7D.4 Quantum Tunnelling — Detailed Lecture Notes (English)

§1 Learning Objectives and Context

Sections 7D.2 (1D box) and 7D.3 (multi-dimensional box) introduced confined motion in infinite wells. We now confront one of quantum mechanics' most counter-intuitive predictions: a particle whose energy $E$ is less than the height $V_0$ of a potential barrier still has a finite probability of appearing on the other side. This is quantum tunnelling.

By the end of this section you should be able to:

Articulate the qualitative difference between classical "blocking" and quantum "leaking" through a barrier.
Write the general wavefunction in each of the three regions (incident, barrier, transmitted) and explain its physical meaning.
Apply continuity of $\psi$ and $\psi'$ at each interface to derive the transmission coefficient $T$.
Distinguish the regimes of validity of the full formula (eq 7D.20a) and the thick-barrier approximation (eq 7D.20b).
Explain the exponential dependence of $T$ on mass $m$, barrier height $V_0$, and width $L$.
Connect bound-state penetration in a finite well (eq 7D.21) to the same wavefunction-leakage idea.
Recognise STM, $\alpha$-decay, and enzymatic hydrogen transfer as a unified family of tunnelling phenomena.

Why it matters: Tunnelling is not abstract. The scanning tunnelling microscope (STM), tunnel diodes and Josephson junctions in semiconductor devices, hydrogen fusion in stars, proton transfers in enzymes, and all $\alpha$-radioactivity rely on the exponential sensitivity introduced here. Mastering this section is a prerequisite for chemical kinetics, catalysis, and solid-state device physics.

§2 Classical vs Quantum: Penetrating a Wall You Can't Climb

2.1 The classical picture

Imagine a ball of kinetic energy $E$ approaching a hill of height $V_0$:

$E < V_0$: not enough energy. The ball must bounce back. Transmission probability is exactly $0$.
$E > V_0$: more than enough. The ball passes over and continues. Transmission probability is exactly $1$.

Classically the answer is binary, governed strictly by the energy.

2.2 The quantum picture

Replace the ball with an electron and the hill with a potential barrier:

$E < V_0$: there is still a small but non-zero tunnelling probability $T > 0$.
$E > V_0$: there is still a non-zero reflection probability $R > 0$ (so-called quantum reflection).

The decisive contrast:
Classical: particles are localised points; energy is a possession; they bounce or pass — never partially.
Quantum: particles are extended waves; energy controls oscillation vs decay character; the wave always leaves a residual amplitude on the far side of any boundary.

2.3 Direct experimental evidence

The following facts cannot be explained without tunnelling:

STM resolves single atoms. The tip is a few Å above the sample; classically that vacuum gap is an insulator, yet a measurable nA-scale current flows.
$\alpha$-decay exists. $\alpha$ particles with kinetic energy $\sim$4–9 MeV escape nuclei whose Coulomb barrier is $\sim$25 MeV — pure tunnelling.
The Sun shines. Protons fuse despite kinetic energies hundreds of times smaller than the Coulomb barrier between them; without tunnelling, stellar nucleosynthesis would not occur.

§3 Setting Up the Rectangular Barrier

To make the idea quantitative, choose the simplest barrier shape — a rectangle:

$$V(x)=\begin{cases}0,&x<0 \quad(\text{Region I})\\ V_0,&0\le x\le L \quad(\text{Region II})\\ 0,&x>L \quad(\text{Region III})\end{cases}$$

A particle of energy $E < V_0$ is incident from the left, propagates through a barrier of width $L$ and height $V_0$, and emerges on the right. Real barriers (a vacuum gap in STM, the Coulomb barrier in $\alpha$-decay) are more complicated, but the rectangular case captures every essential physical feature and admits an analytical solution.

3.1 Wavelength and decay scales in each region

Regions I and III ($V=0$): $\displaystyle k=\frac{\sqrt{2mE}}{\hbar}$
Region II ($V=V_0>E$): $\displaystyle \kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar}$

$k$ is a wavenumber; $\kappa$ is a decay constant. Note that $V_0-E>0$ in the radicand for $\kappa$, so $\kappa$ is purely real — this is the mathematical reason solutions in Region II decay exponentially rather than oscillate.

