Consider the pendulum discussed on Lecture 3, where the potential energy is given by V(0) = mgl(1 - cos 0). (a) derive the classical Hamiltonian of the pendulum without making the approximation 1 - cos 0 ~ 02/2. Use q = 0 as the generalized coordinate. (b) Expand the cosine in the Hamiltonian up to fourth order in 0 and replace the canonical coordinates with operators: 0 - 0, p - p to obtain a quantized Hamiltonian. Note that this is now an anharmonic oscillator, since there are terms that are higher than second order. Write the Hamiltonian using the harmonic ladder operators mizw a = 0 + 2h m 1 2@ ] m12 w a = e - i 2h m 1 2 w -P where w = Vg/I as in the lecture notes. (c) Calculate the expectation values (0|H|0), (1|H|1) and (2|H|2) using the relations a In) = Vn+ 1+1) an) = VnIn -1), a |0) = 0, (n|m) = on,m. The interesting observation from the above result is that the energy differences AE12 = (2|H|2) - (1[H|1) and AEol = (1/H|1) - (0|H|0) are equal in the case of the harmonic oscillator, but different for the anharmonic oscillator