Consider a particle of mass m moving in the potential:

Show that this potential has a bound eigenstate described by the wavefunction:

and find the corresponding eigenenergy. Normalize ψ0 and sketch it. This turns out to be the only bound state for this potential.

(b) Show that the wavefunction is:

where  solves the Schrödinger equation for any positive energy E andnear ±∞ the asymptotic of ψk(x) has the plane wave form. Determine the transmission coefficient if it is defined as the square of the ratio between the amplitude of the coming wave (at -1) and that of the going out wave (at +1). What physical situation does ψk represents? Student can find a simulation of this problem at http://www.kfunigraz.ac.at/ imawww/thaller/visualization/vis.html.

Transcribed Image Text:

1 V(x) = m cosh (ar)