Quantum Tunneling Made Practical
Quantum tunneling is a core idea in wave mechanics. A particle can cross a barrier even when its kinetic energy is lower than the barrier height. Classical physics would predict no passage. Quantum physics gives a finite chance. That chance is called transmission probability.
Why the Inputs Matter
This calculator uses the common finite rectangular barrier model. It is useful for homework, lab checks, and quick concept testing. The model needs particle energy, barrier height, barrier width, and particle mass. These inputs shape the size of the wave decay inside the barrier.
A narrow barrier gives a larger probability. A lighter particle also tunnels more easily. A higher barrier lowers the result. A wider barrier lowers it even more. Width is very important because the approximation uses an exponential term. Small width changes can create large output changes.
Exact and Approximate Results
The exact expression is better for formal work. It accounts for boundary matching at both sides of the barrier. The WKB style approximation is faster to read. It shows the main trend clearly. The calculator reports both when the particle energy is below the barrier.
When particle energy is above the barrier height, the result is not simple free passage. Wave reflection can still occur. The calculator switches to the above barrier expression. It uses a sine term inside the barrier. This gives oscillating transmission values.
Reading the Output
The reflection probability is one minus transmission probability. A high reflection result means most particles bounce back. A high transmission result means many pass through. In real systems, many particles are tested. Probability becomes a measurable fraction.
Units matter in this problem. The tool converts electron volts to joules. It also converts nanometers, picometers, angstroms, and meters. Mass presets help common particles. Custom mass lets you model ions or special particles.
Model Limits
Use the output as a model result, not as a full device simulation. Real barriers may be rounded, layered, or time dependent. Materials may add effective mass values. Still, the rectangular barrier gives a strong starting point. It explains scanning tunneling, alpha decay, tunnel diodes, and nanoscale transport. It also builds intuition for how wave behavior differs from everyday motion.
For teaching, it helps compare exact mathematics with estimates. For planning, it reveals which input controls the final result.