What is quantum tunneling?

Quantum tunneling is a purely quantum effect where a particle has a non-zero probability to cross a potential barrier even if its energy is lower than the barrier height. In classical physics the particle would always be reflected, but in quantum mechanics the wavefunction extends into and beyond the barrier, giving a finite transmission probability.

Which formula does this quantum tunneling calculator use?

This calculator uses the standard one-dimensional rectangular barrier model from introductory quantum mechanics. For energies below the barrier height it computes the exact transmission probability using the textbook formula with hyperbolic sine, and also shows the WKB exponential approximation. For energies above the barrier it uses the oscillatory sine formula for over-the-barrier transmission.

When is the WKB approximation accurate for tunneling?

The WKB approximation is accurate when the barrier is wide and smooth on the scale of the particle's de Broglie wavelength and when the barrier height is significantly larger than the particle energy. In the calculator this corresponds to large values of the dimensionless parameter kappa times barrier width. When that parameter is small, the exact formula should be preferred.

What are some real-world examples of quantum tunneling?

Quantum tunneling appears in alpha decay of nuclei, electron tunneling in scanning tunneling microscopes, Josephson junctions in superconducting qubits, tunnel diodes, and proton or electron transfer in some biochemical reactions. In stars, tunneling allows nuclear fusion to proceed at temperatures lower than the classical Coulomb barrier would suggest.

Quantum Tunneling Probability Calculator (Rectangular Barrier)

How this quantum tunneling calculator works

This tool models a standard one-dimensional rectangular potential barrier of height \(V_0\) and width \(a\). A particle of mass \(m\) and energy \(E\) is incident from the left. Solving the time-independent Schrödinger equation with appropriate boundary conditions gives the transmission probability \(T\).

Case 1: Tunneling (E < V₀)

For energies below the barrier height, the wavefunction decays exponentially inside the barrier. Define

\[ k = \frac{\sqrt{2mE}}{\hbar}, \quad \kappa = \frac{\sqrt{2m(V_0 - E)}}{\hbar}, \quad a = \text{barrier width}. \] \[ T = \frac{1}{1 + \dfrac{V_0^2 \sinh^2(\kappa a)}{4E(V_0 - E)}}. \]

The calculator evaluates this expression in SI units (joules, meters, kilograms) after converting from eV and nm.

Case 2: Over-the-barrier transmission (E > V₀)

When the particle energy exceeds the barrier height, there is still partial reflection due to wave interference:

\[ k_1 = \frac{\sqrt{2mE}}{\hbar}, \quad k_2 = \frac{\sqrt{2m(E - V_0)}}{\hbar}, \] \[ T = \frac{1}{1 + \dfrac{V_0^2 \sin^2(k_2 a)}{4E(E - V_0)}}. \]

WKB approximation

For thick or high barriers, the exact formula simplifies to the WKB exponential approximation:

\[ T_{\text{WKB}} \approx \exp\left(-2 \kappa a\right), \quad \kappa = \frac{\sqrt{2m(V_0 - E)}}{\hbar}. \]

The parameter \(\kappa a\) is a dimensionless measure of barrier strength. When \(\kappa a \gg 1\), tunneling is extremely suppressed and the WKB approximation is very accurate.

Interpreting the results

Transmission probability T: Fraction of particles that make it through the barrier.
Reflection probability R: Fraction reflected back; for this idealized model \(R = 1 - T\).
Regime: Indicates whether you are in the true tunneling regime (\(E < V_0\)) or above the barrier.
κa: Larger values mean stronger suppression of tunneling.

Example: Electron tunneling through a thin barrier

Consider an electron with \(E = 1\ \text{eV}\) incident on a barrier with \(V_0 = 5\ \text{eV}\) and \(a = 0.5\ \text{nm}\).

The calculator finds a small but non-zero transmission probability \(T\).
Increasing the width to \(a = 1\ \text{nm}\) dramatically reduces \(T\) (roughly exponentially).
Reducing the barrier height \(V_0\) increases \(T\).

This sensitivity to barrier width and height is why tunneling currents in devices like tunnel diodes and scanning tunneling microscopes are extremely sensitive to nanometer-scale distances.

Limitations and assumptions

One-dimensional, single rectangular barrier.
No dissipation, decoherence, or inelastic scattering.
Non-relativistic Schrödinger equation (valid when \(E \ll mc^2\)).
Applies to idealized textbook scenarios; real materials may require more detailed models.

Frequently asked questions

Is quantum tunneling really random?

In standard quantum mechanics, tunneling is inherently probabilistic: the theory predicts only the probability that a particle will tunnel, not which individual particle will do so. Over many trials, the observed frequencies match the calculated probabilities.

Why can small particles tunnel more easily?

The tunneling exponent \(\kappa a\) contains the particle mass \(m\). Heavier particles (larger \(m\)) give larger \(\kappa\) and therefore much smaller \(T\). Electrons tunnel relatively easily; protons and nuclei tunnel much less, which is why nuclear tunneling processes are typically very slow.

Can this calculator model double barriers or resonant tunneling?

No. This tool focuses on a single rectangular barrier, which already captures the essential physics of tunneling. Double-barrier and resonant tunneling structures require solving a slightly more complicated boundary-value problem with multiple regions.

Quantum Tunneling Probability Calculator

Tunneling Calculator (Rectangular Barrier)

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