Wavefunction in Forbidden Region Calculator
The wavefunction tail on the left is the exponentially decaying solution in the forbidden region.
Formula first
Overview
In a classically forbidden region, the physically acceptable branch must decay away from the barrier rather than oscillate.
Apply it well
When To Use
When to use: Use this when the wavefunction must be matched across a finite barrier or finite well.
Why it matters: The tunneling picture explains why wavefunctions oscillate in allowed regions and decay exponentially in forbidden regions.
Avoid these traps
Common Mistakes
- Using an oscillatory solution where the energy is below the barrier.
- Forgetting to match both the wavefunction and its derivative at the boundaries.
- Underestimating how quickly the tunneling signal drops with barrier width.
- Using an oscillatory form where the energy is below the barrier.
One free problem
Practice Problem
In a classically forbidden region where E < V, why does the wavefunction exhibit exponential behavior rather than oscillatory behavior?
Solve for:
Hint: Consider the sign of the term (E - V) in the Schrodinger equation.
The full worked solution stays in the interactive walkthrough.
References
Sources
- Engineering LibreTexts, finite square well and tunneling-barrier notes, accessed 2026-04-09
- Peverati, The Live Textbook of Physical Chemistry 2, quantum weirdness/tunneling section, accessed 2026-04-09
- Engineering LibreTexts, field enhanced emission and tunnelling effects, accessed 2026-04-09
- Griffiths, David J. (2018). Introduction to Quantum Mechanics (3rd ed.). Cambridge University Press.
- NIST CODATA Value: Reduced Planck constant (ħ)
- Liboff, Richard L. (2003). Introductory Quantum Mechanics (4th ed.). Addison-Wesley.
- Engineering LibreTexts, finite square well and tunneling-barrier notes