Although $k$ and $\kappa$ look symmetric, their meanings differ entirely. $1/k$ sets a wavelength scale ($\lambda=2\pi/k$); $1/\kappa$ sets a penetration depth — the wave amplitude in Region II falls by a factor of $1/e$ over a length $1/\kappa$.

§4 Wavefunction in the Three Regions

Within each region $V$ is constant, so Schrödinger's equation reduces to:

Regions I and III: $\displaystyle -\frac{\hbar^2}{2m}\psi''=E\psi \Rightarrow \psi''=-k^2\psi$ (oscillatory)
Region II: $\displaystyle -\frac{\hbar^2}{2m}\psi''+V_0\psi=E\psi \Rightarrow \psi''=+\kappa^2\psi$ (exponential)

4.1 General solutions

$$\psi_I(x)=A\,e^{ikx}+B\,e^{-ikx}\qquad(x<0)$$ $$\psi_{II}(x)=C\,e^{\kappa x}+D\,e^{-\kappa x}\qquad(0\le x\le L)$$ $$\psi_{III}(x)=A'\,e^{ikx}\qquad(x>L)$$

4.2 Physical meaning of the coefficients

Coefficient	Meaning
$A$	Incident wave amplitude from the left (often normalised to 1)
$B$	Reflected wave amplitude back into Region I
$C, D$	The two exponential modes inside the barrier
$A'$	Transmitted wave amplitude into Region III

Why no $e^{-ikx}$ in Region III? We impose the physical boundary condition that "no particles arrive from the right at infinity," so only the right-moving (outgoing) wave survives. This is a physical, not mathematical, choice.

4.3 Both exponentials in Region II must be kept

Some texts retain only the decaying $D\,e^{-\kappa x}$ piece in the "thick-barrier" approximation. The exact solution requires both exponentials; the rising $C\,e^{\kappa x}$ contributes important corrections when $\kappa L$ is not large.

§5 Boundary Matching: Continuity of $\psi$ and $\psi'$

5.1 Why must both $\psi$ and $\psi'$ be continuous?

$\psi$ is continuous because $|\psi|^2$ is a probability density and must be single-valued and finite — a discontinuity would imply a non-conservation of probability.
$\psi'$ is continuous because the Schrödinger equation contains $-\hbar^2\psi''/2m$. A jump in $\psi'$ would imply a Dirac $\delta$-function in $\psi''$, which is admissible only if the potential itself contains a $\delta$. For a finite step-potential like ours, $\psi'$ must be continuous.

The continuity of $\psi'$ assumes the potential is finite. The infinite well of 7D.2 is the exception: $\psi$ is forced to zero outside the wall, and $\psi'$ may jump there without violating any physical principle.

5.2 Matching at $x=0$

Continuity of $\psi$: $\quad A+B=C+D$
Continuity of $\psi'$: $\quad ik(A-B)=\kappa(C-D)$

5.3 Matching at $x=L$

Continuity of $\psi$: $\quad C\,e^{\kappa L}+D\,e^{-\kappa L}=A'\,e^{ikL}$
Continuity of $\psi'$: $\quad \kappa(C\,e^{\kappa L}-D\,e^{-\kappa L})=ik\,A'\,e^{ikL}$

5.4 Counting unknowns and equations

Unknowns: $B, C, D, A'$ (four, with $A$ taken as a normalisation). Matching conditions: four. The system is exactly solvable — the standard structure of a quantum scattering problem.

§6 Deriving the Full Transmission Coefficient

6.1 Definition of the transmission coefficient

Transmission coefficient $T$: the ratio of probability flux transmitted past the barrier to the probability flux incident from the left. For equal $k$ in Regions I and III, $$T=\left|\frac{A'}{A}\right|^2$$ The reflection coefficient is $R=|B/A|^2$, with $T+R=1$ (probability conservation).

