The particle in a box and quantum tunneling

Equipment

• “Level −” / “Level +” buttons selecting the well eigenstate
• “Barrier width” and “Relative height” sliders
• Wave-packet-at-barrier figure, rotatable by dragging
• Qualitative estimate readout of the transmission T

Procedure

1. Count the nodes to identify the level n

   Press “Level +” repeatedly and count the wave-function nodes inside the well: level n has exactly n−1 nodes, giving the de Broglie wavelength $\lambda_n = 2L/n$. Compare with $E_n = n^2\pi^2\hbar^2/(2mL^2)$: the energy grows as n².

2. Send the packet at the barrier and read T

   In the second figure, let the packet with $E < V_0$ hit the barrier: the amplitude decays exponentially inside but does not reach zero, so part tunnels through — quantum tunneling. Read the T estimate on the readout and compare with $T \approx e^{-2\kappa a}$, $\kappa = \sqrt{2m(V_0-E)}/\hbar$.

3. Scan the barrier width and height

   Increase “Barrier width” then “Relative height”, recording T each time. Check that T falls exponentially with both quantities — which is why tunneling only matters at atomic scales, and the mechanism behind alpha decay and the scanning tunneling microscope.

Simulation

Experiment history

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