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 . 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

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