Principles of Quantum Mechanics · Problem 7.5.3

The Path Integral Formulation of Quantum Theory — R. Shankar

Physics — Quantum Mechanics · hermite-polynomial-three-term-recursion

Problem

Starting with a+a†=21/2ya + a^\dagger = 2^{1/2}y and (a+a†)∣n⟩=n1/2∣n−1⟩+(n+1)1/2∣n+1⟩(a+a^\dagger)|n\rangle = n^{1/2}|n-1\rangle + (n+1)^{1/2}|n+1\rangle and Eq. (7.3.22), derive the relation Hn+1(y)=2yHn(y)−2nHn−1(y)H_{n+1}(y) = 2yH_n(y) - 2nH_{n-1}(y)

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