Tunnelling

Tunnelling is a phenomenon caused when energy goes negative.

Recall TISE

[−ℏ22md2dx2+V(x)]ϕ(x)=E ϕ(x)\left[-\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x)\right]\phi(x) = E\,\phi(x)

Let there be a 1D potential with an interaction zone between xLx_L and xRx_R asymptoting to constant VLV_L and VRV_R outside. Energy is kinetic + potential

Classically,

E≥VE\geq V

In quantum mechanics,

E≥VminE\geq V_{min}

but tunneling allows E<VE<V locally

Looking at TISE above,

d2ϕdx2=−2m(E−V(x))ℏ2ϕ\frac{d^2\phi}{dx^2} = -\frac{2m(E-V(x))}{\hbar^2}\phi

This is a second order ODE

Classically, The allowed region E−V(x)>0E-V(x)>0 This means solution oscillates around 0

ϕ(x)=e±ikx,k=2m(E−V)ℏ2\phi(x) = e^{\pm ikx}, \qquad k = \sqrt{\frac{2m(E-V)}{\hbar^2}}

The forbidden region E−V(x)<0E-V(x)<0 This causes the solution curve to exponentially grow or decay. A classic particle can't be here due to negative KE but QM allows a decaying Wave function known as tunneling

ϕ(x)=e±bx,b=2m(V−E)ℏ2\phi(x) = e^{\pm bx}, \qquad b = \sqrt{\frac{2m(V-E)}{\hbar^2}}