A particle with an energy E is tunnelling through a potential barrier (see figure below). Note that such a tunnelling process takes place for example at a metal-insulator-semiconductor thin film structure used in semiconducting industry. Sketch the form of the wave function expected in each of the 3 regions and give the functional forms of these wave functions. Explain the choice of the wave function for each region in a few words. What are the mathematical boundary conditions for the

1 Problem of the Week A particle with an energy E is tunnelling through a potential barrier (see gure below). Note that such a tunnelling process takes place for example at a metalinsulatorsemiconductor thin lm structure used in semiconducting industry. V(x) E/2 Figure 1 (i) Sketch the form of the wave function expected in each of the 3 regions and give the functional forms of these wave functions. Explain the choice of the wave function for each region in a few words. (ii) What are the mathematical boundary conditions for the waves when the particle enters and leaves the barrier? (iii) Derive the equations that must be solved to obtain the constants appearing in the wave func tions (iv) If the potential energy in Region 3 is exactly E/ 2, calculate the ratio of the wavelength in Regions 1 and 3