(Solved): 4. Quantum Oscillator (15%). The eigenstate wave functions of the simple harmonic oscillator take ...

4. Quantum Oscillator (15%). The eigenstate wave functions of the simple harmonic oscillator take the following form, 2 */(x) = exp(-+) H?(x). 2 where H? is the Hermite polynomial defined through the generating function g(x, t), g(x, t) = exp(2xt - t²) = ? n=0 tn n! Hn(x). Find the eigenstate wave function 3 (x) of the quantum oscillator.