Quantum Mechanics

Quantum Tunneling

V(x) Barrier
0.2500
E Particle
1.0000
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Parameters

Particle Energy (Freq)10
Barrier Height (Intensity)50
Barrier Width (Thickness)50
Transmission
0.00%
Reflection
0.00%
Total Probability ∫|ψ|²dx
1.0000

Awaiting Telemetry

Governing Dynamics

Quantum tunneling occurs when a particle passes through a potential barrier that it classically lacks the energy to surmount (E < V₀).

The time evolution is governed by the time-dependent Schrödinger equation:

iℏ∂Ψ(x,t)∂t=[−ℏ22m∂2∂x2+V(x)]Ψ(x,t)i\hbar\frac{\partial\Psi(x,t)}{\partial t} = \left[ -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x) \right]\Psi(x,t)

For a plane wave, the exact transmission probability T when E < V₀ is:

T=[1+V02sinh⁡2(κL)4E(V0−E)]−1T = \left[ 1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0 - E)} \right]^{-1}

Where the decay constant inside the barrier is:

κ=2m(V0−E)ℏ\kappa = \frac{\sqrt{2m(V_0 - E)}}{\hbar}