explicitly as “*” or “†” for operators and implicitly when |ψ⟩ is converted to ⟨ψ| and vice versa.

All other aspects are utterly different.
Quaternion multiplication         Bra-ket
Central thing:  3-vector (subspace of ℍ)         operator
Side things:   quaternions (ℍ)              vectors
Result:     same type as the central thing       scalar (ℂ)

The bra-ket notation is just a way of representing the natural map $V \to V^\ast$ you have for an inner product space $V$. Inner products are fairly ubiquitous in math, so there's no reason to think there's any deep connection between two places where they happen to show up.