rvanveshana.com
T = 0.78%
Barrier Uโ‚€Total Energy ERegion I (E > V)Region II (E < Uโ‚€)Region III (T > 0)

Live Telemetry & Summary

Observe the wave function behavior. If particle energy (E) is below the barrier height (Uโ‚€), the wave decays exponentially inside the barrier, emerging on the right with a smaller amplitude. Decreasing barrier width (L) exponentially boosts tunneling probability!

Particle Energy (E):4.0 eV
Barrier Height (Uโ‚€):7.0 eV
Barrier Width (L):0.40 nm
Transmission Probability (T):0.783 %
Reflection Probability (R):99.217 %

Variable Adjuster

Particle Energy (E)4eV
110
Barrier Height (Uโ‚€)7eV
512
Barrier Width (L)0.4nm
0.150.75

Quantum Tunneling Lab

TUNN

Quantum Tunneling is a phenomenon where a wave-like quantum particle can penetrate and pass through a potential barrier that is higher than its total energy. Classically, the particle would always reflect. In quantum mechanics, the wave function decays exponentially inside the barrier but remains non-zero, allowing the particle to emerge on the other side.

Tโ‰ˆeโˆ’2ฮบL,ฮบ=2m(U0โˆ’E)โ„T \approx e^{-2\kappa L}, \quad \kappa = \frac{\sqrt{2m(U_0 - E)}}{\hbar}

Whiteboard Solver Steps

Step 1

Wave Number inside Barrier (ฮบ)

For energy less than the barrier height, the wave function decays exponentially inside the barrier.

ฮบ=2m(U0โˆ’E)โ„โ‰ˆ3.57.00โˆ’4.00=6.062\kappa = \frac{\sqrt{2m(U_0 - E)}}{\hbar} \approx 3.5 \sqrt{7.00 - 4.00} = 6.062
Step 2

Transmission Probability (T)

T is the probability that the particle will cross the barrier (tunnel through). Reflection probability is R = 1 - T.

T=eโˆ’2ฮบL=eโˆ’2โ‹…6.062โ‹…0.40=0.78%T = e^{-2\kappa L} = e^{-2 \cdot 6.062 \cdot 0.40} = 0.78\%