2. Simple harmonic oscillator is always an approximation. Real problems always have po- tential of the...

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2. Simple harmonic oscillator is always an approximation. Real problems always have po- tential of the form V(r) ki + bà" + ci + . The contribution beyond ki² are 1 called anharmonic terms. (a) Show that to leading order the nth energy eigenvlaues of oscillator is changed by hw 2mw -(30n² + 30n + 1) if all other terms are ignored except bi". (b) Calculate the corresponding energy eigen value if only ci survived. (c) Calculate the time evolution of creation and annihilation operator ie (à(t)) and (at(t)) 2. Simple harmonic oscillator is always an approximation. Real problems always have po- tential of the form V(r) ki + bà" + ci + . The contribution beyond ki² are 1 called anharmonic terms. (a) Show that to leading order the nth energy eigenvlaues of oscillator is changed by hw 2mw -(30n² + 30n + 1) if all other terms are ignored except bi". (b) Calculate the corresponding energy eigen value if only ci survived. (c) Calculate the time evolution of creation and annihilation operator ie (à(t)) and (at(t))