Tunneling and Barrier Penetration

Learning Objectives

By the end of this section, you will be able to:

1. Explain physically why a quantum particle can cross a barrier that classically blocks it
2. Solve the time-independent Schrödinger equation in each region of a rectangular barrier and match boundary conditions
3. Derive the closed-form transmission coefficient $T(E)$ and interpret each term
4. Estimate tunneling probabilities for arbitrary smooth barriers using the WKB integral $T \approx e^{-2\int_a^b \kappa(x)\,dx}$
5. Connect tunneling to alpha-decay, scanning tunneling microscopy, fusion in stars, and Flash memory
6. Implement both an exact and a differentiable transmission calculator in Python and PyTorch

The Big Picture: Walls You Can Walk Through

"If you bang your head against the wall hard enough — and long enough — quantum mechanics says some tiny fraction of you ends up on the other side."

Roll a ball at a hill that is taller than its kinetic energy will allow it to climb. The ball rolls up, stops, rolls back. That is the classical story, and it is the story of every encounter you have ever had with a wall. Quantum mechanics tells a different story. A particle aimed at a barrier it cannot classically surmount will almost always reflect — but a tiny, exponentially small piece of its wave function slips through and emerges on the other side.

The phenomenon is called quantum tunneling. It is what lets uranium nuclei decay, what makes the Sun shine, what writes the bits inside the Flash chip in your phone, and what gives the scanning tunneling microscope the ability to see individual atoms. None of it happens classically. All of it is forced on us by the wave nature of matter and a single sign change in the Schrödinger equation.

Classical vs Quantum: Why Walls Aren't Walls

Start with the time-independent Schrödinger equation in one dimension:

Rewrite it to expose the sign of $\psi''$:

Stare at the prefactor $V(x) - E$. There are two regimes — and they live in entirely different mathematical universes:

Classically, a particle with $E < V$ has "negative kinetic energy" in that region, which is nonsense — so it never enters at all. Quantum mechanically, "negative kinetic energy" is translated into exponential decay rather than forbidden. The wave function takes a hit, but it does not get to be zero. And anything nonzero at the far edge of the barrier becomes the seed of an oscillating, transmitted wave on the other side.

The Rectangular Barrier

The simplest geometry that exhibits all the tunneling physics is the rectangular barrier: a region of constant potential sandwiched between two flat zero-potential regions.

We will solve the time-independent Schrödinger equation in each of the three regions and stitch the pieces together with continuity conditions at the edges $x = 0$ and $x = a$.

Region I (x < 0): incoming + reflected wave

A plane wave comes in from the left and partially reflects:

The complex number $r$ is the reflection amplitude: $|r|^2 = R$ is the probability of being reflected.

Region II (0 ≤ x ≤ a): inside the barrier

For $E < V_0$:

Both exponentials are kept. The growing exponential $e^{+\kappa x}$ is not crazy here because the barrier has finite width — there is no infinity for the wave to blow up at.

Region III (x > a): transmitted wave

Only a forward-moving wave: no reflection in Region III because nothing scatters back from there. The complex number $t$ is the transmission amplitude: $|t|^2 = T$.

Matching Conditions and the Wave Function

At every step in $V(x)$, the wave function must obey two conditions to stay a physical solution:

1. Continuity of $\psi$: $\psi(x^-) = \psi(x^+)$. Otherwise the probability density jumps at a point.
2. Continuity of $\psi'$: $\psi'(x^-) = \psi'(x^+)$. Otherwise $\psi''$ contains a Dirac delta the Schrödinger equation can't balance (with a finite potential step).

Apply these at $x = 0$ and $x = a$:

Four linear equations in four unknowns $\{r, A, B, t\}$. Grinding through the algebra — the kind every physics student does once, and only once — gives the closed-form transmission amplitude:

Square the magnitude. The phase factor $e^{-ika}$ drops out, and the cross term from $(\cosh + i \cdot \sinh)$ assembles into:

Interactive: Watch the Wave Tunnel

The figure below plots $\text{Re}\,\psi(x,t)$ (blue) and the probability density $|\psi(x)|^2$ (green) for the stationary scattering state. The pink strip is the barrier. Drag the sliders to change the particle's energy, the barrier height, and the barrier width — and watch the transmitted amplitude shrink or grow.

