Q2) One dimensional harmonic oscillator with the Hamiltonian p2 1 Ho : + 2m mw?x² is perturbed by A H, = 1 x4 a) Calculate the first order corrections to all excited state energies. b) Calculate the first order correction to the ground state wave function.

7. Consider a particle in an infinite square well centered at x = 0 in one of its stationary states. For this problem, you may look up any integrals. Some useful ones are given in Harris. a) Compute (x) and (pr) for arbitrary n. Do this by direct computation but then describe how you could have found these results using symmetry (the symmetry can either be symmetry in the physical system, such as the shape of the wave function, or symmetry related to the expectation value integral, such as the shape of the integrand). b) Using your answer to part a), show that the uncertainty in the momentum is Apx nh for arbitrary n. Do this two ways: (i) first by using your answer to part a) and directly computating (p2) (via an integral) and (ii) by using your answer to part a) and relating (p2) to the kinetic energy operator. c) Show that the uncertainty principle holds for the ground state. 2L -

A quantum mechanical particle of mass m moves in a 1D potential where a) Estimate the ground state energy of the particle. b) Sketch the wave function to the best of your ability.

Q3: For a quantum harmonic oscillator in its ground state. Find: a) (x) b) (x²) с) Ох