PHY3702
Assignment 2
Exceptional Solutions
Due 30 June 2025
, PHY3702
Assignment 2
Due date 30 June 2025
Quantum Mechanics in Three Dimensions
Solutions Based on Chapter 4 & 5
Chapter 4:
Exercise 4.3 (10 Marks)
Problem Statement: An electron with kinetic energy E= 10 eVapproaches a potential step
at x= 0, where the potential is:
V( x) = {0 forx≤ 0 2, 0 eV forx >0 .
Since E < V, this is a quantum tunneling scenario. We need to: (a) Write the Schr¨odinger
equation in both regions, (b) Describe the shapes of ψ( x), (c) Calculate the electron’s
wavelength in region I ( x <0) and decay constant in region II ( x >0), (d) Apply boundary
conditions at x= 0, (e) Compute the ratio of probabilities at x= 10−10 mand x= 0.
Constants:
• E= 10 eV= 1 6. 02 × 10−18 J.
• m= 9 1. 09 × 10−31 kg.
• ¯ h= 1 0
. 55 × 10−34 J· s.
• V= 20 eV= 3 2. 04 × 10−18 J.
1
Assignment 2
Exceptional Solutions
Due 30 June 2025
, PHY3702
Assignment 2
Due date 30 June 2025
Quantum Mechanics in Three Dimensions
Solutions Based on Chapter 4 & 5
Chapter 4:
Exercise 4.3 (10 Marks)
Problem Statement: An electron with kinetic energy E= 10 eVapproaches a potential step
at x= 0, where the potential is:
V( x) = {0 forx≤ 0 2, 0 eV forx >0 .
Since E < V, this is a quantum tunneling scenario. We need to: (a) Write the Schr¨odinger
equation in both regions, (b) Describe the shapes of ψ( x), (c) Calculate the electron’s
wavelength in region I ( x <0) and decay constant in region II ( x >0), (d) Apply boundary
conditions at x= 0, (e) Compute the ratio of probabilities at x= 10−10 mand x= 0.
Constants:
• E= 10 eV= 1 6. 02 × 10−18 J.
• m= 9 1. 09 × 10−31 kg.
• ¯ h= 1 0
. 55 × 10−34 J· s.
• V= 20 eV= 3 2. 04 × 10−18 J.
1