Notice three things:

1. The wave does not stop at the barrier wall. The oscillating Region I wave smoothly converts into an exponentially decaying Region II wave at $x = 0$.
2. The transmitted amplitude is small but nonzero. On the right side $|\psi|^2 = T < 1$ is flat — the surviving probability current carries the particle off to $+\infty$.
3. The left side shows a standing-wave ripple. The incoming and reflected waves interfere. The depth of the ripple measures how much got reflected: $R = 1 - T$.

The Transmission Coefficient T(E)

Look at the structure of the formula. Two limits make the physics obvious:

Thick barrier ($\kappa a \gg 1$)

For large $\kappa a$, use $\sinh(\kappa a) \approx \tfrac{1}{2} e^{\kappa a}$ and the formula reduces to:

The $e^{-2\kappa a}$ is the headline. The amplitude in the barrier decays as $e^{-\kappa a}$; the probability picks up a factor of 2 from squaring. Doubling the width $a$ squares an already exponentially small number. Tunneling is hypersensitive to barrier thickness.

Thin barrier ($\kappa a \ll 1$)

Expand $\sinh(\kappa a) \approx \kappa a$ and use $\kappa^2 = 2m(V_0-E)/\hbar^2$:

For very thin barriers transmission approaches 1: a quantum particle barely notices a sufficiently flimsy wall.

Interactive: Mapping T(E)

The curve below plots $T(E)$ from the exact formula. The vertical pink dashed line marks $V_0$. Notice how T crosses $V_0$ smoothly — there is no discontinuity there — and then oscillates with Ramsauer–Townsend resonances at $qa = n\pi$.

Try the following experiments:

• Halve the barrier width. Watch T(E) below $V_0$ jump by many orders of magnitude — that is what $e^{-2\kappa a}$ looks like in practice.
• Slide the probe energy up to a resonance peak. T = 1 exactly: the wave that bounces back from the far wall destructively interferes with the wave reflecting from the near wall.
• Raise $V_0$ above the probe energy. The exponential suppression kicks in.

Worked Example: Electron Through 1 nm of Oxide

Let's plug in real numbers from Flash memory: an electron tunneling through a thin SiO₂ tunnel oxide layer during the program/erase cycle.

The WKB Approximation: Smooth Barriers

Real barriers are not rectangles. The Coulomb barrier outside a uranium nucleus is roughly triangular; the potential a Flash electron faces is bent by the applied field; the Coulomb screen between two fusing protons in the Sun is parabolic. We need a way to handle arbitrary smooth $V(x)$.

Here is the key intuition. Take a smooth barrier and slice it into narrow vertical strips of width $dx$. In each strip the potential is approximately constant, so the wave is approximately a local exponential $e^{-\kappa(x)\,dx}$ with $\kappa(x) = \sqrt{2m(V(x) - E)}/\hbar$. The total amplitude that emerges is the product of all those strip-by-strip decay factors:

Square for the probability:

Interactive: The WKB Integral in Action

Pick a barrier shape (parabolic, gaussian, or triangular), slide the energy and watch the classically forbidden region (pink) shrink and the WKB transmission grow accordingly. The two green dots mark the turning points $a$ and $b$. The readout shows $\int_a^b \kappa(x)\,dx$ and the resulting $T \approx e^{-2\int\kappa\,dx}$.

Things to try:

• Raise the energy until it touches V₀. The turning points merge, the integral goes to zero, and T → 1. The barrier has disappeared.
• Compare shapes at the same peak and width. The triangular barrier transmits more than the gaussian, which transmits more than the parabolic — area under $\kappa(x)$ is what matters, not just peak height.

Applications: Stars, Microscopes, and Memory Cells

Connections to Machine Learning

1. Energy landscapes and barrier crossing

Stochastic gradient descent on a non-convex loss surface is, at high learning rate, formally like a particle moving in a noisy potential. Escaping a poor local minimum requires "tunneling" through (or thermally hopping over) the surrounding barrier. The same exponential suppression $P_{\text{escape}} \sim e^{-\Delta E / T}$ governs both — Arrhenius for thermal, WKB for genuine quantum.

