Ladder operator

Let $𝑁 \in 𝖵 𝖾 𝖼 𝗍 ℂ (𝑉, 𝑉)$ be an operator and $| 𝑛 ⟩ \in 𝑉$ be an eigenvector such that $𝑁 | 𝑛 ⟩ = 𝑛 | 𝑛 ⟩$ . A ladder operator of $𝑁$ is an operator $𝑋$ such that $[𝑁, 𝑋] = 𝑐 𝑋$ where $𝑐 \neq 0$ . #m/def/linalg It follows that

𝑁 𝑋 | 𝑛 ⟩ = (𝑋 𝑁 + [𝑁, 𝑋]) | 𝑛 ⟩ = 𝑋 𝑁 | 𝑛 ⟩ + 𝑐 𝑋 | 𝑛 ⟩ = 𝑋 𝑛 | 𝑛 ⟩ + 𝑐 𝑋 | 𝑛 ⟩ = (𝑛 + 𝑐) 𝑋 | 𝑛 ⟩

i.e. $𝑋 | 𝑛 ⟩$ is either zero or an eigenvector. A raising operator is a ladder operator for which $𝑐$ is positive and real, likewise a lowering operator is a ladder operator for which $𝑐$ is negative and real.

Properties

If $𝑁$ is a Hermitian operator and $[𝑁, 𝑋] = 𝑐 𝑋$ then either $𝑐$ is real or $𝑋 | 𝑛 ⟩ = 0$ ; and $[𝑁, 𝑋 †] = - 𝑐 𝑋 †$ and thus $𝑁 𝑋 † | 𝑛 ⟩ = (𝑛 - 𝑐) 𝑋 † | 𝑛 ⟩$ .
A (pseudo)Vector operator $ˆ 𝐕$ has raising and lowering operators for $ˆ 𝑉 𝑧$ by $ˆ 𝑉 \pm = ˆ 𝑉 𝑥 \pm 𝑖 ˆ 𝑉 𝑦$ . #to/prove

Proof of 1

Let $| 𝑛 ⟩$ be an eigenvector such that $𝑁 | 𝑛 ⟩ = 𝑛 | 𝑛 ⟩$ , where $𝑛$ is real by ^P2. Assuming $𝑋 | 𝑛 ⟩ \neq 0$ , then $𝑛 + 𝑐$ is also an eigenvalue of $𝑁$ which must also be real. Now

[𝑁, 𝑋 †] = 𝑁 𝑋 † - 𝑋 † 𝑁 = (𝑋 𝑁 - 𝑁 𝑋) † = [𝑋, 𝑁] † = - [𝑁, 𝑋] † = - 𝑐 𝑋 †

proving ^P1

#state/tidy | #lang/en | #SemBr