(a) Generalize Problem 7.2, using the trial wave function14for arbitrary n. Partial answer...

(a) Generalize Problem 7.2, using the trial wave function14

for arbitrary n. Partial answer: The best value of b is given by

(b) Find the least upper bound on the first excited state of the harmonic oscillator using a trial function of the form

Partial answer: The best value of b is given by

(c) Notice that the bounds approach the exact energies as n → ∞ Why is that? Hint: Plot the trial wave functions for n = 2,n = 3, and n = 4, and compare them with the true wave functions (Equations 2.59 and 2.62). To do it analytically, start with the identity