Consider a particle of mass m freely propagating within the box x ∈ [0, R]. Prepare the particle in the state corresponding to the wave function
$ \psi (x) Asin(\frac{3\pi x}{2R}) cos(\frac{\pi x}{2R}) $
inside the box, and vanishing outside. – Show that the state corresponding to ψ(x) belongs to the Hilbert space of the system.
– Determine a complex number A such that the state is normalised
– How many different energy-eigenstates are contained in the state corresponding to ψ(x)? What are the energies which one can therefore measure when the system is in the state corresponding to ψ(x)?
Am not sure how to proceed with part 1 and part 3, so if abyone can show me how to?
Also for part 2,
I think i do this
$\int_{0}^{R} | \psi|^{2} dx = 1 $
$|A|^{2} \int_{0}^{R}sin^{2}(\frac{3\pi x}{2R}) cos^{2}(\frac{\pi x}{2R})$
using this identity $ sin\theta + sin\phi = 2sin(\frac{\theta+\phi}{2})cos(\frac{\theta-\phi}{2})$
I get
$<x| \psi> = \frac{A}{2}[sin(\frac{\pi x}{R}) + sin(\frac{2 \pi x}{R})]$
how to proceed from here to normalise ? Also if anyone can show me how to do the first and last part of the question.
thank you very much,