Question
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
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 ex*x/a*a = a
to x2 ex*x/a*a = a3
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