Consider a finite potential step as shown below, with V = V0 in the region x < 0 and V = 0 in the region x > 0. Particles with energy E < V0 are incident on this barrier from the right-hand side. found in image a) Write down and solve the time-independent Schro ̈dinger equation in the region x > 0. Identify which part of your solution corresponds to a left-moving wave, and which part corresponds to a right-moving wave. b) Write down and solve the time-independent Schro ̈dinger equation in the region x < 0. Explain the condition that ψ should obey at x → −∞, and use this to set one of the terms in your solution to zero. c) Write down the boundary conditions that ψ must obey at x = 0, and use your solutions from parts a) and b) to evaluate them in the present case.
Consider a finite potential step as shown below, with V = V0 in the region x < 0 and V = 0 in the region x > 0. Particles with energy E < V0 are incident on this barrier from the right-hand side.
found in image
-
a) Write down and solve the time-independent Schro ̈dinger equation in the region x > 0. Identify which part of your solution corresponds to a left-moving wave, and which part corresponds to a right-moving wave.
-
b) Write down and solve the time-independent Schro ̈dinger equation in the region x < 0. Explain the condition that ψ should obey at x → −∞, and use this to set one of the terms in your solution to zero.
-
c) Write down the boundary conditions that ψ must obey at x = 0, and use your solutions from
parts a) and b) to evaluate them in the present case.
-
d) Give a general definition for the probability current j, and use your definition to calculate
the incident and reflected probability currents for the situation described above.
-
e) Use your answers from questions c) and d) to determine the value of the reflection coeffi- cient R for the situation described above. What is the value of the transmisson coefficient T in this case?
Step by step
Solved in 5 steps with 21 images
