Linear algebra MOC

Ladder operator

Let be an operator and be an eigenvector such that . A ladder operator of is an operator such that where . #m/def/linalg It follows that

i.e. is either zero or an eigenvector. A raising operator is a ladder operator for which is positive and real, likewise a lowering operator is a ladder operator for which is negative and real.

Properties

  1. If is a Hermitian operator and then either is real or ; and and thus .
  2. A (pseudo)Vector operator has raising and lowering operators for by . #to/prove
Proof of 1

Let be an eigenvector such that , where is real by ^P2. Assuming , then is also an eigenvalue of which must also be real. Now

proving ^P1

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