2. Variational quantum eigensolvers

Modern neural networks trained as variational ansätze for the Schrödinger equation must accurately represent the exponentially decaying tails of $\psi$ inside forbidden regions. Failing to do so means missing tunneling-dominated phenomena like double-well splittings — a known failure mode of low-capacity ansätze.

3. Diffusion-bridge sampling

The Schrödinger-bridge formulation of generative diffusion models is named after exactly the same equation. Computing the most-likely path between two distributions is mathematically analogous to computing the WKB instanton trajectory between two minima of a potential.

Python Implementation

Start with plain NumPy to make the formulas concrete, then upgrade to a differentiable PyTorch version that lets us run gradient-based inverse design — solve for the barrier width that yields a target transmission probability.

1import numpy as np
2import matplotlib.pyplot as plt
3
4# Natural units: hbar = m = 1
5hbar = 1.0
6m = 1.0
7
8
9def transmission_rect_barrier(E, V0, a):
10    """
11    Exact transmission probability for a 1D rectangular barrier.
12
13    Region I  (x < 0):     psi = e^{i k x} + r e^{-i k x}
14    Region II (0 <= x <= a): inside the barrier
15    Region III (x > a):    psi = t e^{i k x}
16
17    Returns T = |t|^2, the fraction of probability flux
18    transmitted through the barrier.
19    """
20    if E <= 0:
21        return 0.0
22
23    k = np.sqrt(2 * m * E) / hbar  # outside wavenumber
24
25    if E < V0:
26        # Classically forbidden — wave decays inside the barrier
27        kappa = np.sqrt(2 * m * (V0 - E)) / hbar
28        sinh_term = np.sinh(kappa * a) ** 2
29        denom = 1.0 + (V0**2 * sinh_term) / (4 * E * (V0 - E))
30        return 1.0 / denom
31
32    if np.isclose(E, V0):
33        # Smooth limit when E -> V0
34        return 1.0 / (1.0 + (m * V0 * a**2) / (2 * hbar**2))
35
36    # E > V0 — propagating inside, partial reflection still
37    q = np.sqrt(2 * m * (E - V0)) / hbar
38    sin_term = np.sin(q * a) ** 2
39    denom = 1.0 + (V0**2 * sin_term) / (4 * E * (E - V0))
40    return 1.0 / denom
41
42
43def thick_barrier_approx(E, V0, a):
44    """Large-kappa-a approximation: T ~ 16 E(V0-E)/V0^2 * exp(-2 kappa a)."""
45    if not (0 < E < V0):
46        return None
47    kappa = np.sqrt(2 * m * (V0 - E)) / hbar
48    prefactor = 16 * E * (V0 - E) / V0**2
49    return prefactor * np.exp(-2 * kappa * a)
50
51
52# --- Tiny worked example: probe one (E, V0, a) ----------------------------
53E_test  = 0.5     # particle energy
54V0_test = 1.0     # barrier height
55a_test  = 1.5     # barrier width
56
57T_exact = transmission_rect_barrier(E_test, V0_test, a_test)
58T_approx = thick_barrier_approx(E_test, V0_test, a_test)
59
60print(f"E = {E_test}, V0 = {V0_test}, a = {a_test}")
61print(f"  kappa            = {np.sqrt(2*m*(V0_test-E_test))/hbar:.4f}")
62print(f"  T (exact)        = {T_exact:.6e}")
63print(f"  T (thick approx) = {T_approx:.6e}")
64print(f"  T (exact) / T (approx) = {T_exact / T_approx:.4f}")
65
66# --- Scan over energy: plot T(E) ------------------------------------------
67energies = np.linspace(0.01, 2.5, 600)
68Ts = np.array([transmission_rect_barrier(E, V0_test, a_test) for E in energies])
69
70plt.figure(figsize=(8, 4))
71plt.plot(energies, Ts, label="exact T(E)", color="tab:green")
72plt.axvline(V0_test, color="tab:red", linestyle="--", label="V0 (barrier top)")
73plt.xlabel("Energy E")
74plt.ylabel("Transmission T(E)")
75plt.yscale("log")
76plt.title("Rectangular barrier transmission")
77plt.legend()
78plt.grid(True, which="both", alpha=0.3)
79plt.tight_layout()
80plt.savefig("transmission_vs_energy.png", dpi=150)
81print("Saved transmission_vs_energy.png")

Now the same physics, but differentiable. Below we use PyTorch's autograd to invert the formula: given a target $T = 0.01$, find the barrier width $a$ that produces it.

