Intuition The one core ideaA tiny particle is not a hard dot but a spread-out wave whose height at each point tells you how likely the particle is to be found there. Because that wave bends smoothly and can never snap to zero, a little bit of it always leaks into and through a wall the particle should not be able to cross — and that leak is quantum tunneling .
This page assumes you have seen nothing . We will build every letter, squiggle, and symbol used in the parent topic one at a time, each resting on the one before it. By the end, the master formula will read like plain English.
Before any symbols, picture the physical situation. A particle comes in from the left, meets a raised "wall" of energy, and we ask: how much of it gets through?
The three flat/raised zones are the three regions the parent talks about. Everything below is a tool for describing what the wave does in each region.
Definition The position variablex x distance along a straight line , measured in metres. Left is small (even negative) x x
Picture: the horizontal axis in the figure above. The barrier sits between x = 0 x = L
Why the topic needs it: tunneling is entirely a question of how the wave changes as you move across the barrier — so we need a name for "how far along" we are.
E E energy the particle carries (its motion energy, since we set the flat ground to zero energy). Think of it as a horizontal water-level line drawn across the whole picture.
The picture that matters: draw the energy level E V 0
If E ≥ V 0
If E < V 0 below the wall top): classically the particle must bounce back. This is the "forbidden" case — and exactly the case where tunneling lives.
V 0 − E height you are short by , V 0 − E
Why the topic needs both: tunneling only exists when E < V 0
ψ ( x ) ψ x height of the matter-wave at that point . Where ψ ψ
Picture: instead of a dot at one spot, imagine a rippling ribbon stretched along the x ψ
∣ ψ ∣ 2 chance of finding the particle near a point is ∣ ψ ∣ 2 squared (so it's always positive). This squaring will matter later (it's where a mysterious "2" comes from).
Why the topic needs it: you can't talk about a particle "leaking through" a wall unless the particle is something that can spread — a wave. ψ Wavefunction and Boundary Conditions and rests on the de Broglie Wavelength (the notion that matter has a wavelength at all).
This is the heart of everything, so we give it a figure.
Definition Oscillating wavee ik x ripples up and down forever — a sine-like wiggle. We write it e ik x travelling wave: a moving particle. Look at the wavy magenta curve.
e − κ x not wiggle. It slides smoothly downward toward zero without ever crossing it — an exponential decay. Look at the smooth violet curve that fades but never dies. This is the "leak".
Intuition Why two different shapes?A wave that curves back toward the axis wiggles (oscillates). A wave that curves away from the axis runs off to a decay. Which one you get depends only on the sign of the energy gap V 0 − E E < V 0
Why the topic needs it: the whole trick of tunneling is that inside the wall the wave decays but doesn't vanish , so a sliver survives to the far side.
Now we earn each letter in e ik x e − κ x
e e ≈ 2.718 e ( something ) e − κ x fraction every step". That constant-fraction fading is exactly the smooth curve in the violet plot above. We use it because it is the only shape whose steepness stays proportional to its own height — precisely what Schrödinger's equation demands inside the wall.
Definition The imaginary uniti i i 2 = − 1 e ik x rippling wave (a compact way of writing sines and cosines). Rule of thumb: i i
k k = ℏ 2 m E k fast wiggle (short wavelength) means big k E
Definition Decay constantκ κ = ℏ 2 m ( V 0 − E ) κ rate the violet curve fades inside the wall. Big κ
κ m ( V 0 − E ) κ heavier (m taller (V 0 slower/less energetic (E κ
Notice k κ k E have ), κ V 0 − E missing ). One makes waves, the other makes decay.
m κ
Definition Reduced Planck constantℏ ℏ ≈ 1.055 × 1 0 − 34 k κ
L x = 0 x = L
In the figure of §1, three arrows appear: one incoming, one bouncing back, one getting through.
A B F heights of the incoming (A B F
Definition Transmission coefficientT T = ∣ A ∣ 2 ∣ F ∣ 2 T fraction of particles that make it through — a probability between 0 and 1. It's "how much got through" divided by "how much came in", each squared because probability uses ∣ ψ ∣ 2 T = 0.024
2 e − 2 κ L amplitude F e − κ L T squared , and squaring e − κ L ( e − κ L ) 2 = e − 2 κ L
sinh sinh ( y ) = 2 1 ( e y − e − y ) combination of the two decay/growth exponentials , and it appears in the exact formula because inside the barrier the wave is really a mix of a fading part e − κ x e + κ x y sinh ( y ) ≈ 2 1 e y e − 2 κ L
ψ ψ ′ meet the wave from the next with no sudden jump (ψ and no sudden kink (ψ ′ ψ ′ A , B , F T Wavefunction and Boundary Conditions .
Why the topic needs it: a wave with a break or a kink would need infinite energy — nature forbids it. Forcing smoothness at both faces is exactly what leaks amplitude to the far side.
Position x and the barrier picture
Energy E vs barrier height V0
Wavefunction psi = matter wave
Sign of gap decides shape
Transmission T = F squared over A squared
Smooth-join of psi and slope
Everything upstream pours into the transmission coefficient T — the one number the parent topic is chasing.
Try to answer before revealing. If you can, you are ready for the parent derivation.
What does x x x = 0 x = L
Which case allows tunnelling, E > V 0 E < V 0 E < V 0
What single quantity is ψ ∣ ψ ∣ 2 ψ ∣ ψ ∣ 2
What tells a wave apart: wiggle vs. decay? An i e ik x i e − κ x
Write k k = 2 m E /ℏ ; big energy
E ⇒ tighter ripples.
Write κ κ = 2 m ( V 0 − E ) /ℏ ; larger mass
m , larger height
V 0 , smaller energy
E .
Why is there a factor of 2 in e − 2 κ L Amplitude fades as e − κ L T
What is T The fraction transmitted, T = ∣ F ∣ 2 /∣ A ∣ 2
What two things must match at each wall face? The wave ψ ψ ′
Why does sinh Inside the barrier the wave mixes e − κ x e + κ x sinh
Recall Where each idea is developed further
The full wave-matching algebra → Wavefunction and Boundary Conditions
Why matter waves exist at all → de Broglie Wavelength
The equation that forces the shapes → Schrödinger Equation
Bouncing vs. crossing at a step → Potential Barrier and Reflection
Real-world payoffs → Alpha Decay , Scanning Tunneling Microscope