The energy eigenfunction for the ground state in a 1-dimensional harmonic oscillator potential is:

₀(x) = (1/a)e^-x*x/2(a*a)

where a is a constant for a given potential and particle mass and x*x = x² and a*a = a².

(a) Show that ₀ has definite parity and state its value.

(b) Sketch the position probability distribution function, P(x)dx, for the state ₀.

(c) Explain why the expectation value for the momentum is

= 0 for a particle in the state ₀.

(d) Calculate the uncertainty on the momentum, p, for a particle in the state ₀.

(e) Explain how the 1-dimensional harmonic oscillator potential can be used to predict the vibrational energy levels of a diatomic molecule. Discuss the limitations of this approximation.

You may use the following integrals without proof in your answers.

to e^x*x/a*a = a

to x² e^x*x/a*a = a³