6.2 Solving the linear system

Combine the four equations of §5, eliminate $B, C, D$, and isolate $A'/A$. Letting $\varepsilon\equiv E/V_0$, the result is

Full formula (textbook 7D.20a): $$\boxed{\;T=\left\{1+\frac{\bigl(e^{\kappa L}-e^{-\kappa L}\bigr)^2}{16\,\varepsilon\,(1-\varepsilon)}\right\}^{-1}\;}$$ Equivalently, since $\sinh(\kappa L)=\frac{1}{2}(e^{\kappa L}-e^{-\kappa L})$, $$T=\left\{1+\frac{\sinh^2(\kappa L)}{4\,\varepsilon(1-\varepsilon)}\right\}^{-1}$$

6.3 Sketch of the derivation

From the two equations at $x=L$, express $C$ and $D$ as linear combinations of $A'$.
Substitute into the two equations at $x=0$, eliminate $B$, and obtain a single expression for $A'/A$.
Compute $|A'/A|^2$ and simplify using $\sinh^2(\kappa L)=\cosh^2(\kappa L)-1$ and identities involving $k\kappa$.

A complete derivation appears in standard quantum-mechanics texts (e.g. Griffiths, Example 2.7); we cite only the result here.

6.4 Quick survey of the formula

$T$ always lies between $0$ and $1$ (probability conservation).
For fixed $\kappa L$, the prefactor $4\varepsilon(1-\varepsilon)$ is maximised at $\varepsilon=1/2$.
$T\to 0$ as $\varepsilon\to 0$ or $\varepsilon\to 1$ because the prefactor vanishes — this does not contradict the intuition "higher energy passes more easily" because $\kappa$ also vanishes as $\varepsilon\to 1$, and the two effects compete.

§7 The Thick-Barrier Approximation and Physical Insight

7.1 Deriving the approximation

In the regime $\kappa L\gg 1$ (tall, wide, or heavy-particle barriers), $e^{\kappa L}\gg e^{-\kappa L}$, so

$\bigl(e^{\kappa L}-e^{-\kappa L}\bigr)^2\approx e^{2\kappa L}$
$\displaystyle T\approx\frac{16\,\varepsilon(1-\varepsilon)}{e^{2\kappa L}}=16\,\varepsilon(1-\varepsilon)\,e^{-2\kappa L}$

Thick-barrier approximation (textbook 7D.20b): $$\boxed{\;T\approx 16\,\varepsilon(1-\varepsilon)\,e^{-2\kappa L}\;}$$ with $\varepsilon=E/V_0$ and $\kappa=\sqrt{2m(V_0-E)}/\hbar$.

7.2 Range of validity

$\kappa L\gtrsim 2$ usually suffices for $<10\%$ error.
$\varepsilon$ must not be close to 1; as $\varepsilon\to 1$, $\kappa\to 0$ and the approximation breaks down.

⚠️ For $\varepsilon\to 1$ or very thin/low barriers, the approximation can yield $T>1$ — unphysical. In that regime one must return to the full formula 7D.20a. The textbook's reminder "this expression assumes $T<1$" reflects exactly this caveat.

7.3 Physical insight: exponential sensitivity

The most important feature of the approximation: $T$ is an exponential function of $\kappa L$. The prefactor $16\varepsilon(1-\varepsilon)$ is of order unity; all the dramatic variation lives inside the exponent.

Take logs: $\ln T \approx \ln[16\varepsilon(1-\varepsilon)]-2\kappa L$.
Differentiate w.r.t. $L$: $\partial(\ln T)/\partial L=-2\kappa$ (linear).
Hence on a semi-log plot ($\log T$ vs $L$) we get a straight line of slope $-2\kappa/\ln 10$.

This exponential sensitivity is the common origin of every quantitative feature later in the section — STM resolution, the 20-orders-of-magnitude spread in $\alpha$-decay half-lives, and anomalous KIE in enzymes. The mantra is simple: tunnelling = exponential.

§8 The Decay Constant $\kappa$: Mass, Height, Width

Decompose $\kappa=\sqrt{2m(V_0-E)}/\hbar$ to read off the dependence on each variable:

8.1 Mass dependence

$\kappa\propto\sqrt{m}$. A proton is 1836 times heavier than an electron, so $\kappa$ is $\sqrt{1836}\approx 43$ times larger. With $T\propto e^{-2\kappa L}$, the proton's transmission coefficient is suppressed by $e^{2\times 42\,\kappa_e L}$ — an astronomical factor.

