Quantum Tunneling is based on Wave-particle duality since it is a result of the wave nature of a particle. The probability of the particle getting through the barrier drops exponentially with the thickness of the barrier.
Y(x) = Y(0)e-kx
Y - the wave function
e – the natural log base number.
k - particles wave number inside the barrier.
x - The distance from the start of the barrier.
k = Ö4pm(E-V)/h
m – Particle’s mass
E – Particle’s Energy
V – Barrier’s Energy
h = 6.626 X 10-34 J * S
When the wave comes in contact with the barrier most of it bounces off the carrier but some passes through the barrier; however it decays exponentially with the thickness of the barrier. Such that the amplitude of the initial wave increases then the amplitude of the wave on the opposite side of the barrier increases. If the barrier thickness increases, the amplitude of the wave on the opposite side of the barrier decreases. If the energy of the initial wave increases, (wavelength decreases) the amplitude of the wave on the opposite side of the barrier increases. If the waves energy greater than that of the barrier tts all passes through the barrier. With a longer wavelength For particles most of them bounce off while a few instantly get through.In conclusion tunneling gets subatomic particles across barriers they normally can not cross. The number of particles that gets across the barrier drops exponentially with the barrier thickness. The number of particles that gets across the barrier increases with the energy of the particle. Quantum tunneling is a purely quantum effect. It has no counter part in Classical mechanics. It could theoretically occur on the macroscopic scale but the odds are too small to ever see it happen