[UPSC IFoS 2010] Schrodinger’s Wave equation and applications (Q196, 20M)

A stream of particles of mass M and energy E is directed from left to a one-dimensional potential well, as shown below :

The potential is -V_0 in the region a \ge x \ge -a and zero elsewhere. Set up a time-independent Schrödinger equation and obtain an expression for the transmission ratio from region I to II. Discuss the result. Show that there is a finite reflection from such a potential well, which is a result of the wave nature of matter.