Particle	$m/m_e$	$\kappa$ ratio	$T$ ($V_0-E=2$ eV, $L=0.5$ nm), order
Electron $e^-$	1	×1	$\sim 10^{-3}$
Proton $p$ (H nucleus)	1836	×43	$\sim 10^{-115}$
Deuteron $d$	3672	×61	$\sim 10^{-163}$
$^{12}$C nucleus	21900	×148	$\sim 10^{-396}$

This is why only electrons (and, in some regimes, hydrogen) display significant tunnelling in chemistry. Tunnelling probabilities for heavier nuclei drop into experimentally inaccessible territory.

8.2 Height dependence

$\kappa\propto\sqrt{V_0-E}$. Doubling $V_0-E$ multiplies $\kappa$ by $\sqrt{2}$ and shrinks $T$ by $e^{-2(\sqrt 2-1)\kappa L}$. Slightly weaker than the mass effect, but still exponential.

8.3 Width dependence

$L$ enters linearly in the exponent: $T\propto e^{-2\kappa L}$. The most dramatic application is STM: at typical metal work functions, $\kappa\approx 1$ Å$^{-1}$, so a 1 Å change in tip height $d$ multiplies the current by $e^{-2}\approx 0.13$. Higher work functions give the canonical "factor of $\sim 10$ per Å."

8.4 The combined formula

$$T\sim \exp\!\left[-\frac{2L}{\hbar}\sqrt{2m(V_0-E)}\right]$$ This compact form encapsulates the three suppressive factors — heavy, tall, wide — all sitting inside the exponent. It is the parent expression for every back-of-the-envelope tunnelling estimate.

Sections ④ (thickness decay), ⑤ (mass effect), and ⑪ (exact vs approximate $T$) of the companion 7D4_visualizations.html let you explore each dependence interactively.

§9 Finite Wells: Bound States and Penetration

9.1 From scattering to confinement

So far we treated a particle incident from outside the barrier (a scattering problem). Now consider a related bound problem: a particle confined to a finite-depth square well, $V(x)=0$ for $|x|\le L/2$ and $V(x)=V_0$ for $|x|>L/2$, with $E

How does this differ from the infinite well of 7D.2? When $V_0=\infty$, $\psi$ is forced to vanish outside the well. With $V_0$ finite, $\psi$ decays as $e^{-\kappa|x|}$ outside the wall but does not vanish — the wavefunction penetrates the classically forbidden region.

9.2 Even/odd decomposition for a symmetric well

Because $V(x)$ is symmetric about $x=0$, every bound state has either even or odd parity:

Inside ($|x|\le L/2$): $\psi=A\cos(kx)$ (even) or $A\sin(kx)$ (odd), $k=\sqrt{2mE}/\hbar$.
Outside ($|x|>L/2$): $\psi=B\,e^{-\kappa|x|}$ (even) or $\pm B\,e^{-\kappa|x|}$ (odd), $\kappa=\sqrt{2m(V_0-E)}/\hbar$.

9.3 Matching → transcendental equations

Let $z=kL/2$ and $z_0=(L/2)\sqrt{2mV_0}/\hbar$. Continuity of $\psi$ and $\psi'$ at $x=L/2$ gives:

Even parity: $\;z\tan z=\sqrt{z_0^2-z^2}$
Odd parity: $\;-z\cot z=\sqrt{z_0^2-z^2}$

These are transcendental equations with no closed-form solution. A graphical analysis nevertheless tells us at a glance how many roots exist — that is the bound-state count.

9.4 The number of bound states (textbook eq 7D.21)

A new bound state appears every time $z_0$ crosses $\pi/2, \pi, 3\pi/2,\ldots$. Re-expressing $z_0$ in the textbook's preferred variable:

$z_0=(\pi/2)\sqrt{8mV_0L^2/h^2}$.

