General Oscillatory Wave Function Calculator
This is the general oscillatory solution in the classically allowed middle region.
Formula first
Overview
Inside the well, the Schrödinger equation gives a sine-and-cosine combination because the particle energy exceeds the local potential.
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 exponentials in the allowed region.
One free problem
Practice Problem
What happens to the form of the wavefunction when a particle moves from a classically allowed region (E > V) to a classically forbidden region (E < V)?
Solve for:
Hint: Consider the sign of the kinetic energy term in the Schrödinger 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
- NIST CODATA
- IUPAC Gold Book
- Griffiths, David J. (2018). Introduction to Quantum Mechanics (3rd ed.). Cambridge University Press.
- Griffiths, David J. Introduction to Quantum Mechanics
- Liboff, Richard L. Introductory Quantum Mechanics