# 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.

Q1:

An electron with an energy of 5.0 eV impacts on a barrier of width 0.30 nm. Find the probability that the electron will tunnel through the barrier if the barrier height is 7.00 eV.

Q2:

An electron with initial kinetic energy 32.0 eV encounters a square potential barrier with height 41.0 eV and width 0.250 nm. Find the probability, as a percentage, that the electron will tunnel through this barrier.

Q3:

An electron with kinetic energy 2.0 MeV encounters a potential energy barrier of height 16.0 MeV and width 2.00 nm. What is the probability that the electron emerges on the other side of the barrier?

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?

Q5:

An electron with an energy of 5.0 eV impacts on a barrier of width 0.30 nm. Find the probability that the electron will tunnel through the barrier if the barrier height is 9.0 eV.

Q6:

An electron with an energy of 5.0 eV impacts on a barrier of width 0.30 nm. Find the probability that the electron will tunnel through the barrier if the barrier height is 13.0 eV.

Q7:

A 13.0 eV electron encounters a barrier with a height of 16.0 eV. The probability of the electron tunneling through the barrier is . Find the width of the barrier.

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.

Q10:

A 5.0 eV electron impacts on a barrier of height 9.0 eV.

Find the probability that the electron tunnels through the barrier if the barrier’s width is 0.60 nm.

• A
• B
• C
• D
• E

Find the probability that the electron tunnels through the barrier if the barrier’s width is 0.30 nm.

• A0.082
• B
• C
• D
• E