Total number $N$ of bound states (eq 7D.21): $$N-1\le\sqrt{\frac{8mV_0L^2}{h^2}}

9.5 Two important limits

$V_0\to\infty$ (infinite well): $N\to\infty$, infinitely many levels — recovering 7D.2.
$V_0\to 0$ or $L\to 0$: always at least one bound state. (A symmetric 1D well always supports at least one even-parity bound state — a special theorem of 1D quantum mechanics.)

9.6 Why does the penetration matter?

It is why the energy levels of real semiconductor quantum wells deviate from the simple infinite-well formula.
When two wells sit close together, their evanescent tails overlap — the foundation of the LCAO picture of chemical bonding.
The penetration depth $1/\kappa$ sets the length scale for electron transfer between molecules and through electrode interfaces.

Sections ⑨ (finite-well bound states) and ⑩ (bound states emerging) of the companion visualisation file animate how new levels appear as $V_0$ or $L$ is varied.

§10 Three Big Applications: STM, $\alpha$-Decay, Hydrogen Tunnelling

10.1 The Scanning Tunnelling Microscope (STM)

Invented in 1981 by Binnig and Rohrer; Nobel Prize 1986. A sharp metal tip is held a distance $d$ (typically 5 Å) above the sample; with a small bias voltage applied, electrons tunnel through the vacuum gap, producing a current $I$.

$I\propto T\sim e^{-2\kappa d}$, with $\kappa=\sqrt{2m_e\phi}/\hbar$ and $\phi$ the work function (typically 4–5 eV).
$\kappa\approx 1\,\text{Å}^{-1}$, so a 1 Å change in $d$ multiplies $I$ by $e^{-2}\approx 0.13$ ($\div 8$).

Exponential sensitivity = atomic resolution. Even a 0.1 Å change in $d$ produces $\sim 25\%$ current variation — easily detectable above instrument noise — making single atoms resolvable.

10.2 $\alpha$-Decay (Gamow, 1928)

An $\alpha$ particle (two protons + two neutrons) preformed inside a nucleus has $\sim 5$ MeV but must escape across a $\sim 25$ MeV Coulomb barrier — classically impossible. Gamow's tunnelling theory:

$\ln\lambda \approx \ln\lambda_0 -\frac{2}{\hbar}\int_a^b\sqrt{2m(V(r)-E)}\,dr$

The integral evaluates to the Gamow factor; $\lambda$ (decay rate) is exponentially sensitive to $E$.

Isotope	$E_\alpha$ (MeV)	Half-life
$^{238}$U	4.27	$4.5\times 10^{9}$ years
$^{226}$Ra	4.87	1600 years
$^{222}$Rn	5.59	3.8 days
$^{218}$Po	6.11	3.1 minutes
$^{212}$Po	8.78	0.3 μs

$E_\alpha$ rises from 4.27 to 8.78 MeV (a factor of $\sim 2$), and the half-life drops from 4.5 billion years to 0.3 μs — a factor of $10^{24}$. This is the exponential sensitivity made tangible.

10.3 Hydrogen tunnelling and enzymatic catalysis (anomalous KIE)

Chemical reactions cross a "transition-state" energy barrier. The classical Arrhenius rate $k\propto e^{-E_a/k_BT}$ usually suffices. But for reactions involving hydrogen-atom transfer (proton transfer, H-atom abstraction), the proton — 1836× heavier than an electron yet 12–16× lighter than C/N/O — sits in a regime where tunnelling is significant but not overwhelming.

Anomalous kinetic isotope effect (KIE): replacing H by D doubles the mass, raising $\kappa$ by $\sqrt{2}$. Semi-classical theory predicts $k_H/k_D\sim 7$; experiments often find $k_H/k_D > 30$, with the excess attributed to tunnelling.
At low temperature, the Arrhenius plot becomes flat — thermal activation has died out, but tunnelling still drives the reaction.
Documented examples include alcohol dehydrogenase (ADH), methane monooxygenase (MMO), and aromatic amine dehydrogenase (AADH); anomalous H/D KIE has become a standard diagnostic for enzymatic hydrogen tunnelling.