1import torch
2
3# Same closed-form formula, but written so PyTorch can differentiate through it.
4# This lets us optimize barrier shapes by gradient descent.
5
6
7def transmission_torch(E, V0, a):
8    """
9    Differentiable transmission for a 1D rectangular barrier.
10
11    E, V0, a can be torch tensors with requires_grad=True. The output is a
12    scalar tensor; calling .backward() gives gradients with respect to all
13    three inputs.
14    """
15    hbar = 1.0
16    m = 1.0
17
18    # Smooth piecewise definition using torch.where so autograd flows
19    diff = V0 - E
20    abs_diff = torch.abs(diff) + 1e-12  # avoid divide-by-zero at E = V0
21
22    # Forbidden branch (E < V0)
23    kappa = torch.sqrt(2 * m * torch.clamp(diff, min=1e-12) / hbar**2)
24    sinh_term = torch.sinh(kappa * a) ** 2
25    T_below = 1.0 / (1.0 + V0**2 * sinh_term / (4 * E * abs_diff))
26
27    # Above-barrier branch (E > V0)
28    q = torch.sqrt(2 * m * torch.clamp(-diff, min=1e-12) / hbar**2)
29    sin_term = torch.sin(q * a) ** 2
30    T_above = 1.0 / (1.0 + V0**2 * sin_term / (4 * E * abs_diff))
31
32    return torch.where(E < V0, T_below, T_above)
33
34
35# --- Inverse design: find the width a that gives T = 0.01 -----------------
36target_T = 0.01
37
38E   = torch.tensor(0.5)
39V0  = torch.tensor(1.0)
40a   = torch.tensor(1.5, requires_grad=True)
41
42optimizer = torch.optim.Adam([a], lr=0.05)
43for step in range(200):
44    optimizer.zero_grad()
45    T = transmission_torch(E, V0, a)
46    loss = (torch.log(T) - torch.log(torch.tensor(target_T))) ** 2
47    loss.backward()
48    optimizer.step()
49    if step % 40 == 0:
50        print(f"step {step:3d}  a = {a.item():.4f}  T = {T.item():.4e}")
51
52print(f"Final width a = {a.item():.4f} gives T = {T.item():.4e}")

Test Your Understanding

Summary

Quantum tunneling is the phenomenon where a particle's wave function penetrates a classically forbidden region and emerges on the other side. It is forced on us by the second-order Schrödinger equation: the sign of $V(x) - E$ flips the equation from oscillating to exponentially decaying — but never to zero.

Key Formulas

Key Takeaways

1. The sign of $V - E$ sets the regime: oscillating ↔ exponential — that one bit of information IS the physics of tunneling.
2. Tunneling probability is exponentially suppressed in barrier thickness: doubling $a$ squares the (already tiny) exponential.
3. Above-barrier reflection is real: even with $E > V_0$, the wave reflects unless $qa = n\pi$.
4. WKB generalizes to arbitrary smooth barriers: $T \approx e^{-2\int\kappa\,dx}$ is the single most useful formula in tunneling.
5. Tunneling is not a curiosity: it powers stars, enables single-atom microscopy, writes Flash memory bits, and is essential to superconducting qubits.
6. The exponential sensitivity is the design surface: tiny changes in barrier height, width, or particle mass swing T over many orders of magnitude — exploited in every tunneling device ever built.

Coming Next: In the next section we will tackle the Hydrogen Atom — the first quantum system rich enough to require all three spatial dimensions, the orbital and magnetic quantum numbers, and the spherical harmonics. We will discover the Bohr energy levels falling out as exact eigenvalues of the Schrödinger equation.