# Worksheet: Quantum Tunneling through Potential Barriers

In this worksheet, we will practice calculating the probability that a particle will tunnel through a potential barrier with energy exceeding that of the particle.

**Q4: **

In scanning-tunneling microscopy (STM), tunneling-electron current is in direct proportion to the tunneling probability and tunneling probability is to a good approximation expressed by the function , where and is the distance of the tip of the scanning-tunneling microscope from the surface being scanned. If STM is used to detect surface features with heights of 0.00200 nm, what percent change in tunneling-electron current must the STM electronics be able to detect?

**Q8: **

In scanning tunneling microscopy, an elevation of the tip above the surface being scanned can be determined with great precision because the tunneling electron current between surface atoms and the atoms of the tip is extremely sensitive to the variation of the separation gap between them from point to point along the surface. Assuming that the tunneling electron current is in direct proportion to the tunneling probability and that the tunneling probability is to a good approximation expressed by the exponential function with per nm, determine the ratio of the tunneling current when the tip is 0.400 nm above the surface to the current when the tip is 0.414 nm above the surface.

**Q9: **

A simple model of radioactive nuclear decay assumes that alpha particles are trapped inside a well of nuclear potential whose walls are the barriers with a finite width 2.50 fm and height 25.0 MeV.

Find the tunneling probability across the potential barrier of the wall for alpha particles having a kinetic energy of 24.0 MeV.

Find the tunneling probability across the potential barrier of the wall for alpha particles having a kinetic energy of 20.0 MeV.