10.4 Other tunnelling phenomena (further reading)

Josephson junctions (Cooper-pair tunnelling): superconducting qubits, SQUID magnetometers.
Stellar fusion: protons tunnel through their mutual Coulomb barrier; without it the Sun would not shine.
Scanning tunnelling spectroscopy (STS): $dI/dV$ at fixed $d$ resolves the local density of states (LDOS).

§11 Worked Examples

[Example 1] Electron tunnelling through a thin vacuum gap

An electron of energy $E=2.0$ eV impinges on a vacuum barrier of height $V_0=5.0$ eV and width $L=0.50$ nm. Estimate $T$ using the thick-barrier approximation.

Solution:

$V_0-E=3.0$ eV $=4.81\times 10^{-19}$ J.
$\kappa=\sqrt{2m_e(V_0-E)}/\hbar=\sqrt{2\times 9.109\times 10^{-31}\times 4.81\times 10^{-19}}/(1.055\times 10^{-34})\approx 8.87\times 10^{9}$ m$^{-1}$ $=8.87$ nm$^{-1}$.
$2\kappa L=2\times 8.87\times 0.5=8.87$.
$\varepsilon=E/V_0=0.4$, prefactor $16\varepsilon(1-\varepsilon)=16\times 0.4\times 0.6=3.84$.
$T\approx 3.84\,e^{-8.87}\approx 3.84\times 1.41\times 10^{-4}\approx 5.4\times 10^{-4}$.

About 1 of every 2000 incident electrons makes it through. In an STM, this is enough to produce nA-scale currents from the aggregate of many electrons.

[Example 2] Effect of a 1 Å increase in barrier width

Same conditions as Example 1, but $L$ increases from 0.50 to 0.60 nm. By what factor does $T$ change?

Solution:

$\Delta L=0.10$ nm, $2\kappa\Delta L=2\times 8.87\times 0.1=1.77$.
$T_{new}/T_{old}\approx e^{-1.77}\approx 0.17$.

A 0.1 nm (1 Å) increase in $L$ drops the current to 17% of its original value (roughly $\div 6$). This is the origin of STM's spatial resolution.

[Example 3] Proton tunnelling out of an acidic O–H

(Textbook Brief illustration 7D.6) An acidic proton experiences a barrier of height $V_0=2.000$ eV and width $W=100$ pm with kinetic energy $E=1.995$ eV. Estimate $T$.

Solution:

$V_0-E=0.005$ eV $=8.0\times 10^{-22}$ J (note how small).
$\kappa=\sqrt{2\times 1.673\times 10^{-27}\times 8.0\times 10^{-22}}/(1.055\times 10^{-34})\approx 1.54\times 10^{10}$ m$^{-1}$.
$\kappa W=1.54\times 10^{10}\times 100\times 10^{-12}=1.54$ — not strictly in the thick-barrier regime; use 7D.20a directly.
$\varepsilon=1.995/2.000=0.9975$, $1-\varepsilon=0.0025$.
$\bigl(e^{1.54}-e^{-1.54}\bigr)^2=(4.665-0.214)^2=19.81$.
Denominator $=1+19.81/[16\times 0.9975\times 0.0025]=1+19.81/0.0399=1+497\approx 498$.
$T\approx 1/498\approx 2.0\times 10^{-3}$.

About 2 of every 1000 protons surmount the barrier — what one observes as a finite hydrogen-exchange rate in solution.

[Example 4] Electron vs proton in the same barrier

For the same barrier ($V_0-E=2.0$ eV, $L=0.50$ nm), compare $T$ for an electron and a proton.

Solution:

Electron: $\kappa_e=\sqrt{2m_e\times 2\,\text{eV}}/\hbar\approx 7.24\times 10^{9}$ m$^{-1}$, $2\kappa_e L=7.24$, $T_e\sim e^{-7.24}\sim 7.2\times 10^{-4}$.
Proton: $\kappa_p=\sqrt{m_p/m_e}\,\kappa_e\approx 43\,\kappa_e\approx 3.10\times 10^{11}$ m$^{-1}$, $2\kappa_p L=310$, $T_p\sim e^{-310}\sim 10^{-135}$.

A mass ratio of 1836 produces a $T$ ratio of $10^{131}$. Proton tunnelling becomes chemically observable only when barriers are very thin ($\le 1$ Å) or very low ($\le 0.1$ eV).

[Example 5] Number of bound states in a finite well

An electron is confined to a symmetric square well with $V_0=10$ eV and $L=1.0$ nm. How many bound states does it support?

Solution:

Compute $P=\sqrt{8mV_0L^2/h^2}$.
For an electron, a useful identity is $\sqrt{8mV_0L^2/h^2}=\sqrt{V_0[\text{eV}]\,L[\text{nm}]^2/0.376}$, since $h^2/(8m_e\,1\,\text{eV}\,1\,\text{nm}^2)\approx 0.376$.
$P=\sqrt{10\times 1.0/0.376}=\sqrt{26.6}\approx 5.16$.
$N=\lfloor 5.16\rfloor+1=6$ bound states.

Compare: an infinite well of the same width would support infinitely many levels. In a finite well, the upper bound is $V_0$ itself.

§12 Key Takeaways

One-page summary:

Tunnelling is the inevitable consequence of the wave's residual amplitude at every boundary — the particle is not "drilling through."
Three-region wavefunction: oscillation (I) → exponential decay (II) → oscillation (III); stitched by continuity of $\psi$ and $\psi'$.
Full formula 7D.20a: $T=\{1+(e^{\kappa L}-e^{-\kappa L})^2/[16\varepsilon(1-\varepsilon)]\}^{-1}$.
Thick-barrier approximation 7D.20b: $T\approx 16\varepsilon(1-\varepsilon)\,e^{-2\kappa L}$, valid only when $\kappa L\gg 1$ and $\varepsilon$ is not close to 1.
$\kappa=\sqrt{2m(V_0-E)}/\hbar$ is the penetration-rate constant; mass $m$, height $V_0-E$, and width $L$ each suppress $T$ exponentially.
Significant tunnelling is essentially restricted to electrons (and, in some regimes, hydrogen). Heavier nuclei have exponentially smaller $T$.
In a finite well, $\psi$ leaks into the classically forbidden region; the total number of bound states $N$ is given by eq 7D.21, with $V_0\to\infty$ recovering the infinite well.
Three flagship applications: STM (exponential sensitivity → atomic resolution), $\alpha$-decay (half-lives spanning 20 orders of magnitude), and enzymatic hydrogen tunnelling (anomalous KIE).

§13 Advanced Discussion Questions

Replace the rectangular barrier with a "double well" — two finite wells separated by a barrier. Use the tunnelling concept to argue why the symmetric ground state has lower energy than that of either isolated well. Connect this picture to the bonding orbital of $\text{H}_2^+$.
Repeat the derivation for $E>V_0$. The wavefunction inside the "barrier" then oscillates rather than decays. Show that the resulting $T$ exhibits oscillations with $L$ — analogous to the Fabry–Pérot transmission peaks of classical optics.
In a finite well, the penetration depth is $1/\kappa_n$ for level $n$. How does $1/\kappa_n$ vary with $n$? Is the lowest state the most or the least penetrating? Justify.
In STM, simultaneously varying tip voltage $V$ (changing $E$) and tip height $d$ allows independent extraction of "topography" (current $\propto$ LDOS height) and "electronic structure" ($dI/dV$ revealing levels). Sketch how you would design such a measurement.
Compare the exponential sensitivity of $\alpha$-decay (half-life spans $10^{24}$) with that of $\beta$-decay (which depends on weak-interaction matrix elements rather than tunnelling). Why is $\alpha$-decay so much more sensitive to $E_\alpha$?
(Advanced) Derive the even-parity matching condition $z\tan z=\sqrt{z_0^2-z^2}$ for a symmetric finite well. Hint: enforce continuity of $\psi$ and $\psi'$ at $x=L/2$ and divide one equation by the other.

Hints: (1) corresponds to the LCAO picture of molecular orbitals — the conceptual bridge from tunnelling to chemical bonding. (6) is a standard graduate-level qualifier exercise.